EXAMPLES IN MATHEMATICS, MECHANICS, NAVIGATION AND NAUTICAL ASTRONOMY, HEAT /.,.' :A.M, AND ELECTRICITY FOE THE UK1S OT JUNIOR OFFICERS AFLOAT. LIBRARY niversity o< Ca IRVINE, FOR OFFICIAL USE. EXAMPLES IN MATHEMATICS, MECHANICS, NAVIGATION AND NAUTICAL ASTRONOMY, HEAT AND STEAM, AND ELECTRICITY. FOR THE USE OF JDNIOE OFFICERS AFLOAT. LONDON: PUBLISHED BY HIS MAJESTY'S STATIONERY OFFICE. To be purchased, either directly or through any Bookseller, from WYMAN AND SONS, LTD., FETTER LANE, E.G. ; or OLIVER AND BOYD, TWEEDDALE COURT, EDINBURGH ; or E. PONSONBY. LTD. 116. GRAFTON STREET, DUBLIN. PRINTED BY EYRE AND SPOTTISWOODE, LTD., EAST HARDING STREET, E.C, PRINTERS TO THE KING'S MOST EXCELLENT MAJESTY. . 1911. Price Two Shillings. N/ PREFACE. This book of examples is divided into the following five parts : I. Mathematics. II. Applied Mechanics. III. Navigation and Nautical Astronomy. IV. Heat and Steam. V. Electricity. In Parts L, II. and III. there is a further grouping of the examples in sections which correspond to divisions of the subject matter. The examples are for the most part taken from papers which have been set in Naval examinations during recent years. Thanks are due to Engineer-Commander Roome, R.N., the Rev. C. H. Salisbury, R.N., Messrs. H. H. Holland, R.N., and J. White, R.N., for the trouble they have taken in selecting the examples and in checking the answers. Thanks are also due to Messrs. T. Slator, R.N., R. H. Whapham, R.N., and J. G. Green, R.N., for checking several of the answers. The work has been carried out under the editorship of Mr. A. J. Parish, R.N. J. A. EW1NG. February, 1911. INDEX. PART I.-MATHEMATICS. SECTION. PAGE 1. Graphs - 5 2. Projections, &c. - 10 3. Algebra 4. Trigonometry - - 18 5. Analytical Geometry - - 26 6. Differential Calculus - - 30 7. Small Corrections - - 43 8. Maxima and Minima - - 45 9. Integral Calculus - 47 10. Areas and Volumes - - 60 11. Centre of Gravity and Moment of Inei-tia - 64 12. Differential Equations - - - 65- PART II. APPLIED MECHANICS. 1. The action of forces on a body at rest - 69' 2. Motion with uniform and with variable acceleration - - 78 3. Composition of velocities, Paths of projectiles in vacuo. Relative velocity - - - 82 4. The action of forces of constant magnitude on a body in motion : i A. Translation - - - 83 B. Rotation - 88 C. Translation and Rotation - - 90 D. Circular orbit - 92 5A. The action of forces of variable magnitude on a body in motion - - 94 SB. The particular case of harmonic motion - 97 6. Friction - 99 7. Mechanism and mechanical efficiency - - 109 8. The steam and gas engines - - 123 9. Stresses in jointed structures - - 130 10. Applications of Hooke's Law - - 142 11. Theory of bending ---.._ 148 12. Theory of torsion - 155 13. Fluid pressure - - 157 14. Heel, change of trim and oscillations of ships - 161 15. Resistance of ships - ... 162 PART III. NAVIGATION AND NAUTICAL ASTRONOMY. 1. Time problems - 164 2. Sunrise, Sunset, &c. - - 167 3. Great circle sailing 1 , , Composite sailing J u (12)5777. Wt. 1743*. 2500. 5/11. E. & S. A 2 SECTION. PAGE. 4. Small errors - - 174 5. Deviation analysed - - 177 6. Tide questions - - 187 7. Course and distance : Instruments - - 189 8. Current questions - 193 9. Scouting problems, &c. - - 195 10. Observations to determine deviation - - 203 11. Position lines by observations of sun, stars, &c. - 207 12. Error and rate of chronometer ... 221 13. Day's work and observations, for use with Charts A, B, C, D. 232 PART IV.-HEAT AND STEAM - - 265 PART V. ELECTRICITY - - - 287 PART L-MATHEMATICS. Section 1. Graphs. 2. Projections, etc. ,, 3. Algebra. 4. Trigonometry. 5. Analytical Geometry. 6. Differential Calculus. 7. Small Corrections. 8. Maxima and Minima. 9. Integral Calculus. 10. Areas and Volumes. 11. Centre of Gravity and Moment of Inertia. 12. Differential Equations. Section 1. GRAPHS. 1. A uniform beam A 1 G 1 G 1 D 1 is originally in the position ACGD. [A& = 3 feet. C 1 G l = 1 foot.] C l is joined by a link C^B, 5 feet long, to the fixed centre B, 4 feet vertically above A. The end A l is constrained to move on a curve AA t E of such a shape that G describes the horizontal 6 straight line GA. Construct the shape of the curve graphically, and check, by calculating and plotting on your diagram, the values of A^ and A,N perpendiculars from A l on AC and AB when the angle A 1 G 1 A is 30 degrees. [AtJtf = 2 feet ; A^N = 4 19 or 4 77 feet.] 2. When a certain instrument is in exact adjustment two observed quantities u and v are equal. The adjustment is effected by varying a certain distance x. The results of two experiments were (i) ac = 164 w = 96'5 v = 101'2; (ii) sc = 171 u = 101 v = 98'6. Deduce, graphically, the best value of x to try next ; verify your result by working the question analytically. [168-6.] 3. Find the real root of the equation 3x 3 - 12'9x 2 + 20'8x - 1.6-1 = 0. [2-3.] 4. Find the values of x between degrees and 360 degrees which satisfy the equation (sin x + ' 5) cos x sin x. [52 13-6' ; 194 14-8'.] 5. Solve the equation 2x 2 ' 3 - x log e x = 10 [2-1515.] 6. Sketch the curve 3y x 3 9x for values of x lying between 3 and 3. Find the slope of the tangent at the points where x = - 2, - 1, 0, 1, 2, and the greatest value of y within the given limits. [l,-2, -3, -2, 1; y = 3-464.] 7. A ship, Q, steams due East along Ox at 10 knots, starting from 0. A cruiser, P, originally 9 miles due N. of 0, gives chase at 20 knots, always heading straight for Q so that PQ is the tangent to the curve described by P. Assume that the equation to the curve described by P is x = 6 - 3y* + i t/ 1 . Draw the curve described by P, either by graphical construction or from the equation. Find approximately by trial how long P will take to close within 4 miles. What is P's course at the moment when y = 4 ? [16-6 minutes ; S. 22 37' E.] 8. The diagram shows a portion of a stream, with certain distances in nautical miles. CD is the edge of a shoal over which there is slack water. TIDE 2 KNOT S ox- D SLACK WATER -2- B A launch, whose speed through the water is 5 knots, is to go from A to B. Find, graphically, the time of the journey if she steers (a) S. 30 E., (b) S. 40 E. until slack water is reached and then straight for B. Find, by trial, the least possible time for the run. [30 -5 min. 29-9 miu. ; 29-89 min.] 2 9. Plot the curve y = be ' c from x = to x = 4, taking a= "08, b = '5. 10. Solve the equation (6 - sin0) 3 = 40. [6 = 0, 2-656.] 11. A hollow closed steel cylinder, sides and ends of the same thickness, is 1 foot long and 3 inches radius (outside measurements) and weighs 31 '4 pounds. Find the equation which determines the thickness of the metal and find the thickness approximately. [A cubic foot of steel weighs 500 pounds.] [/s _ $ + T s tf t -01 ; t = -036 ft.] 12. The equation e + ' e w= 9/200 determines, on certain assumptions, the time, t seconds, which a bomb, let fall from a height x feet, will take to reach the ground. Find, graphically, the value of t when x = 1,000 feet. Verify by solving the quadratic in [8-32 seconds.] 8 13. A wire hangs between two points in the same horizontal line 200 feet apart, and sags 45 feet in the centre. Assume that the length I of the wire is given by the formula 100 100 .. (1), where c is determined by the equation (100 _1\ _e'+e-' 7 ) .... (2). Find the values of the right and left hand sides of (2) when c = 115, and when c 120. Deduce the value of c which satisfies the equation, and show that the corresponding value of I is 225 feet roughly. ["161-29 ) 164-14) [l60- f 165- \ 14. The perimeter of a triangle is 8 inches and its area 2 square inches, the base is a mean proportional between the other two sides. Prove that the base is the smallest positive root of the equation x 3 32x -f- 65 = 0, and find its value. [2 547. One root is inadmissible, being negative ; the other roots are 4 and 2'5 nearly, but b = j~ac . . b < ^~Y^ . ' . 36 < a + b + c, u i.e., 9 or < 100 ; (ii) all values between 9 and 100.] 19. For what real values of x is */42 llx x 2 real ? [All values of x between 3 and 14.] 20. What must be the value of L in order that the roots of the equation Lx 2 + Rx + 7 - - k may be (i) real, (ii) equal, and (iii) imaginary ? r ,.. T ^Wk .... r Wk r .[(i)i<- r ; (n) L = -J-; (in) L 21. Express as partial fractions 3x + 5 1 l'2x x 2 +'4x-4'37. a* 2 + x - 1 x (x - 2) (x + 3). _ _ _ '1 + x) (1 - x 2 ). U (1 a?) ~ 4(1 + x) 2(1 +V) r 1 r l ' 6 4 J ] i L~, i .Q- A- ^~ 1 *7 1 * _1_ O '-t* X 1" O 1 L ftr + 2 (x 2) 4 3 _,_ 3(a: 4- 3)* 15 2x -f 1 3_ I 3 (x + l) 2 (x + 2). [ * + l (*+l) 2 * + 2- J 2x 3 - 7x 2 + 6x + 9 2 9 (x - 2) 2 . + x - 2 + (x- 2) 2 --l 2x 3 - 2x 2 - 3x + 2 i o_ j_ i (x + 1) (x + 4) 10 7.r + 2 /Y3 I L a/ v 1 \ ^f v2 -i. T -U 1 VJ tJLs -L Ol *t ' 1 I Ol *C (^ i*. |^ 1 I ~,2 . ^ r_ , l X -, (X - I) 2 (X 2 + 1). L 20' 1) + 2(a- I) 2 ~~ 2(ar 2 + 1)' J 1 r 1 r 5 4 \0x 1 r 1 [" 5 4 l(Xr 7 "] - L 16 U + 1 + (x + I) 2 ~~ 2x 2 x + 1 J - (x + I) 2 (2x 2 - x + 1). L 16Lr + 1 ^ (x + 1) 22. Employ the binomial theorem to show that when x is a small fraction /I + xV i,2 . 2 2 . 22 3 (T^) =1+ r + 9 x+ 8i x ' approximately. 23. Prove that in the expansion of (1+x) 12 , where x = 0'003, no term "after the third is greater than one-hundredth part of the preceding term. By means of the expansion evaluate (1 ' 003) 12 to four decimal places. [1-0366.] 24. When x is so small that its cube and higher powers may be neglected find the value of C\ _ i^Y \ v > ***> J r 2_ 13 r _ 841 T2 n L 2T 108* 1728* 25. Apply the binomial theorem to find (0-9f 01 to four places of decimals, and verify by a logarithmic calcu- lation. [9989.] 26. Find the coefficient of x r in the expansion of (a + x) n by the binomial theorem. What is the ratio of this term to the one which precedes it ? What is the ratio of the terms containing x 10 and x 20 in the expansion of {a + x} 30 if a= n 10 27. Prove that the formula t _ / 1 -f n 2 x , nx may be replaced by 2 if x is sufficiently small ; and find approximately the error involved. r n(n + 3)(l-iQ . -. ~8~ 28. Establish the logarithmic series ~2 ~3 4 lop- (1 4- y\ y - _ i L _ . _1_ iog,^j. T- SG; 234 Show that the difference of the Napierian logarithms of two numbers which differ by one per cent, of the smaller number, is approximately O'Ol, whatever the numbers. Hence show that if a slide rule with a ten-inch scale can be set and read to one-hundredth of an inch, the error arising in the multiplication of two numbers does not exceed one-half per cent. 29. Expand e x and a x in series of ascending powers of x. From the expansion of e x find the series for (e e~ ) and calculate the value of this series to three places of decimals. [1-175.] 30. Prove that for any fixed value of x, however large, the terms in the series for e* t namely 1 _1_ [ " " O T2i3 will ultimately begin to decrease in magnitude. At what term will the decrease begin if x = 100 ? [102 nd .] 31. Assuming the expansion of log e (l + x), find log e l'01 to six places of decimals. [-009949.] 32. Prove 1 ( m ~ n V 3\m + n/ -j- n Hence calculate the value of log,10 to four places. [2-3026.] 33. Given log e n, obtain a series for calculating log e (n -f 1), where n > 1. f \\ [log e (w + 1) log e n = log! i + -] = A better series is log (n + 1) log n = 2 / + e ^ b ' \ 2n + 1 + 3(2n + I) 3 + 5(2 + 1) + '}] 17 34. Prove that /128\ , f 3 I/ 3 \ 3 > log < (125) = 2 {2-53 +3(253)+ j Hence, given Iog 10 2 = ' 30103, find the modulus for changing from Napierian to common logarithms. 128 2 1C f --- -. L 125~ 10 3 Hence by given series 10 log 2 3 log 10 = 023716 /. 10 Iog 10 2 3 Iog 10 10 = /* x -023716 0103 , 35. Show that the logarithm of the n power of a number is equal to n times the logarithm, of the number. Prove that Having found the logarithms of a series of numbers to the base e, show how to calculate their logarithms to the base 10. 36. Prove that the logarithm of the product of two numbers is the sum of their logarithms, and that the logarithm of the quotient of two numbers is the difference of their logarithms. Using logarithms, find as decimals correct to three places the values of x, y, and z in the following cases : (ij 3* = 4(5)*. [-2-714.] g. + 1 (11)+1 = 8'. [--024.] (iii) z = (3'4) (2 ' 5)1 ' 6 [(3-4)*' 332l = 200-656.] 37. Solve the equation 3-21"= -718. [--2841.] Evaluate 2-7 1 ' 4 [4-017.] (25'6)- 2 ' 3 [-0005768.] C0037)' 81 [-01072.] (037)- 2 ' 7 [7342.] 59(lo ge '261) 3 l-\&.-\ 076(log256f 3 [-5737.] ae~ k \ where a = 5, k - 300, t = '001. [3-704.] 38. Find the coefficient of a- 8 in the expansion of u 5777. 18 39. Find the coefficient of so 4 in the expansion of (2* 2 - aa + ir 1 - [31.] 40. If as is so small that its square may be neglected, prove that approximately 13 4/27=6i + #32+1085 L ' 360"' 41. Using logarithms, calculate the values of when w has the successive values 1, 5, 10, 10 2 , 10 3 , 1C 4 , verifying that as ra increases increases and gets nearer to the value 2 '71828. 42. Assume that, if in the long run one shot in three is a hit, then the chances of 20, 19, 18 ... hits in 20 shots are given by the successive terms in the binomial expansion of (I + I) 20 . Plot a graph showing the relative -chances of 6, 7, 8, 9 ... hits. [Hint. Take 10 inches to represent the chance of 6 hits. Find the ratio of the chance of 7 hits to the chance of 6 hits, and so on.] i 43. If x be positive, prove that x x is less than e. Prove that 2 J 1 , _ + Find the value of e* + e '* and of e e . [2 cos x ; 2e sin x.~\ Section 4. TRIGONOMETRY. 1. Find the value of ae~ sin (nt + g), when a =5, k = 50, t = "005, n 1000, g = - '1745, the angle being measured in radians. [-3-87.] 2. If = 1 -0565 log, 5=1^ +9 x 10~ 7 (| 2 - 502 96 A+ 0-0902, find the value of < when t = 326 '1. [177.] 19 3. If T, = T 2 e~^, find T 2 when p= '54, SB = 5, and 2\ = 350. [5209.] 4. If y 1 y p = constant, and if p = 3500 when t 400 and y = 1 '41, find p when t = 350. [2211.] 5. The scale of a slide rule is 50 cm. long and the numbers on it run from 1 to 10 : (i) find the length in mm. of that part of the scale which lies between 1 ' 9 and 2 ' 0. [11-1.] 6. A fixed length, I cm., is laid off on any portion of the scale, the scale readings being N at one end and M at the other. Show that the ratio of M to N is the same wherever the fixed length is laid off, and that in the slide rule in Question 5 the ratio is given by the equation 7. If find v when p = 300, 1= '04, and Vl = '0122. [-01875.] 8. Assuming the expressions for the sines and cosines of the sum and difference of two angles, deduce the expressions for sin A + sin B ; sin A - sin B ; cos A + cos B ; cos A - cos B ; as products. Simplify the expression (cos 6 - cos 30) (sin 80 + sin 20) (sin 50 - sin 0) (cos 40 - cos 60)' [1.] 9. Establish the following identities : n sin 6 + sin 30 -f sin 50 + sin 70 _ . fi ' cos + cos 30 + cos 50 + cos 70 ~ (sin 10 - sin 50) (cos 40 - cos 60) 2 . (cos 70 + cos 50) (sin 40 + sin 60) = 10. Establish the following relation : sec A + tan A tan ( 45 + sec A - tan A = / A tan45 - u 5777. 20 11. A certain construction for dividing an angle 3a into three equal parts is correct when tan a = sin 3a . 2 + cos 3a Reduce this equation to its simplest form and show that it is satisfied when a = 0, but by no other value of a less than T - u [2 sin o = sin 2a.] 12. Express as sums or differences of sines or cosines the following products : (i) sin mx sin nx. (ii) cos mx cos nx. (iii) sin mx cos nx. [(i) TS[COS (m n) x cos (/ + n)x] ; (ii) i[cos (m ri) x + cos (m + )i)x] ; (iii) |[sin (tn nj x + sin (m + )#].] 13. Express cos (d + ) in terms of sin a and cos a if 2 cos cos (j> cos a. 2 sin # sin < = sin a. Put the equations into a form suitable for finding 6 and when a is given, and give numerical results if a = 27 11'. [cos (0 + 0) = ^(cos a sin a) and cos (0 * ) = ^(cos a + sin o) ; 62 35-8' ; 14 54- 7'.] 14. Show, without using tables, that sin 10 sin 30 sin 50 sin 70 = T V- 15. Plot the curve y = sin x from x TT to x = -\- IT. Construct also the line 4y = x -f 2 with "reference to the same axes. From these graphs find the values of x which satisfy the equation 4 sin x = x -\~ 2. [2-9117; '763; 1-848.] 16. Graph the equations y= l'25x* -Q'5x - 0'45, y sin TTX, to the same axes and to the same scale. Use your graphs to obtain approximate values of the roots of the equation 20 sin TTX = 25x 2 - lOx - 9. [ -121, -941.] 17. The following quantities, measured in a laboratory, are thought to follow the law y b* = c. Try if this is so, and, if so, find 21 the most probable values of the constants 6 and c. There are errors of observation. X ' o-i 0-2 0-4 0-6 1-0 1-5 2-0 y 700 632 240 126 25-72 5.14 0-85 equation [6 = 39-81 ; c= 1174.] 18. Solve the equation 3 log e x - 2x + 5 = 0. [4-88.] 19. Find a value of between Oand -^ which satisfies the equation tan = 1 + 2 sin 0. [1-2377.] 20. Find values of between and IT which satisfy the 8 sin - 3 sin 20 = 5. [1-1397,2-7643.] 21. Draw carefully the graph of y = sin x from x = to x = , taking as scale for x, 1 inch 1 radian, for y, 1 inch = ' 5. By taking measurements only from this quadrant, complete the graph as far as x = 2ir. Since all sine curves are alike, make the graph just drawn represent s = 6 sin (5t + '85) by merely altering the scales and graduating the reference lines for s and t. [Scale for *, 1" = 3 ; for t, 1" = -2 ; the zero for t is 0-85" to the right of the zero for x. The graduations of the points where the graph cuts the reference line are t = -17, -4583, 1 -0866.] 22. Draw a complete wave-length of each of the curves 2/ = 2 '5 sin (1-885 - 1-4661*) y = sin (10 4 t + '7854). 23. Establish the relations (i) tan" 1 5 + tan" 1 ^ = sin ~ 1 - 7 ^ + cot" l 3 = 45, 6 O +/D (ii) tan" 1 + tan" l $ + tan" 1 \ = tan" l ., (iii) 2 tan' 1 i + tan" 1 ! + 2 tan" 1 ^ = , o / o 4 and verify them by a graphical construction. c 2 and 22 24. Show that in any spherical triangle ABC cos a cos 6 cos c + sin 6 sin c cos A. The straight lines OA, OB, OC make with one another angles AOB = 65 ; BOC = 70 ; GO A = 85. Find the angle between the planes AOB, AOC. [70 H-6'.] 25. A sphere, containing lines of latitude and longitude marked upon it, is enclosed in a cubical box whose edge is equal to the diameter of the sphere ; one of the points of contact is on the meridian of Greenwich at a place whose latitude is X, and a second point of contact is on the equator. Find the positions of the other points of contact. [(i) 0, 90. (ii) X, 180. (iii) 90 - X, 0. (iv) 90 X, 180. (ii) and (iii) are in the opposite hemisphere to that in which the first of the given points of contact is.] 26. Three planes meet at a point, and, taken two by two, they make angles with one another of 76 30', 85 20' and 38 40'. Find the smallest of the three angles that the lines of inter- section of the planes make with one another. Prove the formula you use. [34 23'.] 27. The funnel of a steamer makes an angle of 80 degrees with the deck. The steamer rolls (without pitching) through 9 degrees on either side of the upright. Sketch a diagram corre- sponding to the position of greatest roll, representing a sphere centre with one radius OA vertical, one radius OB at right angles to the deck, and one radius OC parallel to the funnel. Calculate the extreme inclination of the funnel to the vertical. [1325'-4.] 28. A rectangle OACB lies on a horizontal floor. It is rotated about the side OA through an angle 6 and then about the side OB through an angle 20. If OA = a and OB = 6, find the final height of C above the floor. \_b sin + a cos sin 20.] 29. Assume the bow of a boat to be a straight line at 6 to- the horizontal and that the sides meet at the bow at an angle a to one another. Find the angle between the water-lines on the two sides at the bow. [2 tan- J f si sin tan 30. Two lines of a coal bed are observed to dip below the horizontal at angles of 15 degrees and 4 degrees and their bearings are North and East respectively. Show that the angle 23 of dip of the line of greatest slope is about 15 degrees, and find its bearing. [N. 14-6 -E.] 31. Given the S.A.T. 19 h 49 m 38 s in lat. 30 50' N. and the sun's declination 7 8' N., find the altitude and bearing of the sun. [27 7' S. 81 46' E.] 32. Given the sun's declination 14 46 '5' N., find his amplitude when setting at a place in lat. 41 35' S. [W. 19 56' N.] 33. Find the initial course and the latitude of the vertex of the great circle joining a place in lat. 45 47' S. long. 170 45' E. to another in lat, 12 4' S., long. 77 14' W. [S. 65 44' E., 50 31' S.] 34. Prove cot \Q cot 6 cosec + cosec J0. 34A. Find all the positive values of less than 360 degrees which satisfy the equation 3 sin 36 5 cos 30 = 2, using an auxiliary angle for the solution. [26 22', 73, 146 22', 193, 266 22', 313.] 35. Simplify cos A + cos B cos ( cos A + cos I sin A si cos sin B + sin (A B) ) ' V 1 + cos (A + B)' [tan - tan _.] 36. The figure shows three bars, AB, BC, CD, hinged to one another at B and C and to fixed centres at A and D. If AB = 10 inches, CD = 4 inches, BC = 8 AD = 14 find the angle through which AB oscillates as CD makes complete revolutions. B D [114 14'.] 37. A billiard ball, diameter 2 inches, moves on a horizontal table along a line making 20 degrees with a cushion. The 24 cushion overhangs so that the point at which the ball strikes it is 1 ' 4 inches above the table. Find the distance along the cushion, between the point of contact and the point apparently aimed at. [2-518 inches.] 38. Assume that, roughly speaking, the earth describes annually a circle, radius 90 X 10 6 miles, about the sun, and the moon describes in -rVth of a year a circle about the earth of radius 24 X 10 4 miles, both circles being in the same plane. Find at what angle the moon's path relative to the sun cuts the earth's path relative to the sun. [1 50'.] 39. A vertical post, height li, stands on a plane whose inclination to the horizontal is a. What is the length of the shadow of the post when the sun's altitude is and its bearing from the direction of the horizontals on the plane ? Show how to check your result simply. rs na assuming the sun to be i n f ront o f tan if> + tan a sin the plane.] 40. A coal bed is found to lie at a depth of 350 feet at a station A, of 585 feet at B which is 600 yards due N. of A, and of 280 feet at C which is 400 yards due E. of A. Taking the bed to be an inclined plane, find its depth at D, which is 150 yards E. and 350 yards N. of A. [460 -8 feet.] 41. The motion of a galvanometer needle is given by the equation n i /~i o-4< . 2?r v = lOe sm TTQ t, JL where B is the angle in radians made by the needle from the zero position at time t seconds. Draw a curve showing the value of 6 at any time during the first four complete oscillations. 42. If is the circular measure of a small angle, show that the limiting value of - is unity, when 6 is indefinitely u diminished. 43. Find, without using tables, the range of a distant object C in front of a base AB whose length is 50 yards ; the angles at A and B being measured as 45' and 12' respectively less than a right angle. [3,015 yards.] find an approximate value of (supposed to be very small). 45. If 6 is small, show that approximately 0(1 - cos 0) _&_ - sin ' 10' If 6 is a small angle, show that approximately 3 sin _ a 2 + cos0 ~ and find the percentage error when 6 = . [044.] 46. If r is small, approximate to its value in the equation sin (< + r) /A sin <^>. [G*-l)ten$.] 47. Prove Huyghens' formula for the approximate length of a circular arc, namely ' ~ , where A = chord of arc and o B = chord of half arc. 48. It is desired to calculate a five-figure table of the sines of angles, the angles being in radians. How far can the table be computed without using more than two terms of the series for the expansion of sin x in powers of x ? Compute sin ( ' 2) to five places. [As far as x = -2268 (nearly 13 degrees) ; * 19867.] 49. Find (not graphically) all the values of x between Q and 360 which satisfy the equation 3* ( *C\ cos x cos 2x = 2 sin ^- ( cos x + sin - j [60, 90, 270, 300.] 50. Find (not graphically) the values of between and 360 satisfying the equation 2 sec = I -f 4 tan 0. [14 59' and 136 57'.] 51. Express 3 sin + 4 cos in the form A sin (0 + x), What is the greatest value of the expression 3 sin + 4 cos ? [^4 = 5, x = 2nir + '9273 ; greatest value, 5.] 26 Section 5. ANALYTICAL GEOMETRY. 1. A point P in a piece of mechanism is required to move so that the sum of the distances OP from a fixed point 0, and PM from a fixed straight line through 0, may always be 12 inches. Find the equation of the locus of P. Find also the direction in which P is moving at the instant when OM = 4 inches. [a? 2 + 24y = 144, O being the origin and OM the axis of a; ; at angle of 18 26' with MO.~\ 2. A triangular set-square APB, with the right angle at P, is placed flat on the paper ; and A is made to describe the fixed straight line Ox, while the opposite edge PB always passes through the origin 0. If AP = a and the angle OAP = 6, express the coordinates x, y, of P in terms of a and 6. Hence deduce the equation of the locus of P. r [xa ^- ; y = a sin 9 ; y* = 3. Find the distance of the point P (coordinates x, y} from the point A (3, 0), and from the point B (0, 6). Find the locus of P if BP = 2AP. Describe the locus and draw a diagram to scale [unit \ inch]. [x 2 + y* Sx + 4y = 0. The locus is a circle passing through the origin its centre is at the point (4, 2) and radius = 2^/5.] 4. The vertices of a triangle are at the points (2, 3), (4, 5), ( 3, - 6) : find the equations of the sides, and the angle between the sides which meet at the vertex (2, 3). Find also the length of the perpendicular from the vertex (2, 3) to the opposite side, and the area of the triangle. \y = 4x + 1 1 ; 5y = 9x 3 ; 1y = x 39. 43 6'; 8-2 ; 29.] 5. Show that the equation of a circle can always be written in the form x 2 + y 2 + 2gx + 2fy + c = 0. Find the coordinates of the centre and the radius of the circle x 2 + ?/ 2 -f Sx Wy = and the equation of the tangent at the point which is furthest from the origin. [(-4,5); 6-4; 5y = 4* + 82.] 27 6. Draw a triangle OAB right angled at and let OA = a, OB = b. Find the equation of AB, taking OA and OB as axes of x and y. If P is a point (coordinates xy) on AB and M is the foot of the perpendicular from P on OA, express the area of the triangle 0PM in terms of a, b, and x and find for what value of x the area is a maximum. [* + f=l; (-*); "] a b 2a 2 7. An ellipse which has its centre at the origin and its major and minor axes along the axes of x and y respectively passes through the points (3, 3 '2) and (4, 2 '4). Find its equation and its area, also the length of the chord through the origin parallel to the line joining the two given points. [f 2 + y*= 1 ; 62-8 ; 8-045.] L 25 16 8. A rod, length 5 inches, is moved so that one end moves along each of two straight lines at right angles to one another. Show that the path traced out by any point in the rod other than the middle point is an ellipse. Find the semi-axes of the ellipse traced out by a point in the rod 2 inches from one end. [3 inches and 2 inches.] 9. Find the points of intersection of the straight line 4x 2y = 3 with the parabola y 2 4x. Write down the equations to the tangents to the parabola at these points and find the angles which the tangents make with the given line. r /9 Q \ /I ,\ 1x , 3 1 1 - 3 I ; /-. 1 I ; y = + ; y =. 2x -- ; L U / ' V* / 32 2 29| ; 53-1.] 10. A point moves so that the square of its distance from a fixed point varies as its perpendicular distance from a fixed straight line. Find its locus. [A circle.] 11. The focus of the ellipse 2 9 x ' 4- y" . - i a 2 r b 2 ~ is 2 inches from the directrix and the eccentricity is ' 5. Find a and b. 12. Construct, with instruments, a parabola to touch the axes OX and OF at A and B, where OA = 2, OB = 1, on a scale of 5 inches = unity ; also find the equation to the curve. 4y 2 x 8y + 4 = 0.] 28 13. The line 3ac + 2t/ = 5 cuts the axes of x and y in A and B. Find the length AB, and show that the line through the point (2, 6) perpendicular to AB passes through the point of intersection oix-{-y+2 = Q and I7x + ly + 54 = and bisects one of the angles between them. [AB = _ ^13. The line is 3y = 2x + 14.] 14. ABCD is a rectangle ; AB = 2f inches, BC = 4 inches, E is the middle point of BC ; F is the middle point of AD. Construct with drawing instruments a parabola to pass through A and D having its vertex at E and axis in the direction EF. 15. Find the equation to the parabola having its focus at (5, 3) and the straight line y = 2x 8 as its directrix. Show that 2y 4x + 1 5 = is a tangent to this parabola and find the coordinates of the vertex. [x* + xy + 4y 2 ISx 46y + 106 = 0(5-2,2-9).] 16. Find the coordinates of the vertex and focus of the parabola whose equation is y = 2 - 3x + 4x 2 . [(ftt); (t>i)-] 17. If the normal PG at each point P of a parabola be produced outwards to a point Q, so that PQ = PG, find the locus of Q. [y 2 = 16a (x + 2a), where y 2 = 4ax is the given parabola.] 18. Prove that the circle whose centre is at the point (2, V) and passes through the point (2, 5) touches the line x + y 3 = and find the coordinates of the point of contact. [0,2).] 19. Find the equation to that diameter of the circle x 2 + y 2 + 2x 4y -f- 1 '= which makes equal intercepts on the axes ; and find the equations to the tangents at the ends of this diameter. [ar + y=l;ar y + 3= 2^/2.] 20. The centre of a circle lies on the straight line 2y x = 4 ; it also passes through the point (0, 4) and touches the axis of x. Find the radius and the coordinates of the centre. [centre at (4, 4), radius = 4 ; or, centre at (0, 2) t radius = 2.] 21. Prove that the curves 2x 2 - y 2 = 1 and 3x 2 - 4xy + 3y 2 = 1 touch each other at two points, and find the coordinates of these points. 1 > J/2 1\ =1 ancl [ = =).i 7 v/7/ V >/7 V7/ J 22. Tangents are drawn to the circle x 2 -f if = 12 at the points where it is met by the circle x 2 + y* - 5x + 3y - 2 = ; find the point of intersection of these tangents. [6, -3-6).] 23. The equation of a curve is ?/ = 9a 3 - x 3 . Find the equations of the tangent and normal at the point at which x = a. [# + ty = 9a ; 4x y = 2a.] 24. Find the equation of the tangent to the curve 7y3= x 3 2a-x at the point where x = x'. 3(2a O? (2a ') 25. Plot a graph of the equation y = x 3 , x ranging from to 1. What is the inclination of the tangent at A (co- ordinates 1, 1), and where does the tangent at A cut Ox ? Find a point on the curve the tangent at which is parallel to OA, and the equation of the normal at this point. RIM.', .-f; (4=-^);, + ,= - 26. Graph the curves (i) 2/ 2 = 4 (x - 1), (ii) 27s/ 2 = 4 (x - 3) 3 . If these curves be intersected at points P and Q respectively by the line y = 2, show that the normal at P to the first curve intersects the tangent at Q to the second curve at a point on the axis of x. [a; + y = 4 and x y = 4 intersect at the point (4, 0).] 27. Find the angle between the two curves x 2 + y 2 = 4 and 5x 2 + 2/ 2 = 5 at their point of intersection. [37 46'.] 28. The line x = 2 cuts the curve y = x 3 + 1 at the point P ; the axis of x is cut at A, T and G by the curve, the tangent at P and the normal at P respectively. Find the lengths PT and PG ; find also the area bounded by the arc AP, the tangent at P and the axis of x. [9-031 ; 108-4 ; 3|.] 30 i i i 29. If the tangent at any point of the curve x -f- y = a ut the axes OX, OY in P and Q, prove that OP + OQ = a. Taking the case of a = 4, show how to construct the curve by means of this property. 30. In the catenary Show that the length of the perpendicular let fall from N, the foot of the ordinate PN, upon the tangent at P = c ; also, if the normal at P meet the axis of x at G, prove that PG = C 31. Find the length intercepted on the axis of a; between the normals to the curves yx* = 36 and yx = 12 at their point of intersection. [5$.] Section 6. DIFFERENTIAL CALCULUS. 1. Sketch the curve o W y2iiiL . t .' . . x and y being measured in inches and x ranging from to 8. Find j^ and the angle which the tangent at the origin makes with Ox. [ cos f; 57-5.] 2. Find if ax (i) y = sin 2 3ac. [3 sin 6*.] (iii) y = x tan x. [tan a? + x sec 2 ^.] 3. Differentiate the following expressions with respect to x : /\ "" *^ 9X *X 9X (i) x am f~ cos 5 + sin Si a L a a a J /..v %/x 1 a* ' fTx' ^2 (1 + ar)V^ (iii) Vl + 2x cos a + x 2 . [(a? + cos ) (1 + 2* cos a + # 2 )~*-] 31 - 4. Find - in the following cases : y = cos' ( 1 -x y = 1 + x = log j x 2 (1 + x ) 1 x (1 - x 2 )* -8* y = e cos /I / - V 1 + cos 5 sin cos x 1 + cos x x 3 log (tan x). [cosec *.] |> 2 [3 log (tan a?) + 2x cosec 2x].J + x 2 7t/ 2 = 0. y% = a s , where a is a constant. | (3 [ A/ ~ -J<& y = sin y =log i X 1 -f x 2 . 1+ Jx 1- +/x ' f L l ar) ^ y = e lte (4 cos 5x 3 sin 5x). [ - e ~ to (2 sin 5# + 39 cos Sx).J - .V + 2x + 6w - 10 = 0. 5y 2a? 3 5. A point moves to and fro in a straight line so that its distance from the starting point t seconds after last leaving it is always t\2 - t) 3 feet. Find the velocity and acceleration of the point when t = 1. [0; -6.] 6. If y = e~** sin 2x, show that 7. If y = e ax sin (bx + c), show that for (i) n = 1, (ii) n = 2, (iii) n any whole number, where r 2 = a 2 + b 2 and tan = - a and a, b, and c are constants. 8. If (i) x = a sin pt + b cos pt for any value of t, where a, b, and p are constants : show that this is the same as x = A sin (pt + q) if A and q are properly evaluated. If and if C = 100 sin 600t, R being 2 and L being ' 005, find V in the form V = A sin (600 + g). [(i) A = t, (ii) s = be nt + ce- nt . [Acceleration = (i) , find the velocity with which the end G is sliding on the door at any instant, [4w tan C.] 41. Draw a rectangle ABGD, with AB horizontal and above CD. Produce BG to E, making GE equal to CD. Draw the diagonal AC. 40 ABCD is a block, sliding in vertical guides ; AC is a groove cut in the face of the block ; EC is part of a rocking lever, clear of the block. The lever turns about E and carries a stud C fitting into the groove. If EC is rotated round E the stud slides in the groove and allows the block to fall. Find the velocity of the block when EC has turned through an angle 0, the angular velocity of EC being B. Take EC = b, CD = a. [(a sin + b cos 0) 0.] 42. The diagram shows part of a proposed attachment for taking an indicator diagram from a fixed cylinder gas engine. A rod AQ is fixed at right angles to the piston. Q is a pin fixed on AQ. Q slides in a slit-bar OP, which turns about 0, a fixed point vertically over M, the centre of the travel of Q. P is a point fixed on the slit-bar. The horizontal motion of P is taken as representing the motion of the piston. Is this correct ? If MO = 4a, OP = a, travel of Q = 4a, find the horizontal travel of P, and its horizontal velocity when MQ = a, if the velocityof Q at that instant is u. [No; -=. -228.] V o 43. If the rectangle in the diagram is rolled up into a cylinder of radius r and length 2nr the diagonal OA becomes a screw thread of uniform pitch, making one turn in n calibres, and the curve OP.A becomes a screw thread of increasing twist. If the equation of the curve is ff 2 ry 4n 2 2w and if a shell is fired with muzzle velocity v from a gun rifled with the increasing twist, what is the rate of twist at the 41 muzzle, and the initial number of revolutions per second of the shell ? 2717 2TTT [one turn in ~ calibres ; ."I .2 wr J 44. The equation cos z = sin S sin I -f- cos S cos I cos h gives the Sun's zenith distance z in terms of the N. declination 8, N. latitude I, and ft the hour angle. Differentiate this equation with respect to S, treating I and z as constant. Deduce that, if the Sun's declination increases by n' during the interval between equal altitudes before and after noon, then the chronometer time of apparent noon is OQ [tan I cosec ft tan S cot ft] minutes of time earlier than the mean of the chronometer times of equal altitudes. r dh L^TsT = tan / cosec h tan <5 cot A.] 45. Let y = a + xy n , where a and n are constants. Differ- entiate this equation as it stands with respect to x. What is the value of -^- when x = ? ax Differentiate again, and find the value of -=-* when x = 0. dx~ Hence expand y in powers of x as far as the term in 2 . Show that one root of the equation is 1 ' 0105 nearlv. 100 [a n ; 2 n a 2n ~ * ; y = a + a n x + n a 2 " - '.r 2 .] 46. The equation sin 9 sin a cos x arises in finding the collimation error of a sextant. 42 Differentiate the equation twice with respect to x, treating a as constant. dO , d*0 > and when =: 0. Deduce the first two terms in the expansion of 9 in a series of ascending powers of x. X 2 [a, 0, tan a ; 6 = a tan a.] A 47. The zenith distance of a star of declination 8 and hour angle h to an observer in latitude \ is given by cos z = sin X sin 8 + cos X cos 8 cos 7i. Differentiate this equation with respect to h, treating X and 8 as constant. Deduce from spherical trigonometry that dz JT = cos X sin (stars azimuth). When is the zenith distance changing most rapidly ? [When azimuth = 90.] 48. The equation sin a = sin 8 sin I + cos 8 cos I cos h gives the sun's altitude a in terms of the hour angle h, the sun's N. declination 8, and the observer's N. latitude I. Differentiate this equation with respect to h, treating 8 as constant, but a and I as variables. Deduce that if a ship is steaming due N. at 20 knots, so that the latitude increases 20' hourly, then the sun's maximum altitude will occur about 16 sin (I -8) minutes before noon. Section 7. SMALL CORRECTIONS. 1. A cylindrical tube, length Z, outside radius ?, inside radius a, open at both, ends, weighs w pounds per cubic inch. What is its weight W, all dimensions being in inches ? Find -j- and express it in terms of W", r and a. If r = 5 inches, a = 3 inches, what fraction (approximately) of the weight will be removed if the outside is turned down to a radius of 4 ' 95 inches ? dlV 'irW [JF=T/io(r a); - - = -^-^l '031.1 > ' ' dr r 2 a? 2. In a triangle, two sides are 140 feet and 80 feet in length. The true angle between them is 77 43', but this is by mistake measured as 78. Find the resulting error in the calculated length of the third side of the triangle. [Calculated length, 146-09 feet ; -371 feet too large.] 3. In solving a plane triangle ABC the base c is measured and the base angles A and B observed. If c is correctly measured, but A and B are subject to small errors, A A radians and AB radians, prove that the resulting errors in (7, a and ft will be - AA -- AB, I cosec C AA + a cot C AB, and a cosec C AB + 6 cot C AA. 4. The base of a triangle is 100 yards and the base angles are observed to be 35 15' and 106 25'. Find the remaining parts and the resulting errors in these parts if the base angles are subject to errors of 5 minutes each. [a = 93-05 yards, b = 154-66 yards. C=3820'. Aa = 7" or 19". A6 = 2M" or 18". Ac = 10'.] 5. In a horizontal sundial for latitude X when the shadow on the dial makes an angle with the N. and S. line, the local apparent time is x hours after noon, where tan (15x) = tan cosec X. Prove this, and find the value of 6 corresponding to 2 p.m. local apparent time in latitude 60, and the error in time if the angle is made too large. [26 34' ; 65 sees.] 44 6. The side c of a spherical triangle has to be calculated from the known values of a, 6 and the included angle C. If a and 6 are correctly measured, but C is subject to an error AC, prove that the resulting error Ac in the value of c will be sin 6 sin A AC. If a = 115, 6 = 70 and C = 65, find the error in c due to an error of 5' in C. [4' (nearly).] 7. Investigate an expression for the change in azimuth of a heavenly body due to a small change in the altitude, and find the change of azimuth due to a change of 10' in the altitude when lat. = 42 N., alt. - 12, az. = S. 60 W. [11' -6.] 8. Prove that in a given latitude all stars, when rising or setting, change their azimuth at the same rate. 9. In latitude 50 S. find the shortest interval of sidereal time in which the altitude of Sirius (dec. 16 35' S.) changes 10. What are the altitudes at the beginning and end of this interval ? [I*2 m 20 s ; 26 47'; 16 47'.] 10. In lat. 42 30' N., when the Sun's dec. is 12 45' N., find the shortest time in which the Sun rises through 1 in altitude. Find also the Sun's motion in azimuth during this interval. [5 m 25'5 s ; 27' 5.] 11. If h and h + Afo, p and p + Ap are the westerly hour angles, and polar distances respectively of the Sun at sunrise on two consecutive days, show that A/i = T 2 3- cot h cosec 2p Ap, approximately. When is A/& positive and when negative ? At a certain place the Sun rises at 8 h O m 48 s a.m. apparent time, when his declination is 22 53' S. Find by the above formula the apparent time of sunrise on the following day when his declination is 22 47' 30" S. [8 h O m 12 s a.m.] .12. In lat. 42 N. find the error in hour angle due to an error of 1' in latitude when the azimuth is N. 70 W., also when the azimuth is N. 125 E. The assumed latitude being too great, state clearly whether the erroneous hour angle is greater or less than the true hour angle. [(i) 2 seconds too great ; (ii) 4 seconds too great.] 13. Supposing that azimuth angles were measured from the north end of a meridian to the eastward from to 360, and Afe, Aa, and AZ represent increments in the hour angle, altitude and latitude respectively, show that A/i = Aa cosec az. sec lat. Al cot az. sec lat. 45 If in lat. 45 N. observations are taken to determine the longitude, when Aa, AZ are + 3', + 10' respectively, find the effects upon the longitude (i) When the Sun bears S. 45 E. (ii) ,, S.30W. [(i) +20. 1'; (ii) -33'.] 14. A ship is steaming S. 20 E. at 18 knots in lat. 50 N., the declination of the Suri is 5 S., decreasing 58" hourly. Find the S.A.T. when the altitude is greatest. [5 m 46 s .] 15. Find the time occupied by the Sun in setting at a place in lat. 40 N. on the day when evening twilight is shortest, given the Sun's semi-diameter = 16'. [declination 5 50' -6 S. ; 2 m 49 s .] Section 8. MAXIMA AND MINIMA. 1. Show from graphical considerations that if. y is a function of x, then the criterion that y shall have a maximum or minimum value is that and prove that it is possible to discriminate between these cases d 2 y by finding the value of -5-^. x(x +1) Find the maximum and minimum values of - ^ and x I distinguish between them. Are these the absolutely greatest and least values of the expression ? Graph the expression from x = - 10 to + 10. [The expression has a "minimum" value of 5 -83 when x = 2-414 and a "maximum" value '171 when x = -414 ; the greatest value is o> and the least is oo (algebraically) or zero (numerically).] 2. Find the maximum value of where y 1'41. [2625 (* being -5266).] 3. The equation of a curve is q 2 x 46 Give a simple reason why y is always numerically less than x. Sketch the curve, and find the maximum value of y. 4. The total cost C of a ship per hour (including interest,' depreciation, wages, coal, &c.) is in pounds where s is the speed of the ship in knots. Express the total cost of a passage of 2,000 miles in terms of s, and find what value of s will make this total cost a minimum. s 6 5. The cost C of a ship per hour (including interest and depreciation on capital, wages, coal, &c.) is in pounds 0-4 where S is its speed in knots relatively to the water. Going up a river whose current runs at 5 knots, what is the speed which causes least total cost of a passage ? [15 -64 knots.] 6. A piece of wire, 12 inches long, is bent up into the perimeter of a rectangle ABCD, the wire passing twice along the side AB so that its course is AB BC CD^-DA AB. Find the length x of BC if the area of the rectangle is a maximum. [3 inches.] 7. An open rectangular tank whose depth is y and base a square side x [inside measurements, in feet] is to have an inside capacity a 3 cubic feet. It is made of two pieces of metal, riveted at the four sides of the base and along one of the vertical sides. If the cost of riveting is 6 per foot length of riveted seam measured inside, find the proportions of the tank for which the cost of the riveting will be a minimum. Give a common-sense reason as to why this cost is a minimum and not a maximum. O=-7937a; y = 1 -5874a.] 8. The stiffness of a beam of rectangular section is preposi- tional to the product of the breadth and the cube of the depth. Find the dimensions of the stiffest beam of rectangular section if the perimeter of the section is limited to 20 inches. [7 '5 inches X 2-5 inches.] 47 9. The characteristic of a series dynamo is 1'2C E -, where C = - and P - C 2 #. .1+ -03C* '05 + E Find what value the external resistance R must have so that the power P given out may be a maximum. [-04282.] 10. The annual cost of giving a certain amount of electric light to a certain town, the voltage being V and the candle- power of each lamp (7, is found to be for electric energy, and m rf for lamp renewals. The following figures are known when C is 10 : V 100 200 A 1500 1200 B 300 500 Find a and 6, m and n. If C is 20, what value of V will give minimum total cost ? [900, 60,000, 404-8, 733-55; V = 358 -36.] Section 9. INTEGRAL CALCULUS. 1. Integrate the following expressions : (1 + x 2 ) x/x dx. dx. x" I +*)** I - x 2 ) 2 dx **+ c.] t ii L - 48 I sin 3x dx. I sin rax dx. I cos 2 x dx. f x dx } VI - x 2 ' f 4x J 1 + x 2 < r 3 + 2x J (1 + 3x + x 2 ) d r l 2 _ i dx. j * f x + 2 I ( 4-1W Z~Ti f x dx ] x 2 - x - 2' f x 2 J x 4 - x 2 - 12 dx ' ( 1 J x T TT da; - [rrf ! + x 2 ' dx 3 + 2x + x 2 ' dx f dx j V5- 2x-x 2 f dx J cos|x' I cos 3 x dx. I sin 3 x dx. sin 2 x dx. [c cos 3a\] fc cos mx.~\ L m sin [2 log, a (1+ ^ 2 i<>g.V^hrt + *' ] )\/3 _j / x + ~ T~ tan" ~TT; 7 \V3 + c. [tan" 1 a; + c.] r ^- *"-' (^) + c - ] ^ L V3 ^--'C-Tr)--] [i tan' 1 (e 2 *);+ ] C 81 "" 1 f^ 1 ) + C -J [log, tan (I +|)+/* + c.] 51 10. Find f~ f~ 6 sin 2 d0 b sin 2 cos0 d0. J o J o [1-525; -0417.] 11. Find the values of f a x f^ x f / 2 , -^dx, ~2 i Q dx, I cos 2 x dx. Jo v + * Jo x + y Jo [414a; -511; *] 12. Find the value of r2 /~36~ \ Z x by putting x = 2 sin 2 0, and explain how to find the limits ior the new variable. 13. ?/ is approximately, but not exactly, equal to 1 x 2 . State, with clear explanatory diagrams, whether you would expect to obtain approximately correct values of (1) ^ W dx (2) y dx by means of the approximate value of y. [(i) not correct ; (ii) correct if the error of approxi- mation is variable. The reverse, if the error is practically constant.] 14. Find Illustrate the second integration by a diagram, and explain, briefly but clearly, the effect on the actual area of the diagram of the scales on which x and - =- are represented. 1 + x 2 15. What is the area in the first quadrant of the curve whose equation is 8 + 2x-x% x + 1 [12-05.] u 5777. E 52 16. Show that a curve whose equation is y = aae 3 + frx -f- c can be made to pass through any three given points by properly choosing a, 6, c. Find the values of a, b, c, it the given points are 2, 1 ; 3, 1 ; 4, 3 ; and the area included between the curve, the extreme verticals, and the horizontal axis. [a = 1 ; b = 5; e = 7 ; area = f .] 17. What is the area in square centimetres of a sine curve in which 1 cm. horizontal- represents 10 and 10 cm. vertical represents sin 90 ? The curve ranges from to 180. At what angle does the curve cut the horizontal axis ? = 115 cm. nearly ; tan- 1 / = 60 "2.] 7T \ 9 / 18. Sketch the curves 2/x 1 - 2 = 2 and i/ae 1 '* = 2 (i) At what angle do these curves intersect each other ? (ii) Find the area of the closed figure bounded by these curves and the ordinate x 2. (iii) Find the coordinates of the centroid of this area. [(i) 4 11'; (ii) -123; (iii) 16; 1-1.] 19. Find the average value of sin 2t cos 3t while t changes from to 2n. [0.] 20. Find the R.M.S. of sin (~ t + '75] for a complete cycle. [707.] 21. Find (i) the mean value, (ii) R.M.S. of a sin - t over half a period. [(0 V ; 00 707 22. The equation of a curve is , . 2 7TX y = b sin . a Find the mean height of the portion for which x lies between and a. 23. Plot the shape of the curve whose equation is y x 3 x 5 for values of x ranging from to 1. Find the maximum value of y, the mean value of y, and the square root of the mean value of y 2 . [186; yV; '107.] 53 24. What is the average length of the ordinate PM of a semicircle of radius a (1) When P is one of a series of equidistant points on the circumference ? (2) When M is one of a series of equidistant points in the base? .-..x TTO, -, * {ll) T' ] 25. In the curve whose equation is a 3 -x 3 11 = - ax find (i) the area, and (ii) the volume formed by the revolution about Ox, of that portion of the curve for which x ranges from ~ to a. [(i) -401a 2 ; (ii) l'4a 3 .] 26. The inner curved surface of an air-container is formed by the revolution about Ox of an arc of the parabola whose equation is y 2 4x (all coordinates being in inches). The ends are circles, radii 12 and 8 inches, and the length is 20 inches. Find the volume of the container. [3-78 cu. ft.] 27. Find the position of the centre of gravity of a solid hemisphere. From a solid sphere 5 inches in radius a segment 2 inches in height is cut off. Find its volume and the position of its centre of gravity. [In middle radius, at distance from centre = f radius 54 '5 in. 3 ; 0'69 inch from centre of base.] 28. Find the area and the centroid of the area cut off from the parabola y 2 = 4x by the straight line y x. 29. The head of a shell is formed by the revolution of a circular arc AB, centre C, about the perpendicular BO. If GA = 2a CO = OA = a LACP = 6 LACQ =0 + A0, what is approximately the volume formed by the revolution of the rectangle PN about OB ? 51 Hence show, by integration, that the volume of the head is 3 ' 2a 3 nearly. [27r 3 (2cos 01) 2 cos 0A0.] 30. The figure shows the plan, and an isometric sketch, of a piece of steel, formed of two equal cylinders of radius r and length I, whose axes are horizontal and intersect at right angles. The dotted lines in the plan indicate a horizontal section at a height x above the axes. What is the breadth of this section of either cylinder ? What is the area pf the square section common to both cylinders ? What is the volume common to both cylinders comprised between parallel sections at heights x and x + Aic ? Hence find by integration the total volume common to both cylinders, and show that if r 1 and I = 10 inches, and steel weighs ' 29 pounds per cubic inch, the weight of the piece is about 16^ pounds. ri_6 r 3 _ volume common to both cylinders.} 31. Mallock's formula for the retardation, or negative acceleration due to air resistance, of a projectile is ~k (v - 850), where v is the velocity in feet per second and k is a coefficient 55 depending on the form and weight of the shell. Express the dv statement by an equation connecting -57 and v. Integrate the equation, obtaining a formula for the remaining velocity after time. [v 850 = (v 850)e*', where V Q = initial velocity.] 32. For a 12-inch gun the numerical value of k in previous question is ' 0748. Find the remaining velocity after 3 seconds, with a muzzle velocity of 2,650 feet per second. [2,288 ft./sec.] 33. A thin uniform rod OA, 6 feet long, swings in a vertical plane about a horizontal axis through one end 0. Express the velocity of a point in the rod, distant r from 0, at the instant when the rod makes an angle 6 with the vertical, in terms M of 5T Hence write down an approximate expression for the kinetic energy of a small portion, length Ar, of the rod, and find, by integration, the total kinetic energy of the rod at this instant. If the rod falls from a horizontal position, show that when OA is vertical the velocity of A is 24 feet per second nearly. dQ \_r -T ; mv 2 , where v = velocity of A.~\ 34. The diagram shows the blades, hinged at 0, of a pair of shears designed to have a constant cutting angle 2a. What angle does the tangent to either blade at P make with OP ? By considering an adjacent point Q on either edge, express this angle approximately in terms of OP = r 0# = r'.+ Ar AOP = d AOQ = B + A0. Integrate the equation which is obtained by supposing Q ultimately to coincide with P. [a ; r = tan a ; r = 35. The blade of a fan consists of a uniform circular disc, centre 0, from which a small portion has been cut away by a chord AB equal to the radius r. 56 If COP = 0, COQ = + A 0, what is approximately the area of the strip P^'P' drawn parallel to AB ? Calculate the distance of the centre of gravity of the blade from AB. 36. Assume a recoil buffer so adjusted that when the gun has recoiled a distance x inches the force resisting recoil is W (1 - 2) tons, where W and a are constants. Find the \ ft / work done in recoiling a total distance 6 inches. [7 ( 1 , ) W inch-tons.! L \ 6a-J 37. A cart weighing 200 pounds, containing 400 pounds of sand, ascends a straight hill rising 40 feet vertically altogether. The sand is assumed to be thrown out uniformly so that the cart reaches the top empty. Write down an expression for the work done against gravity, after the cart has risen a vertical height of h feet, in rising a small additional distance A/i. Find by integration the whole work done against gravity in the ascent; [(600 10k) &h ; 16,000 foot-lbs.J 38. Why is it obvious from a graph of a sine or cosine curve that = f 2ir sin 20^0= 0? Jo A torque acts on a shaft. When the shaft has turned through an angle the torque is G sin sin (0 a). What is approximately the work done in turning through a small additional angle A0? Find by integration the work done per revolution. [irG cos .] ! 39. A uniform disc, radius R, thickness a, is mounted on a cylindrical spindle of radius r, length over all I. Find the moment of inertia of the combination of disc and spindle, about the axis of the spindle, if their combined mass is M. f 27r cos d0 = ("* cos 20 d0 = Jo Jo r-a -, 2 ( R*a + r 2 / a) ) '- 40. A pulley is in the form of a uniform circular disc, radius a, thickness 26, with a V~ sna P e d groove, depth 6, breadth 26, cut all round the rim. 57 the total weight of the pulley, if the material weighs w pounds .per cubic inch. Find also the moment of inertia abdut the axle by integration, assuming that the moment of inertia of a uniform circular disc of mass ra and radius r is 9 WlV **' ~- about an axis through the centre perpendicular to the plane a of the disc. a - (a - 6 41. A trough of water has a V-section, the ends are vertical and the top is horizontal. One end ABC is movable, but is kept in position, by a horizontal bar DE fixed outside the trough.. If the water is -, 10 inches deep, find where this bar should be placed in order that the movable end should have no tendency to tilt. If the sloping sides are unequal, will the bar have the same position as when they are equal ? [See diagram.] [5" from the bottom. Yes.] 42. The Figure represents a watertight door ABCD, with certain dimensions in feet. Sea water (weighing 64 pounds per cubic foot) has risen on one side of the door to the line LL' '. What is approxi- mately the pressure on a very thin strip PQP'Q' (PQ = Ax) ? Express the resultant pres- sure on the door in the form of an integral, and find it. Find also the depth at whicl} the resultant pressure acts. [4,800 pounds ; 7 7 feet, below LV.~\ 58 43. The diagrammatic sketch shows a primitive planimeter. A carriage runs on rails RR. A horizontal disc turns on a pivot C fixed on the carriage. A fixed axle D is parallel to RR. A roller A can turn freely on D, and rests on the disc BO that when B turns A turns also, but when the carriage moves B slides under A without turning A. A cord round the rim of B goes off, always at right angles to the rails, to a pencil P. The cord tries to fly back, urged by a spring (not shown). Show that if P is taken round the boundary of the rectangle, sides y and AE, the total angle turned through by A varies as 2/Ax. Deduce that, if P is taken round the boundary of a curve, A turns through an angle proportional to the area of the curve. 44. The rod AB, of length a, is the axis on which a roller of radius r can turn freely. The roller rests on the paper, and B is kept on the line Ox, while A is taken right round the boundary of the rectangle PQMN, height y, base Ax, starting and finishing at the same point. Through what angle does the roller turn as it rolls on the paper (1) as A moves from P to Q ; (2) as A moves round PMNbacktoP? 59 Deduce that by taking A round the perimeter NPQRSTUVN the total area included may be inferred from the total angle through which the roller turns. u S _.(? T o V M N [(i) ff^ ; (ii) y5 ; (iii) roller turns through ar ar ar 45. Prove that the work done in compressing a gas from a volume v l to a volume v 2 is equal to fi pdv, ii where p denotes the pressure. 46. One cubic foot of dry air at atmospheric pressure is rapidly compressed to half its original volume : find the work done in foot-pounds. We suppose the compression so rapid that no heat is lost, in which case pv 1 ' 4 remains constant. [Atmospheric pressure is 2,117 pounds per square foot.] [1,693 - 6 foot-pounds.] 47. Explain how the work done by a variable force may be represented by an area. A spiral spring is pulled out slowly. The total extension is 3 inches, and the final value of the pull is 22 pounds. Assuming that at each moment the pull exerted is proportional to the extension, find by integration the total work done. [3-21 foot-poimds.] 48. A cork, 2 inches long, is drawn slowly from the neck of a bottle. Find the work done supposing that at each instant the force exerted is proportional to the area of the surface of the cork in contact with the neck of the bottle. Initially the upper end of the cork is flush with the top of the bottle, and the pull at start is 42 pounds. [3 '94 foot-pounds.] 60 Section 10. AREAS AND VOLUMES. 1. The curve y = a + be* passes through the three points x = 0,y = 26-62; x = 1, 7/ = 35'70; x = 2, y = 49 '81 : find a, 6, and c. What is the area of the curve from the ordinate at x = to the ordinate at x = 2 ? [10-22, 16-4, T55; area = 73.] 2. Find the volume formed by the revolution about Ox of that portion of the curve whose equation is a* + 0* - 1, which is the first quadrant. [~='209.] 3. Sketch the curve whose equation is y* = x (x - I) 2 . Find, for the portion for which < x < 1. (i) the area, (ii) the coordinates of the centre of gravity of the area, (iii) the volume of the solid formed by the revolution of the curve about the axis of x, (iv) the coordinates of the centre of gravity of this volume. [(i) 1; (ii)|,0; (iii) ^; (iv) |,0.] 4. In the curve y = ax n if y = 2'34 whenx = 2 and y 20 ' 62 when x 5 find a and n. Let the curve rotate about the axis of x, forming a surface of revolution. Find the volume of the slice between the sections at x and x + dx. What is the volume between the two sections at x = 2 and x = 5 ? [>=-4511; rc = 2'375; 1155-7.]' 5. The equation of a curve is Qy = lOx* - x\ all dimensions being in inches. Plot the graph, x ranging from to 10, and find (i) the area included between the plotted curve and Ox, (ii) the x coordinate of the centre of gravity of this area, (iii) Find the volume generated by the revolution of the plotted curve round Ox. [(i) 14-055 sq. in. ; (ii) 3 T 5 (>) 72 ' 72 inA ] 61 6. The arc of the parabola y* = 4ax, cut off by the double ordinate at x = a, is rotated about the tangent at the vertex. Find the volume enclosed by the surface of revolution so generated and by the two planes where y = 2a and y = 2a. 7. Trace roughly the curve 4i, = ae showing that it has the form of a figure of eight. Find its whole area and the volume obtained by revolving it about the axis of x. [21$; 53-6.] 8. A frustum of a prism whose edges are vertical stands 011 a horizontal triangular base ABC, the edges AA f , BB', CC' are 7, 11 and 3 feet respectively, and the sides of the base are each 3 feet long. Find the volume of the frustum and the total surface area. [27-27 cubic feet ; 78 square feet.] SA. A circular reservoir, in the form of a frustum of a cone, has a diameter of 45 at the bottom and 61 feet at the top, and the length of the sloping edge is 17 feet. Find the volume of water in it if it is filled to a depth of 13^ feet. [29,075 cubic feet.] 9. What fraction of the surface of a globe 10 feet in diameter is visible to an eye at a distance of 5 feet from the nearest point of the surface ? CM 10. A tangent cone to a sphere of radius r has its vertex at a distance 4r from the centre : compare the area of the conical surface between the surface of contact with the area of the included spherical cap. [5 : 2.] 11. Two circles, diameters 10 inches and 5 inches, have their centres 10 inches apart. Find the area contained by the external common tangents and the outer portions of the arcs of the circles. [126-4 square inches.] 12. A pyramid stands on a plane base which is a triangle whose sides are 7, 9 and 13 feet respectively. Find the area of the section parallel to the base and drawn (1) through the centre of gravity of the pyramid, (2) at a distance from the vertex = % height of pyramid. [(1) 16-85; (2) 3-328 square feet.] 13. A double convex lens, 3 inches diameter and f inch in thickness, is formed of two spherical surfaces of equal radius. Find its volume and weight, given that it is made of flint glass of specific gravity 3 ' 07 and that a cubic inch of water weighs '036 pound. [3-04 cubic inches ; -336 pound.] 62 14. The coordinates of points on a curve are given "by the following table :- X i 1-5 2 2-7 3-8 4-7 5-1 5-7 6 y 0-9 1-87 2-89 3-2 2-85 2-23 1-7 1 Plot the curve, choosing as the scale for x 1" = 1, for i/l" '5, and find by Simpson's rule the area of the curve between the curve, the axis of x, and the ordinates x = 1 , x = 6. [11-3.] 15. The half areas of equidistant transverse sections of a ship up to the loadwater-line are in square feet 7*3, 59 "1, 158-2, 237-9, 276-3, 261 '2, 192 '7, 85 '6, and 9 '3, including the end sections. If the sections are 23 feet apart calculate, by Simpson's rule, the displacement of the ship in cubic feet. [58,976 cubic feet.] 16. The cross-section of a tree (A square inches) at a distance x inches from one end is as follows : X 10 1 30 50 70 90 110 130 150 A 120 123 129 129 131 135 142 156 What is the volume of the tree in cubic feet, its total length being 160 inches ? [Use Simpson's rule and check result by using mid-ordinate method.] [12-3 cubic feet.] 17. Plot that portion of the arc of a curve given by the following values of x and y, on a scale of 1 inch = " 5 : - 5 1 1-25 1-5 2 2-5 3 - 3-3 3-47 3-5 3-45 3-35 3 2 5 18. Find the equation to the parabola of the form y = a + bx -\- ex 2 , which passes through the three points on the above curve whose abscissae are 1, 2, and 3. On the same axes and with the same scale as the given curve draw, as accurately as possible, this parabola, calculating the values of y for the same values of x as given in above table. Do you consider that the area of the given curve between the curve, the axis of x and the ordinates x = 1 and x = 3 may be taken to be the same as that of the parabola between the same ordinates ? 5 [Equation is y = 2 + ^x a; 2 .] 63 19. Prove that the area of the segment of a parabola cut off by any chord is two-thirds of the area of the triangle formed by the chord and the tangents at its extremities. [If oblique axes may not be used, take y 2 = 4ax as a a equation to parabola y = mx + and y = nx + (a 2a\ , (a 2a\ as tangents ; ^- a , -J and (^ , -J as the points of contact; their point of intersection (a a a\ mn ' m ~^~ n)' equation to the chord 2(mnx + ) of contact is y = - ---- ; area of segment a 2 / 1 1Y of parabola = ^1 --- 1 ; area of triangle _Yi_iy ~ 2 \n m) . ' . a> = a> o C-y/.]} .u 5777. F 68 17. A flywheel is brought to rest by a constant frictional torque T. Find the velocity at any instant. 18. In an electric condenser of capacity K discharging through a large resistance R we have v__ -gdv Ti~ dt' Integrate this equation to find v, the potential difference of the condenser coatings at any instant, and thence prove the formula for the leakage resistance of a condenser. 19. A vertical iron rod supports a weight W at its lower end. Find the cross-section y at any height x from the lower end if the tensile strength -j. is everywhere the same. 117 w r y _ ^_ e f x , where w = weight per unit volume.] / 20. Solve the equation d V - a _ itf d^~ by > where a and 6 are positive. A ship, mass 10,000 tons, is steaming at 10 knots against a constant wind-resistance of 1 ton weight and water-resistances of 25 tons weight. If the latter vary as the square of the speed, how far will the ship move after the engines are stopped, and for what time ? [log, ( /vX _ + N/6 ' y ) _ 2^/ a b7x + C; 0.95 miles. 24 minutes.] \v/a >/ I/' v/ 21. If (l-\i y = Ae tx + Be'* prove 4 - ~k\j = 0. y = Ae k * + Be>* - (h + fc) + ^ = 0. ii y =Acosnt J t-Bsinnt i^ + n^y^Q. 22. Solve d' 2 x n ^ + ^x= 0. If the earth were a homogeneous sphere of radius 4,000 miles, prove that a body would fall in vacuo down a straight tunnel to the Antipodes in about 43 minutes. PART II APPLIED MECHANICS. Section 1. The action of forces on a body at rest. ,, 2. Motion with uniform and with variable acceleration. 3. Composition of velocities. Paths of projectiles in vacuo. Relative velocity. 4. The action of forces of constant magnitude on a body in motion : A. Translation. B. Rotation. C. Translation and Rotation. D. Circular orbit. SA. The action of forces of variable magnitude on a body in motion. SB. The particular case of harmonic motion. 6. Friction. 7. Mechanism and mechanical efficiency. 8. The steam and gas engines. 9. Stresses in jointed structures. 10. Applications of Hooke's Law. 11. Theory of bending. 12. Theory of torsion. 13.- Fluid pressure. 14. Heel, change of trim, and oscillations of ships. 15. Resistance of ships. Section 1. THE ACTION OF FORCES ON A BODY AT REST. 1. The accompanying sketch illustrates a method of erecting a tall flagstaff. The pole DE, 60 feet long, is first of all placed in position and strongly stayed. The pole carries a pulley at E, and over this passes a rope attached to the flagstaff at 0. By hauling on this rope the mast is raised, its lower end butting against the vertical face of a trench which has been previously dug in the ground. If the flagstaff has its centre of gravity distant 50 feet from A, and if AC is 60 feet, find the pull on the rope when the angle 6 is 45 degrees. F 2 70 Taking several values of 0, plot a curve showing how the tension of the rope varies as the flagstaff is raised from a horizontal to a vertical position. It-io-J [66 W.] 2. A wheel, whose weight is 45 pounds, carries an additional weight of 5 pounds on its rim. The wheel rests in equilibrium on a rough horizontal plane. The plane is then tilted very gradually until its inclination 9 to the horizon is given by sin 6 = rV Show that the wheel will then be in equilibrium in a position in which the 5-pound load is at the same level as the centre of the wheel. 3. A circular disc of weight 20 pounds stands on .a rough inclined plane whose inclination is 15 degrees, the plane of the disc being in the plane of greatest slope. The disc is supported by a tangential force applied by means of a string at its highest point. Find the tension in the string and the magnitude of the frictional force at the point of contact. [2-63 pounds ; 2-63 pounds.] 71 4. A plank, 20 feet long, of negligible weight, is balanced horizontally across a fixed cylindrical log 1 foot in diameter, the axis of the log being horizontal and at right angles to the plank. Weights of 120 pounds are carried at the ends of the plank. If the balance is upset by adding 2 pounds to one of these loads, find the inclination of the plank to the horizontal in the new position of equilibrium. [9 28'.] 5. The accompanying sketch illustrates a device commonly employed for lifting heavy blocks of stone. Calculate the horizontal thrusts exerted by the curved bars against the sides of the recess. [3-23 tons.] 72 6'. ABC, DBE are two of the legs of a four-legged table- These legs are hinged together at B, and hinged to the table top at A and D. The table top weighs 100 pounds, and the table stands on a smooth horizontal surface. Calculate the reaction at the hinges A, D, and B. [67 '3, 67 '3, and 62*5 poun9s.] 7. The accompanying sketch shows diagrammatically a Ramsbottom Duplex Safety Valve. The effective diameter of each valve is 3 inches, and the spring is adjusted so tjiat steam . begins to blow off at valve B when the pressure reaches 180 pounds per square inch. Find the pull in the spring. Find also what vertical force must be applied at the point C in the lever so that, with a pressure of 150 pounds per square inch, the steam may just blow off (1) at valve A, (2) at valve B. [2,550 pounds ; 71 pounds down ; 4'8 pounds up.] 73 8. AB and CF are two horizontal levers, which, can turn about the fixed points E and A. They are connected together by the vertical link DB, which is freely hinged at its extremities to the two levers. The weight of AB is 10 pounds, its e.g. is G. The weight of CF is I 5 pounds, its e.g. is H. The weight of DB is 2 pounds. Calculate the load which must be suspended from C to maintain the levers in a horizontal position. If after this counterweight is applied a force of 100 pounds is applied vertically at K, calculate the weight which must be hung from F in order to preserve the balance. -4 5. -o A 6AB 40 inches. AK = 8 inches. AG = 20 inches. CD = 8 inches. DE = 6 inches. CF = 56 inches. CH = 28 inches. [12f, 6i| pounds.] 9. Find the force which must be applied vertically at F in- order to lower the railway signal shown in the figure. G is the e.g. of the arm, and A is its pivot point. GA = 24 inches. AB = EC = 6 inches. EF = CD = 12 inches. The weight of the signal arm is 90 pounds. The weight of the vertical rod BC is 64 pounds. The weight of the mass at D is 112 pounds. ' B o [20 pounds.] 10. The weight of a locomotive is supported by springs, which transmit the pressure to the axles as shown in the Figure below. The fulcrum of the lever ABC is fixed to the frame. _*_ o ou * 00 If the total weight taken by the three wheels is 15 tons and acts along DE, find the portion of this weight taken by each wheel. [From the left 4 -38, 6-42, and 4-2 tons.] 11. The mechanism of a platform weighing machine is shown in the accompanying diagram. 75 AEF is a lever pivoted at E, BCD is a lever pivoted at C. The platform bears directly on the ends A and B of the two levers by means of the short struts shown in the diagram, and the end D of the lever BCD presses upwards against the lever AEF at the middle point of EF. The load W is balanced by a weight P carried at F. Show W that if BC = CD and EF = 2EA then P = y for all positions of W on the platform. 12. The accompanying sketch gives the dimensions of a crank shaft having disc-shaped webs. Calciilate how far the e.g. of the crank shaft is distant from the axis of the shaft. 1 i i i ' \f "XO , > < ^ " - O^-, 'j 1 1 , L _ 1 1 -' -' _ 1 I i 1 ' V r K> **- H "' fr4'-> 'to CM i i i f k Y _ 1 [3-82 inches.] 13. To determine the height of the e.g. of a locomotive above the rail level, one rail is slightly elevated above the other, and the load carried by each rail is separately measured by weighing machines. In a particular case the difference of rail level is 6 inches, the loads on the upper and lower rails are 18 and 32 tons respectively. Taking the distance between the centre lines of the rails as 5 feet, show that the height of the e.g. of the locomotive above rail level is 7 feet approximately. 14. The section of a pipe is as shown in Figure below, the centres of the inside and outside bores being eccentric to the extent of ' 01 inch. Calculate how far the e.g. of the section is distant from the centre of the outside bore. [.0676 inch.] 76 15. The Figure below shows three concentric steel tubes, fitted with a solid steel plug. Find, to the nearest inch, the position of the centre of gravity of this arrangement. __ 2.0 1 10' 1 W//S I I 2" hove i V a "^ t. d I MOtekwe&s o? CcicV> tube "1 SoM S\'ee\ Plug [6 ft. 5 in. from large end.] 16. Show how to determine the shift of the centre of gravity of a figure due to a shift of a portion of the figure. A vessel of 6,000 tons displacement has 150 tons of coal shifted so that the centre of gravity is moved 21 feet transversely and 6 feet verti- cally. Find the shift of the centre of gravity of the vessel. [6^ inches.] 17. ABCDEF is a regular hexagon (see the Fig. attached). G and H are the middle points of EF andCB. Find, by graphical methods, the magnititde and line of action of the resultant of the 6 forces shown in the Figure. [7 Ibs.] 18. The Fig. attached shows the magnitude and lines of action of forces acting on a linear foot of a retaining wall arising from water pressure. Determine, by means of a force polygon and a funicular polygon, the magnitude and line of action of the result- ant water-pressure. - 15- H* 19. A locomotiA'e weighing 30 tons is carried on 3 axles with independent springs, the distance between axles 1 and 2 being 5 feet, between 2 and 3, 4 feet, while the resultant weight falls between axles 1 and 2 at a distance of 4 feet from axle 1. Give a graphical -construction for determining the loads on 2 and 3 for any assumed load on 1, and calculate the limits between which the load on 1 must be if neither of the other axles is to be entirely relieved of load. [6 and 16f tons.] 78 Section 2. MOTION WITH UNIFORM AND WITH VARIABLE ACCELERATION. 1. A train passes another on a parallel track : the former is running at a uniform speed of 50 miles an hour, the latter has a speed of 35 miles an hour, and a uniform acceleration of |- foot per second per second. How soon will the latter train overtake the former, and how far will the trains have moved in the meantime ? [88 seconds ; 2,151 yards.] 2. An engine driver puts on his brakes and shuts off steam when he is running at full speed : in the first second afterwards the train travels 86 feet and in the next 82 feet. Find the original speed of the train, the time that elapses before it comes to rest, and the distance it travels in this interval, assuming the brakes to cause uniform retardation. [60 miles an hour ; 22 seconds ; 323 yards.] 3. An express train, timed to run at a full speed of 55 miles per hour over a certain section of its journey, is checked by signal to 15 miles per hour over a mile of road under repair. The train takes one mile from rest to get up full speed, and a quarter of a mile to pull up. Find how much in time the train will lose by this check, assuming uniform acceleration and retardation. [3 m 38 s .] 4. From a vessel at rest another is observed to be 7 miles distant and steaming 21 knots directly away from the first. If the first vessel starts at once in pursuit, what time will elapse before the second comes within a range of 5 miles, supposing the first can attain a full speed of 23 knots in 10 minutes uniformly from rest ? What is the greatest distance between the two vessels ? [117^ minutes ; 8J-| miles.] 5. A projectile leaves the muzzle of a 12-inch gun with velocity 2,400 ft./sec., the travel of the projectile being 30 calibres and the twist of rifling one turn in 35 calibres. Find the angular velocity of discharge. [68 57 revolutions per second.] 6. The connection between velocity and time for a particular motion is known to be of the form v = a sin pt. The maximum velocity occurs 10 seconds after the start and its magnitude is 40 feet per second. Construct the velocity-time curve for the first 20 seconds and deduce thence the distance traversed during this period. [509 feet.] 79 7. A train starts from a station A with, an acceleration '8 feet per second per second and this acceleration decreases uniformly for 2 minutes, at the end of which time the train has acquired its full velocity, which is maintained for a further period of 2 minutes. The brakes are then applied and pro- duce a constant retardation of 3 ' 5 feet per second per second, bringing the train to rest at station B. Draw the acceleration-time curve and thence deduce the velocity-time curve and the distance-time curve. Find the maximum velocity attained and the distance between the stations A and B. [48 ft. 'sec. ; 3,310 yards.] 8. A and B are two points 10 feet apart, P is a point which starts from A and moves to B in 10 seconds, its acceleration being a maximum at A and decreasing uniformly with the time to nil at B : find its velocity when halfway to B. If P's acceleration were proportional to its distance from B, what would be its velocity when halfway to B ? [l-32ft./sec. ; 1 36 ft./sec.] 9. A ship increases her speed while steaming 10 cables as follows : . ! i Speed (knots) - j 10-0 12-1 13-7 14-8 15-6 16-0 M I i Distance (cables) 1 2 4 6 8 10 Draw the speed-distance curve and deduce the acceleration- distance curve. Draw also the speed-time curve and state the time taken over the run. [4 m 24 s .] 10. The Figure below is the velocity-time graph for a given motion. Draw to scale the corresponding acceleration-time graph, and calculate the total distance traversed in the sixty seconds. w Sees The length of the given diagram is to be taken as 6 inches and the height as 1^ inches. [= 1,258 feet.] 11. The graph of the speed of a train during a 10 minutes' run is given below. Tabulate the value of the average accelera- tion or retardation during each minute and plot the acceleration- time curve, paying particular attention to the abrupt change at the highest point of the speed curve. 30 ^ \ [Acceleration. 1 -244 ft./sec 2 . 2 -208 3 -159 4 -098 5 -073 6 -318 7 -195 8 -147 9 -073 10 -049 .] 1O 25 / \ 20 / \ 15 / 7 \ 10 / \ \ 2 \ / 1 2 3 . 4 5 6 1 8 . Time in minutes. if) V S- i.. LL 12. The Figure below is the velocity-time graph for a given motion. Deduce the distance-time graph and plot it accurately to the scales 40 feet to 1 inch, and 4 sees, to 1 inch. The given graph is symmetrical above and below the horizontal line. 24- Sec 81 13. The relation between distance and time for an electric tram-car starting from rest is given in the table : Time in seconds - 10 20 30 40 50 60 70 Distance in feet - 36 160 395 660 880 1,040 1,160 i Plot the distance-time curve, and deduce the velocity-time and acceleration-time curves for the first 60 seconds. 14. The relation between acceleration and time for an electric tramcar starting from rest is given in the table : i Time in seconds - 10 20 30 i 40 50 60 70 Acceleration in 4 9 1-1 -4 4 55 57 4 ft./sec. 2 Plot the acceleration-time curve and deduce the velocity- time and distance-time curves. 15. The relation between distance and velocity for an electric tramcar starting from rest is given in the table : Distance in feet - 1 100 i 200 400 600 800 1,000 1,200 Telocity in ft./sec. 15 i i 21 26 26 22 16 11 Plot the velocity-distance curve and deduce the acceleration- distance and time-distance curves. 16. The relation between acceleration and distance for an electric tramcar starting from rest is given in the table : Distance in feet - ' 100 200 400 600 800 1,000 1,200 Acceleration in 1-5 1-2 9 . -35 3 625 45 25 ft./sec. 2 Plot the acceleration-distance curve and deduce the velocity- distance and time-distance curves. 17. Define simple harmonic motion, and show that in a motion of this character the time of an oscillation is independent of the amplitude. 82 In a S.H.M. the complete travel is 10 inches, the velocity at the mid-point of the motion is 8 feet per second. Calculate how many complete oscillations are made per minute. In a S.H.M. the velocity at the mid-point is 8 feet per second, the velocity at a point 1 foot away from this position is 4 feet per second. Calculate the travel of the motion and the number of complete oscillations made per minute. [183 nearly ; 2-31 feet ; 66 nearly.] 18. A point executes harmonic motion in a straight line, the whole length of the motion being 6 inches and the number of complete oscillations per minute being 100. * What is the acceleration of the point when it is moving away from the centre and is distant 1 inch from the centre, and what period will elapse before it returns to the point, travelling in the reverse direction ? [9-13 ft./sec. 2 ; -235 sec.] 19. A ship is rolling with angular motion which is approxi- mately simple harmonic. The ship makes 10 complete rolls per minute and one particular roll amounts to 15 degrees on either side of the upright position. Find the angular velocity with which the ship passes through the upright position ? [7*85 degrees per second.] 20. In a simple harmonic motion the period is one second and the travel one foot. Draw the velocity-time curve. Section 3. COMPOSITION OF VELOCITIES. PATHS OF. PROJECTILES IN VACUO. RELATIVE VELOCITY. 1. Find the maximum velocity of an ice-yacht sailing at right angles to a wind of 20 miles an hour, if the angle between the sail and the keel is 15 degrees (neglecting all resistances to motion along the keel). [74 '64 miles per hour.] 2. A ship A is steering S. at 8 knots, and a ship B is steering E. at 10 knots. Viewed from A, B is in a direction S. 30 E. and it is noticed that this bearing alters 2 degrees in 17 seconds. Calculate the average distance between the ships during this period. [3,500 yards.] 3. The finish of a race between two boats A and B over a 10-cable course is a line between the mainmasts of two ships C and D, 2 cables apart, at right angles to the course. A 1-knot tide runs from C to D ; at the start A is 10 cables from C, and B 10 cables from D. What time-handicap should be allotted if the average time for a mile is 10 minutes ? [8 seconds.] 83 4. A shell, having a velocity of 2,500 feet per second, bursts 20 feet before reaching a target. It is noticed that the frag- ments strike the target over an area which is circular in form and 5 feet in diameter. Deduce from this that the added velocity given to the fragments of the shell cannot have exceeded 310 feet per second. 5. From a 6-inch gun in a cruiser, steaming North 20 knots, a destroyer bears S.E., distant 2,000 yards, steaming West 30 knots. A shot is fired from this gun with horizontal velocity 2,000 ft./sec. at the midship section of the destroyer, with elevation and training correct, but deflection not allowed for. How far aft of the midship section will the destroyer be hit ? [260 feet.] 6. A wheel, 6 feet in diameter, is rolling along a horizontal straight line at a speed of 45 miles per hour. Sketch the curve traced out by a point on the rim of the wheel, and calculate the vertical and horizontal velocities of the point when it is at a height of 4 feet above the lowest point of the wheel. [62-2 and 88 ft./sec.] 7. Find the vertical height of a projectile after 3 seconds if the whole time of flight is 20 seconds. [821 feet.] 8. From a fighting-top, 100 feet above the water-line, a shot is to be fired with muzzle velocity 2,000 feet per second at a target's water-line, distant 300 yards. At what angle to the horizontal should the gun be fired if the resistance of the air be neglected ? If the gun is lowered to the deck, 20 feet above the water- line, by how much will its maximum range be decreased ? [6 10' ; 80 feet.] Section 4A. THE ACTION OF FORCES OF CONSTANT MAGNITUDE ON A BODY IN MOTION. TRANSLATION. 1. The inlet valve of a petrol engine is kept on its seating by a spring of strength 20 ozs. The valve opens downwards and its full lift is r %- inch. If the weight of the valve is 2 ozs., find how long the spring will take to close the valve. [0118 seconds.] 2. A 12,000-ton ship is steaming 12 knots and is brought to rest during a collision in 5 seconds. Estimate the force which comes into play tending to shear one of her 47-ton guns off its mounting. [6 tons.] u 5777. 84 3. A plumb-line is suspended from the roof of a railway carriage. If the railway train has an acceleration a, show that the plumb-line will deviate from the true vertical. In par- ticular, if the acceleration is '7 feet per second per second, calculate this angle of deviation. [1 15'.] 4. Water is supplied to the cylinder of an hydraulic engine along a straight horizontal pipe, 3 inches bore and 80 feet long. The bore of the cylinder is 6 inches and the piston starts forward with an acceleration of 8 feet per second per second. If the water-pressure at the far end of the pipe is 60 pounds per square inch, calculate the water-pressure at the cylinder end, assuming the frictional losses in the pipe are zero. [25-31bs./ins. 2 ] 5. Water is flowing through a pipe 20 feet long with a velocity of 10 feet per second. If the flow be stopped in one- tenth of a second, and the retardation during stoppage is assumed constant, show that the intensity of pressure produced is equivalent to a head of about 62 feet of water. 6. (a) A mass of 20 pounds is resting on a spring balance. The spring balance is placed in a lift, and as the lift starts upwards it is observed that the reading of the spring balance is 25 pounds. Calculate thence the upward acceleration of the lift. (6) A ship is coming steadily to rest in smooth water. It is noted that a weight at the end of a 20-feet plumb line deflects 6 inches towards the bow of the ship. Deduce thence the retardation of the ship in ft. sec. units. [8 ft./sec. 2 ; ft./sec. 2 ] 7. Below is the plan view of a sailing boat. AB is the trace of the plane of the sail. Calculate the relative velocity of the wind and the sail for the direction normal to the sail. The sail area of the boat is 250 square feet. The weight of a cubic foot of air is '076 pound. Calculate the propulsive force on the boat. Indicate the assumption made in the calculation. [10-1 ft./sec. ; 15-9 pounds.] 85 8. It has been found, from observations taken at the Skeny- vore Lighthouse, that the force of the waves during a storm may reach the value of 3,000 pounds per square foot. Assuming that this pressure is due to the destruction of the momentum of a stream of water, calculate the velocity with which the water must have hurled itself against the obstacle. [38-8ft./sec.] 9. A rope coiled on the deck is being wound on to a drum at a speed of 8 feet per second. If the rope weighs \ pound per foot run, calculate the horse-power required to set the rope in motion. [-014.] 10. The front of a racing motor offers 15 square feet of area to wind-pressure. Establish a formula for calculating approximately the resistance offered by wind-pressure for a given speed. If the density of the air is ' 078 pound per cubic foot, calculate the H.P. absorbed in overcoming this wind-resistance when the car is travelling at a speed of 80 miles per hour. [107.] 11. A particle of mass 3 pounds is moving in a straight line with a velocity of 8 feet per second. After a period of 10 seconds it is moving with this same velocity in a direction perpendicular to its former direction. Calculate the magnitude and direction of a steadily applied force which will bring about this change in this time. Plot the curve traced out by the particle during this period. [105 pound ] 12. A boat is sailing 6 knots with a 10-knot following breeze. Her single sail of 500 square feet area is spread so that its plane is approximately vertical and athwartships. Estimate the steady pressure of the wind on the sail and the corre- sponding horse-power. Take 13 cubic feet of air to weigh 1 pound. [54*6 pounds ; 1 h.p.] 13. Some cubical blocks of stone are resting on a break- water when it is swept by a heavy sea. The velocity of the wave is estimated at 30 feet per second. If the blocks weigh 120 pounds per cubic foot, show that the impact of the water will be sufficient to overturn a block weighing as much as 180 tons. 14. A block is suspended by four vertical parallel strings of equal length so that the block is perfectly free to move hori- zontally in any direction. A bullet "weighing half ounce is fired horizontally into the block with a velocity of 1,500 feet per second. The bullet passes through the block, and emerges with a velocity of 1,200 feet per second. Find the magnitude G 2 86 and direction of the velocity of the block just after the bullet has emerged (a) when the direction of the bullet's motion is unchanged, and (6) when its direction is still horizontal but it is deviated by 30 degrees from the line of fire. Assume the block to weigh 100 Ibs. [094 ft./sec ; -236 ft./sec. at 52| degrees.] 15. The mass of a ship is 8,000 tons. Its speed with engines stopped drops from 6} to 5f knots in a distance of 90 feet. What is the average force resisting the passage of the ship during this motion ? What effective H.P. must be developed by the engines to increase the speed from 5f to 6y knots in a distance of 140 feet ? [23-8 tons; 632.] 16. A cyclist on allowing his machine to run freely down a uniform gradient of 1 in 20 finds that his speed eventually becomes constant. He knows, moreover, that his machine will just move forward on a one per cent, gradient. Calculate thence the opposing force due to wind-pressure when running at full speed on the steep gradient, taking the weight of machine and man together as 200 pounds. [8 pounds.] 17. When turning steel in a lathe the resistance, measured in pounds weight, encountered by the tool is given approximately by the expression 18,000 bd., where 6 is the breadth and d the depth of the shaving, measured in inches. Taking a cubic foot of steel to weigh 480 pounds, find what weight of steel can be removed in an hour when the expen- diture of energy at the tool is at the rate of 50 H.P. [8-2 tons.] 18. A motor-car is maintaining a steady speed of 30 miles per hour along a level road. On reaching a 1 in 18 descent the car is allowed to " free wheel." It is noticed that the speed still remains 30 miles per hour. The total weight of the car is 37 cwt. Calculate the effective horse-power exerted by the engine when running at 30 miles per hour on the level. 19. A railway truck of mass 10 tons moving at a speed of 4 feet per second collides with a similar stationary truck. The collision causes the buffers to compress. The buffer springs are such that the full range of movement of one truck relatively to the other is 9 inches and the , reaction between the trucks when the buffers are fully compressed is 5 tons. Calculate whether the buffers will be fully compressed in this case, and, if not, find the maximum amount of the compression produced. [Each buffer is compressed 3 '686 inches ; 4 '09 tons.] 87 20. A truck of mass 10 tons moving at 4 feet per second collides with, a truck of mass 12 tons moving at 3 feet per second in the same direction. The buffer springs of the trucks are compressed 1 inch by 1 and 1-g- tons respectively. Find the maximum reaction during collision, and the time of collision. [1-67 tons ; -32 eecs.] 21. A 45-ton gun fires a 900-pound projectile with velocity 2,500 feet per second, the travel of the projectile in the bore of the gun being 50 feet. Find the time of discharge and the recoil distance during this time, assuming no brake to act. [0285 sec.; 5 -31 inches.] 22. The work done on a projectile as it travels up the bore of a 45-ton gun is 41,940 foot-tons, and 3 per cent, of this is expended on friction and rotating the projectile. The mass of the projectile is 800 pounds, and its travel in the bore of the gun is 36 feet. At the instant the projectile leaves the muzzle a brake comes into action which exerts a constant force of 160 tons. Find the muzzle velocity of the projectile, the force which causes the gun to recoil, the maximum velocity of recoil and the whole distance of recoil. [2,700 ft./sec., 1738 tons, 21 -4 ft./sec., 2 ft. 3-4 in.] 23. Two locomotives, each weighing 80 tons, are attached to a train of 400 tons and draw it up an incline of 1 in 200 at a steady rate of 40 miles per hour. If the frictional resistance is 21 pounds per ton, find the pulls in the couplings between (1) the locomotives, (2) the locomotive and train, when the two engines are working at the same rate. Find what this H.P. actually amounts to. [6,440 pounds ; 12,880 pounds ; 962.] 24. An engine can exert a uniform pull of 4 tons. The resistance of the train it pulls is estimated at 20 pounds to the ton and the brake power, when applied to its full amount, is an additional 400 pounds per ton. The train has to be pulled from one station to the next, distant 1 mile, in 3 minutes. What is the weight of the heaviest train which can be taken in this time? [203 tons.] 25. The block in a simple ballistic pendubim is supported by four parallel cords, each 16 feet long. The weight of the block is 1,700 pounds and the weight of the shot, 9 pounds. The hori- zontal displacement of the block is 6 feet and the penetration is found to be 20 inches. Find- (1) The initial velocity of the shot, (2) The velocity of the shot at the moment of impact, (3) The average resistance to penetration. [(1) 1,643 ft./sec.; (2) 8-65 ft./sec. ; (3) 101 -2 tons.] 26. The penetration of a 4-ounce bullet at velocity 500 feet per second in a fixed block of wood is 5 inches. 88 If the bullet strikes a block of the same wood, 3 inches thick and of mass 1 pound, at the same velocity, show that if the block be free to move it will be perforated, and find the velocity with which the bullet will issue. [300 ft./sec.] 27. A smooth hemisphere rests on a horizontal plane. A particle starting from rest on the top of the hemisphere slides down the surface. Show that the particle will leave the surface 7" when it has descended a vertical distance > where r is the radius of the hemisphere. Calculate the position of the point where the particle will strike the horizontal plane. [ H>is vvew Hoe stValps QS.SP art View Q \00\<\V)Q 0V) B A and B. Two pins, P and Q, project from A, and two pins, R and /S, project from B. Four straps of equal length connect the 89 pins P, Q, R, S, so that the centres of these pins are at the corners of a square whose side is 10 inches long. If the engine shaft is transmitting 30 H.P. at 1,000 revolutions per minute, calculate the tension in the two straps which are transmitting the H.P. [94 '5 pounds.] 2. An engine is running at 240 revolutions per minute. Steam is suddenly shut off and the load removed at the same instant, and the engine then runs 300 revolutions before coming to rest. The mass of the flywheel is 2,000 pounds and the radius of gyration 3 feet. Find the moment of resistance of the engine, assuming it constant at all speeds. [94J lb. ft.] 3. In order to turn a flywheel in its bearings it is found necessary to exert a torque of 350 lb. ft. The mass of the flywheel is 5 tons and the radius of gyration 3 ft. 6 ins. If the driving power is cut off when the flywheel is running at 120 revolutions per minute, find how long it will take to come to rest. [2 m 34 3 .] 4. A 1,000 pound flywheel has a radius of gyration 2 feet. What steady tangential force acting 1 foot from the axis of rotation will in 10 seconds set up a speed of 6 revolutions per minute against an opposing frictional torque of 2 lb. ft. ? [9 '86 pounds.] 5. In a double-acting steam engine, with a stroke of 18 inches and a cylinder diameter of 9 inches, the average effective pressure is 20 pounds per square inch in either end of the cylinder. Find how much work is done in one revolution of the engine, neglecting diameter of piston rod. The engine has a flywheel weighing 2 tons, whose mass may be regarded as concentrated in a ring of 7 feet diameter. The engine is running without a load, but is using up 15 per cent, of the work done by the steam in overcoming friction, and is storing the rest in the flywheel. Find in how many revolutions the speed will increase from 100 to 120 revolutions per minute, and the time taken to effect this change of speed. [3,817 ft. IK; 12|; 7 sec.] 6. A riveting machine is worked by a 3 H.P. electric motor. The moment of inertia of the flywheel is 1,000 lb. ft. 2 and at the commencement of an operation the flywheel is making 200 revolutions per minute. If closing a rivet occupies 1 second and corresponds to an expenditure of energy of 5,000 ft. Ibs., find the reduction in speed of the flywheel. How many rivets can the machine close per hour ? [57 rev./min ; 1,188.] 90 7. A uniform horizontal semicircular trap-door, of diameter 4 feet, is hinged about the diameter, and its centre of gravity is '85 feet from the centre. Find the velocity with which its lowest point will reach the vertical plane through the hinges when the door is dropped. [14-9 ft. per sec.] 8. A ship is heeled 15 degrees from the vertical and a door of a transverse bulkhead, which is wide open against the bulk- head, swings to. Neglect friction and find its angular velocity on closing. The door is 5 feet high and 4 feet broad. [3*5 radians/sec.] 0. The door of a railway carriage is open and perpendicular to the carriage. The train starts with acceleration 2 ft./sec. 2 Consider the door as uniform, its width being 30 inches, neglect friction and find the velocity of the outside edge on closing. [3 -87 ft./sec.] Section 4C. TRANSLATION AND ROTATION. 1. Two masses M and m, suspended from a wheel and axle of radius a, fr, do not balance. Show that the acceleration of is (Ma mb) ag Ma 3 + mb 2 + r where I is the moment of inertia of the machine about its axis. 2. A flywheel consists of a solid disc, 8 inches in diameter and 2 inches thick, and weighs ' 25 pounds per cubic inch. The wheel is mounted on a short horizontal axle, 2 inches in diameter, which turns without appreciable friction. A thin cord, wrapped round the axle, carries a mass of 1 pound, which is allowed to descend under the action of gravity. Calculate the velocity of the wheel when the mass has descended 5 feet from rest. [115 revg./min.] 3. A flywheel is mounted on a horizontal axle 1 ' 25 inches in diameter. A thin cord, wrapped round the axle, is attached to a pin on the axle and the other end is attached to a mass of 4 pounds. The mass falls from rest through a distance of 66 inches and then rises a distance of 59 '7 inches, when it again comes to rest. It is also found that the mass falls a distance of 60 inches from rest in 14 '8 seconds. Find the moment of inertia of the flywheel, taking friction into account. [7-21b.ft. 2 .] 91 4. The Figure attached illustrates a trolley, driven by an electric motor through a belt drive. The weight of the whole machine is 2,000 pounds. The mass of each pair of wheels and axles is 200 pounds, the mass of the rotating part of the motor is 1,000 pounds. The radius of the wheels is 1 foot and the radius of gyration of a pair of wheels and axle is 9 inches. The speed of the motor is three times the speed of the wheels. The radius of gyration of the moving parts of the motor is 1 foot. The trolley is placed upon an incline of 1 in 100. It runs down this incline under the influence of gravity, the wheels driving the motor. Calculate the velocity acquired after running a distance of 200 feet. Wind-resistance and friction are to he ignored. [4-77 ft, sec.] 5. Four hundred feet of thin flexible rope, weighing 1 ounce per foot, is coiled up to a circumference of 10 feet ; one end is held firmly in position and the coil falls with its plane vertical, unwinding as it drops. Estimate the loss of potential energy after a fall of 100 feet, and use the principles of " Energy " to get the velocity of the coil then. [2187-5 ft. Ibs. ; 61 1 feet per second.] 6. A truck is running on the level with a velocity of 20 feet per second when the brakes are applied and the wheels locked. If the coefficient of sliding friction between the wheels and rails be yV> find how far the truck travels after the brakes are applied. The weight of the whole truck, including the wheels and axles, is 10,000 pounds, and the weight of each pair of wheels and axle is 500 pounds. The radius of gyration of a pair of wheels and axle is a foot and the diameter of each wheel is 3 feet. Find how far the truck travels if the wheels are not locked when the brakes are applied, having given that the tangential force between the brake block and wheel is 100 pounds per wheel. [125 feet; 164 feet.] 7. A tripod consists of three uniform straight rods, each weighing 10 pounds and each 10 feet long, one extremity of each being freely hinged together. A weight of 50 pounds is concentrated at the apex. The tripod stands upon a smooth 92 horizontal plane, and it is supported with each of the legs inclined at 60 degrees to the horizontal. If the tripod is then allowed to collapse, use the principle of energy to find the velocity with which the apex strikes the plane. [24 '5 feet per second.] 8. A plank AB, 20 feet long and weight 50 pounds, rests upon two solid cylindrical rollers P and Q, each 1 foot in diameter and each weighing 40 pounds. Initially the plank rests with P under A and Q under C, and the system is placed on an incline of 1 vertical to 10 horizontal. Use the principle of energy to compute the velocity acquired by the plank when B comes over the roller Q. [10 -7 ft. sec.] 9. A flywheel having an axle 3 inches in diameter is placed between two horizontal rails, with the axle resting on the rails. A rope, 1 inch in diameter, is coiled round the axle, one end being fastened to the axle, and on the free end a mass of 500 pounds is hung. Under the action of the weight of the 500 pounds the system rolls 16 feet in 10 seconds along the rails, starting from rest. Find the moment of inertia of the flywheel, if its mass is 1,000 pounds. Assume that the rope remains vertical. [1,004 lb./ft. 2 .] 10. A uniform baulk of timber, 12 feet long and of mass 700 pounds, is used as a ballistic pendulum by being suspended from an axis 2 feet from its upper end. A bullet, of mass 8 ounces, strikes it at the centre of percussion and deflects the lower end through 18 inches. Find the velocity of the bullet. Neglect the cross-sectional area of the baulk. [1,798ft. sec.] Section 4D. CIRCULAR ORBIT. 1. An elastic string, whose unstretched length is 18 inches, has a mass of 4 pounds attached, and makes 60 revolutions per minute as a conical pendulum. The string is then 20 inches long. Find the tension in the string and the kinetic and potential energy of the system. [8-2 lb,i 5 -2 ft. Ib. ; 3'1 ft. lb.-] 93 2. In a travelling crane the lifting ropes run at speed of 3,000 feet per minute. Calculate the tension set up in the ropes due to centrifugal action, given that the rope weighs 2 pounds per foot run. pounds.] 3. An electric tramcar turns a sharp corner of mean radius 30 feet at a speed of 10 miles an hour, the road being horizontal : find the least height of the centre of gravity in order that the inner wheels may not leave the rails. The gauge is 4 ft. 6 ins. [10 feet.] 4. A motor bicycle ascends a slope of 1 in 19 at a speed of 45 miles per hour. When the top of the hill is reached the road descends at a slope of 1 in 19. Show that if the bicycle is to keep on the ground the crest of the hill must be rounded off to a circle of about 135 feet radius, the rounded-off portion extending over a length of about 14 feet of the road. 5. A motor-car is rounding a curve of radius 60 yards on a level road. What is the maximum speed at which this is possible if the wheel gauge is 4 ft. 6 ins. and the centre of gravity is in the middle line of the motor and 2 feet from the ground ? What is the least coefficient of friction between road and tyres which will prevent side-slip at the maximum speed ? If the car crosses a bridge on which the roadway is in the form of an arc of 18 feet radius, at what speed would the wheels leave the ground at the crown of the arc ? [54-9 miles/hr. ; 1 125 ; 17 railes/hr.] 6. Calculate the super-elevation of the outer rail of a line 4 ft. 8|- in. gauge, the radius of the curve being 900 feet and the speed 40 miles per hour. If a 40-ton coach runs at 60 miles an hour round this curve, what is the side-pressure on the outer rail ? [6-75 inches ; 5 '97 tons.] 7. The crank arms and crank pin of a crank shaft are equivalent to a mass of 650 pounds at 1 foot radius. The shaft is supported in two bearings, 5 feet centre to centre, and the central plane of the crank is 2 feet from the centre of one of the bearings. Find the dynamical load on each bearing at 240 revolutions per minute. [2-29 and 3 '45 tons.] 8. The radius of curvature of a trajectory at a point in the rising branch is 6 nautical miles and is inclined at 30 degrees to the vertical. Find the velocity of the projectile at this point. How much higher will the projectile rise if there is no air- resistance ? [1,005 ft./sec. : 3,950 feet.] 94 9. An arm AB is capable of rotation in a horizontal plane about a vertical axis through A. A mass of 20 pounds is capable of sliding freely along this arm AB. The arm is caused to rotate and the mass is checked in its tendency to fly out by a spring which connects the mass to the point A. The natural length of the spring is 2 feet and it requires a pull of 10 pounds to produce an elongation of 3 inches. By means of a curve trace the variations in the extension of the spring as the speed of rotation increases up to 80 revolutions per minute. Show that there is one particular speed at which the extension of the spring tends to become excessively large. [Critical speed 76 '4 rev./min.] Section 5A. THE ACTION OF FORCES OF VARIABLE MAGNITUDE ON A BODY IN MOTION. 1. A variable force P pounds is the only unbalanced force acting on a mass of 30 pounds. Its value after t seconds is given in this table : t * 4 6 8 10 12 14 16 P 93-3 73-7 55-3 42-3 30-9 21-5 13-1 6-3 Draw a graph showing the relation between P and t during these 16 seconds, and state what the area between the curve and the axes measures. Find the velocity of the mass after these 16 seconds, assum- ing it was initially at rest. [612 ft./sec.] 2. A cage, weighing 4,000 pounds, is wound up a shaft. The relation between the tension, T pounds, in the rope, and the distance, x feet, which the cage has risen is given in the following table : * 10 20 30 40 50 60 70 80 ! T 6,000 5,900 5,660 5,200 4,500 3,500 2,650 2,130 2,000 Plot a curve for tension and distance, and find the work done during the 80 feet given and the kinetic energy of the cage at the end of that distance. At what point is the kinetic energy greatest, and what is then its value? ~' -' [335,000 and 15,000 ft. Ib. ; 45 ft., 62,000 ft. lb.] 95 3. A weight of 500 pounds is lifted by a vertical force which varies continually as the weight is raised ; the velocity corre- sponding to various elevations is given in the following table : Height above the ground in feet 5 10 15 20 25 Velocity in ft./sec. " - 5-2 6-4 5-5 3-1 Deduce from this table the average lifting force during each interval, and sketch a curve showing approximately the variation of this force during the complete operation. 4. The following table gives the total powder pressure, P tons, on the base of a projectile, when it has travelled x feet along the bore, the whole travel in the bore being 5 feet : X 5 75 1-0 1-25 1-5 2-0 2-5 3-0 3-5 4-0 4-5 5-0 P 3 93 118 125 119 102 78 60 48 38 29 24 23 Draw the force-space curve, and hence find the total work done on the projectile. Assuming that 5 per cent, of this work is expended in over- coming the friction of the driving-band and in rotating the shot, find the muzzle- velocity. The mass of the projectile is 12 ' 5 pounds. [1,860 ft./sec.] 5. A and B are two points on a railway line, 2,000 yards apart. The line between A and B first rises and then falls again to the same level, the actual form of the bank being given in the following table, which states the heights in feet above AB measured every 100 yards along the tract : A 100 200 300 400 500 600 700 800 900 1,000 08 2-4 4-8 7-5 11 15 18-7 22 25-1 27-6 (Continued) 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 B 29-3 30 29-8 28-5 25-3 21-2 16-5 11-6 6-5 ! A train arrives at A at a speed of 30 miles an hour, and during the journey from A to B the engine exerts a constant 96 draw-bar pull of 1,000 pounds, while the weight of the train, not including the engine, amounts to 300 tons. Assuming that the energy expended by the engine is all accounted for in kinetic and potential energies, draw to scale a graph showing the variation of train-velocity during the journey. 6. The pull exerted upon a train, together with the resistance to motion experienced by the train, are given in the following table, at intervals of 5 seconds from rest : Pull in tons - 5 4-2 3-75 3-47 3-25 3-13 3 Resistance in tons - 1 75 6 45 4 5 7 Time in seconds 5 10 15 20 25 30 Draw a velocity-time curve for the motion and state the train's velocity at the end of 30 seconds. The mass of the train is 200 tons. [14-5ft./see.] 7. A length of steel hawser is subjected to a tensile test. The extensions were measured over a 30-inch length, and the connection between the loads and extensions is given in the following table : Load in pounds 8,000 16,000 24,000 32,000 40,000 48,000 56,000 64,000 Extension in inches (I 91 1-52 2-04 2-52 2 94 3-32 3-71 4-02 On increasing the load very slightly the hawser snapped without appreciable further extension. Plot the load-strain diagram, and compute thence the work which can be stored per foot run in such a cable. A battleship of 14,000 tons is being checked by such a steel hawser ; when the hawser is taut the tension in it is 8,000 pounds, and the vessel still has a speed of 1 foot per sec. Cal- culate the least length of hawser which will be capable of bringing the ship to rest. [3,390 ft. Ib. ; 150ft.] 8. The I.H.P. of a ship of 11,000 tons when steaming at full speed, 20 knots, is 14,000. If 50 per cent, of this is effective in propelling the ship and the resistance varies as the square of the speed, find the acceleration when the ship is moving at half speed and the engine is exerting its full torque. Add 10 per cent, to the mass of the vessel to allow for the mass of the moving water. [101 ft. /sec 2 .] 97 9. After allowing for all resistances to motion the E.H.P. of a 10,000 ton ship is estimated to be 1,000. Assume the rate of work to remain constant while the speed changes from 10 knots to 13 knots, and plot the speed-time curve, and find the distance run during the change of speed. [814 yards.] Section 5B. THE PARTICULAR CASE OF HARMONIC MOTION. 1. A long spiral spring, weighing 3 pounds, has a mass of 11 pounds hung from the lower end and this is observed to make 50 complete vertical oscillations per minute. Find the static elongation of the spring when an additional weight of 2 pounds is hung on, (a) neglecting the weight of the spring, (6) taking it into account. [1-38, 1-26 it.] 2. The area of a dock is 72,000 square feet and the water- plane area of a 16,000-ton ship is 28,000 square feet. Find the period of a small vertical oscillation of the ship when in and when out of dock. [3-88 sec.; 4-97 sec.] 3. In the Fig. attached the under side of the piston of a lifting pump is 8 feet above the surface from which the water is being raised. Calculate the greatest upward acceleration which Bore 6 Bore can be given to the piston without causing the water to lag behind the piston. The friction of the water in the vertical pipe can be ignored. 98 If the suction of the pump is a simple harmonic suction with a travel of 2 feet, calculate how many double strokes can be made per minute, the water remaining in contact with the piston throughout the stroke. [25 ft./sec 2 . ; 47f .] 4. A ship is making 10 complete rolls per minute. One of these rolls amounts to 20 degrees on either side of the upright position. A man is at the mast-head, 120 feet above the axis of rotation of the ship. If the man weighs 12 stone, what force will he have to exert to prevent being thrown off the mast ? [60 pounds.] 5. A pendulum, which beats seconds at a place where the acceleration due to gravity is 32 ' 2 ft./sec 2 ., is transferred, without change of length, to a place where the acceleration is 32 ' 19 ft./sec. 2 How many seconds are gained or lost per day ? [13-4 lost.] 6. The centre of gravity of a connecting rod, 6 feet long centre to centre, is 4' 2 feet from the small end. The rod, suspended so that it is free to swing about an axis through the small end centre, swings in unison with a plumb-line 5 ' 4 feet long. Find the radius of gyration about a parallel axis through the centre of gravity. If the rod is suspended so that it is free to swing about an axis through the big end centre, find the length of the plumb- line which will swing in unison. [2 -24 feet; 4 -6 feet.] 7. The Figure attached represents a connecting rod. When set oscillating about a horizontal knife-edge bearing on A, the time of a complete oscillation is 2 ' 6 seconds. When, inverted and sup- ported with B on the knife-edge the time of an oscillation is 2 '4 seconds. The distance between A and B being 6 feet, determine the position of the e.g. of the rod and the radius of gyration with reference to an axis through the e.g. parallel to the knife-edge. B I [l_ft. 8| ins. from /? ; 2-24 feet.] 8. A flywheel, is balanced upon a knife-edge parallel to the axis of the wheel and inside the rim at a distance of 30 inches from the axis of the wheel. If the wheel oscillates with a period of .2 '58 seconds, find its radius of gyration. - [32-4 inches.] 9. A certain well-balanced wheel turns freely about a horizontal axis. It is loaded by attaching a very small mass of 10 ounces at a distance of 12 inches from the axis, and is then observed to make 20 complete oscillations per minute. Find the moment of inertia of the wheel. [3-931b. ft. 2 .] 10. A plank, 5 feet long, rests with its middle point on a fixed horizontal cylinder of diameter 3 feet. If the plank be slightly displaced and there be no slip, find the time of oscillation. [1'3 seconds.] 11. A uniform rod of length 2 feet is suspended horizontally by two equal parallel strings, each of length 6 feet, from two points in the same horizontal plane. The rod is turned through a small angle about the vertical line through its middle point and then let go. Find the time of oscillation. [1 -57 seconds.] 12. A small vertical engine, 6-incli stroke and 1 ton weight, is supported on a platform on girders which it deflects \ inch. The mass of the reciprocating parts of the engine is 100 pounds and they may be assumed to move in a simple harmonic manner. Find the amplitude of the forced vertical vibrations set up at (a) 200, (b) 400 revolutions per minute. [ 18 and ' 24 inches.] Section 6. FRICTION. 1. A and B are two vertical uprights driven firmly into the ground ; C is a horizontal cross-piece which fits freely between them. C is held in position by metal straps which hinge freely about bolts passing through this timber. The forces supporting C are supplied by the friction called into play between the outer bolts and the outside faces of the uprights. The bolts are all 1 inch in diameter. The distance between the centres P and Q is 25 ' 5 inches. u 5777. H 100 Find whether this arrangement will be satisfactory when the coefficient of friction between the uprights and the outside bolts is 0'3. B -ELEVATION 12 u--^Ql/* B -PLAN- r A [Yes.] 2. The Figure illustrates a clamp for holding a specimen between two centres. The left centre is attached to a piece which can slide freely along the horizontal bar. Show that if the coefficient of friction is greater than this sliding piece automatically locks itself by the friction set up at A and B, so that it is impossible to move it by means of a force applied along the line of centres. jra 101 3. A light arm CD is fitted with a tube AB at one extremity and this tube slides an easy fit on a vertical post. The arm CD, which is then constrained to a horizontal direction, carries a weight shown at the point P in the accompanying Figure. Show, graphically or otherwise, that if CP is less than a certain amount the arm will descend, but if greater than this critical value the tube will jamb on the post. Calculate this critical length CP when AB is 9 inches long and 3 inches in diameter. The coefficient of friction is * 5. B o [7 '5 inches.] 4. A sliding door rests upon two wheels which run on a horizontal rail. The e.g. of the door is 5 feet above the rail level and the two wheels are spaced 3 feet on either side of the vertical through the e.g. The door is pushed forward by a horizontal force applied at a height of 4 feet above the rail level. The front wheel is seized in its bearings and refuses to turn, the other wheel rotates freely. Calculate the force required to move the door. The weight of the door is 800 pounds, and the coefficient of friction for the wheel sliding on the rail is ' 2. [92-3 pounds.] 5. The accompanying Figure represents a grab used for lifting stone blocks. The flat grips, E and C, are pressed against the side of the block by levers, ABC, DBE, freely hinged at B, and, if the mechanism is suitably designed, the friction between the grips and the block is sufficient -to lift the blook. The block shown is a cube having an edge of 4 feet. Show that if the coefficient of friction between the grips and the H 2 102 block is f the grab will be able to lift the block. Calculate also the least coefficient of friction for which this is possible. [474.] 6. The cylinder of a vertical steam engine is 2 feet in diameter, the crank is 9 inches, and the connecting rod is 2 ft. 3 ins. long. The crosshead of the engine works in guides,, and the coefficient of friction between the crosshead and guide bar is '08. Take the position when the crank and the con- necting rod are at right angles on the down stroke. Assume a difference in pressure between the top and underside of the piston amounting to 80 pounds per square inch, and suppose that a force of 300 pounds is absorbed in forcing the piston rod through the gland. Calculate the thrust on the con- necting rod in pounds and the turning couple on the crank shaft in foot-pounds. [36,800 pounds ; 27,600 Ib. ft.] 103 7. ABOD is a rectangular door working in two vertical grooves AB, DC. The weight of the door is '72 tons, and its depth, AB, is 6 feet. The sliding door is being raised by a vertical force of ' 8 ton in the plane of the door, acting at a distance of 9 inches from the centre of the door, and in con- sequence the door is slightly tilted in its grooves so that contact only occurs at G and A. Show that the reactions at these points are each ' 1 ton and calculate the coefficient of friction between the door and groove. [*-] 8. The total weight of a four-wheeled railway truck is 5 tons, and its centre of gravity is situated midway between the front and back axles, which are 10 feet apart. The draw- bar is at a height of 3 feet above rail level. The axles are supposed frictionless, but the brakes can be applied so as to lock either the front or rear axle or both. Calculate the draw-bar pull required to move the truck in the following three cases : (1) Both axles locked ; (2) Front axle only locked ; (3) Rear axle only locked. The coefficient of friction beween the wheel and rail may be taken as ' 4. [(I) 2 tons; (2) 1-13 tons; (3) -893 tons.] 9. The Figure attached represents the form of steering gear known as a Rapson's Slide. Assuming that the coefficient of friction between the slide and the tiller is ' 08, calculate the pull which must be applied to the wire rope to move th.3 tiller to the left in the position shown, the couple to be overcome amounting to 40 tons-feet. [5-23 tons.] 104 10. A leather-faced cone clutch is represented in the accompanying sketch. The mean diameter of the area in contact is 18 inches and the angle of the cone is 10 degrees. The clutch is required to transmit 20 H.P, at 1,000 revs, per minute. If the two parts of the clutch are forced together __ "" "to": -ss-e: by the thrust of a spring acting along the axis of the shaft, cal- culate the least possible value of this thrust. The coefficient "of friction may be taken as ' 25. [97 pounds] 11. A footstep bearing is as shown in the Figure attached. The load on the block is 3 tons, and the taper of the wedge is 1 in 10, the coefficient of friction being 0'2. Find the 105 force in tons weight acting in the line of the arrow (a) to raise, (6) to lower the block. In raising and lowering, the friction is between the surface indicated by the darker lines. [1-62 and -87 tons.] 12. A pair of wheels and an axle are rolling along a pair[of level straight rails A and B. The lowest points P and Q of the ^HL a -*^:%fc flanges just rub against the rails as shown in the Figure. The pressure between the rails and the points P and Q is 100 pounds in each case, and the coefficient of friction is 0'18. Find how much energy is dissipated by friction in one revolu- tion. Calculate in what distance the system will be brought to rest, if its initial velocity is 20 miles per hour. The total mass is 500 pounds, and the radius of gyration about'the axis of rotation is 12 inches. [37 -7 ft. Ibs. ; 2,540 feet.] 13. The bell-crank shown in the Figure below is mounted on a pin of 4 inches diameter. Forces P and Q are applied at right angles to the arms of the lever and in each case at 4 inches from the centre of the pin. The coefficient of friction at the pin is equal to tan 30 degrees. If Q is 10 pounds weight, find by a construction the value of P, and the resultant force on the 106 pin (1) when P is just overcoming Q and (2) when Q is just overcoming P. Find also the values of P required to overcome Q when the direction of P is (1) 45 degrees above and (2) 45 degrees below the given direction. [14 '5 and 7 pounds, 19 and 26 pounds.] 14. A flywheel is mounted upon a horizontal shaft which can turn in V-shaped bearings as shown in the Figure below. The angle of the V is 90 degrees. The mass of the flywheel and axle is 100 pounds. The radius of the wheel is 10 inches. The radius of the axle is 1 inch. The coefficient of friction is ' 1. Calculate the magnitude of the least weight w suspended from the edge of the flywheel which will produce rotation. [1-42 pounds.] 15. A wheel of radius R, exerting a normal pressure W on a horizontal muddy road, forms a rut. The horizontal distance of foremost point of the tyre in contact with the road from the vertical through the wheel centre is K. Prove that, if K is small compared with R, the tractive force at the axle necessary to overcome the resistance thus set up is , i WK approximately Z5 -. H Find the horse-power absorbed in this way in the case of a 2-ton motor with 24-inch wheels running at 20 miles an hour on roads over which K = ' 1 inch. [2-] 16. A rope is coiled round two fixed bollards in the manner shown and one end is held with a force of 40 Ibs. Calculate the greatest force which can be applied at the other end with- 107 out causing the rope to slip. The coefficient of friction between rope and bollard may be taken as ' 3. [632 pounds.] 17. Taking /u, between a rope and bollard to be 0' 3, find how many turns must be taken round the bollard that a pull of 1 ton may be resisted by 50 pounds. Further, how many turns should suffice if the rope fits in grooves on the bollard, the angle of the grooves being 45 degrees ? [2 and 77 turns.] 18. Find the necessary width of belt, a quarter of an inch thick, to transmit one H.P., the belt embracing 45 per cent, of the circumference of the smaller pulley, and running at 300 feet per minute. The coefficient of friction is '25, and the stress in the belt is to be limited to 300 pounds per square inch. [2-89 inches.] 19. A flywheel of mass 20 tons is rotating 120 times per minute. The radius of gyration of the wheel about the axis of rotation is 10 feet. A belt hangs over the flywheel, one end being attached to a fixed point and the other end carrying a load of 200 pounds. The arc of contact is 180 degrees, the coefficient of friction is 108 ' 25 and the diameter of the wheel is 25 feet. Find how long this brake will take to bring the wheel to rest (1) when the 200 pound weight pulls in the direction of rotation, (2) when the 200 pound weight opposes the direction of rotation. [9-82 and 21-56 minutes.] 20. A band brake is fitted to the winding drum of a steam winch in the manner shown by the accompanying sketch. The ends of the steel band are attached to the points B and C of a continuous lever PBAC, which can turn about the pivot point A. The engine-man applies the brake by pressing down with his foot at the point P. If the pressure applied is 120 pounds, calculate the tension in the rope which can be resisted by the brake. The coefficient of friction between the band and brake-drum may be taken as ' 1. [333 pounds.] 21. A belt is to transmit 2 H.P. between two shafts running al 100 revolutions per minute, the belt running over two equal pulleys each 2 feet in diameter. If the coefficient of friction be '16, find the initial tension in the belt necessary to prevent slipping, neglecting the mass of the belt. [214 pounds.] 22. An 8-inch leather belt, f inch thick and weighing 0'8 pound per linear foot, connects two pulleys, each 4 feet 109 diameter, on parallel shafts. Slip is found to commence when the moment of resistance is 700 ft./lbs. and the revolutions 400 per minute. Taking the coefficient of friction between the pulleys and the belt to be ' 25, estimate the greatest and least unital tensions when on the point of slipping. [273 and 156 pounds per square inch.] 23. A leather belt, 7 inches wide and '4 inch thick, connects two pulleys of the same diameter on parallel shafts. The belt weighs 60 pounds per cubic foot and the coefficient of friction is '3. If the maximum stress intensity allowed is 330 pounds per square inch, find the maximum horse-power that can be transmitted and the corresponding speed of the belt. [67'4 H.P. at 92 feet per second.] Section 7. MECHANISM AND MECHANICAL EFFICIENCY. 1. The Fig. below represents the section of an hydraulic lifting-jack. The ram is kept tight against the water-pressure by means of a cup-shaped leather. The diameter of the ram is 4 inches, the water-pressure is 1,000 pounds per square inch, the coefficient of friction between the leather and the cylinder walls is 0'15, the depth of the leather exposed to water-pressure is 1 inch. Calculate the upward thrust exerted by the lifting-jack and find the efficiency of the machine. [4*77 tons ; 85 per cent.] 110 2. The Fig. below shows the mechanism of a pneumatic riveter. Air under pressure acts upon the piston C. The movement of the piston is communicated to the punch D through the crank AB and two connecting rods. Consider the position in which AB make an angle of 15 degrees with the vertical ; neglect the obliquity of the connecting rods, and find the velocity-ratio of D to C. If the thrust on the piston is 2,000 pounds, find the thrust exerted by D. The efficiency of the machine is 75 per cent. [268; 2'5 tons.] 3. In the case of a lifting-crab, in which the velocity ratio is 15 to 1, it is found that a force of 30 pounds will lift a load of 128 pounds and a force of 40 pounds will lift a load of 210 pounds. Calculate the probable load lifted by a force of 100 pounds, and estimate the efficiency in this case. [700 pounds ; 47 per cent.] 4. In a screw-jack one turn of the screw raises the load of an inch. When the screw is turned by a lever 4 feet long it is found that a force of 35 pounds will just raise a load of 1 ton and a force of 57 pounds will just raise a load of 2 tons. Calculate what load you might expect to raise by a force of 80 pounds, acting at the end of a lever 6 feet long. Determine the efficiency of the machine in each of these three cases. [4|tons; 5 -3, 6-4 and 7 '4.] 5. Show that a machine whose forward efficiency is less than 50 per cent, is irreversible. Ill In a test of a crane with a velocity-ratio of 300 the following values of load and effort were observed : Load in tons 1 2 3 * 5 Effort in pounds 31 49 67-5 86 104-5 123 Show, by means of a curve, the relation between load and effort, and deduce an analytical expression for this connection. Plot a curve showing the connection between load and efficiency. [Ps= -00818 IF +.13 pounds.] 6. In a Weston's purchase the number of sprockets in the two sheaves of the compound wheel is 7 and 8. Find the velocity-ratio for raising and for lowering. In an experiment with the purchase the following results were obtained : Load in pounds 50 150 250 350 Effort in pounds - 8-4 22-5 36-2 50 Find the law of the machine and plot the friction-load line and efficiency-load curve. [16 and 14; P= -14 W + 1-5 pounds.] 7. In a hyper-acme purchase the number of sprockets in the effort wheel is 12, and 39 links of the effort chain measure 31 inches. This wheel is mounted on a shaft carrying a double- threaded worm, which gears with a wheel having 23 teeth ; concentric and compound with this wheel is the load wheel, having 6 sprockets ; and 25 links of the load chain measure 22 inches. Find the velocity-ratio of the purchase. In an experiment with the purchase the following results were obtained : Load in pounds 50 150 350 Effort in pounds - 6-5 15 23-5 32 Find the law of the machine and plot the friction-load line and efficiency-load curve. [20-8; P= -085 IF +2 '2 pounds.] 8. The following is the description of the winding gear of a lifting-crab. A handle with a leverage of 2 feet turns a shaft,. 112 which is fitted with a bevel wheel of 12 teeth engaging in another of 25 teeth. This latter wheel is keyed to a shaft which carries a worm engaging in a worm-wheel of 50 teeth. The worm-wheel shaft carries also a pinion of 15 teeth, which engages in a large spur wheel of 110 teeth. The winding drum, which is 3 feet in diameter, is carried on the shaft of this last wheel. Find how many revolutions of the first shaft correspond to one of the last wheel, and, assuming an efficiency of 55 per cent., find the force in pounds which must be exerted at the handle to lift a load of 5 tons. [764 nearly ; 20 pounds.] 9. The gearing of a lifting-crab is shown in outline in the Figure attached. The crank F is 20 inches long and the drum E is 6 inches in diameter. The crab may be used in single gear, A gearing directly with B, or in double gear, A gearing with C and D with E. The wheel teeth are, on A 8, on B 63, on C 27, and on D 8. It is found that in single gear a force at F of 25 pounds weight lifts a load of 4 cwt., and in double gear the same force lifts 10 cwt. Find the efficiency in each case. [34-1 and 25-3 per cent.] 10. The Figure attached represents an oblique acting pulley. The pulley wheel A is guided by frictionless restraints v^\\\\\\^\^^^ ^^^. - O so that its centre moves in a vertical line. The centre of the pulley wheel B is fixed. The cords under the movable pulley 113 make angles of 30 degrees and 60 degrees with the horizontal. Calculate the ratio of P to W for this position, and use the principle of work to deduce the velocity-ratio of haul to lift. [-732; 1-366.] 11. Show that the propulsive force on a bicycle is the turning moment given to the crank shaft divided by the semi- gearing of the machine. A bicycle and its rider weigh 200 pounds ; the high gear is 80 inches and the low 60 inches, while the effective crank length is 6 inches ; if the road and frictional resistances absorb 3 pounds of the propulsive force, find what steady force at the pedal at right-angles to the crank will keep the bicycle going at uniform speed up a slope of 1 in 40 (1) on the high gear, (2) on the low. [53^ and 40 pounds.] 12. The engine of a motor-car is giving out 20 B.H.P. at 800 revolutions per minute. Calculate the torque in the shaft connecting the engine to the gear-box. A spur-wheel, with 11 teeth, is keyed to this shaft, and engages in the gear-box with a second shaft carrying a spur- wheel with 16 teeth. This second shaft drives the back axle through a bevel-gear, the number of teeth on the bevel being 10, and the number in the wheel attached to the back axle being 24. The diameter of the road-wheels is 36 inches, and it is found that the propulsive force exerted by these road- wheels is 220 pounds. Calculate the efficiency of the gearing, and the speed of the motor-car, in miles per hour. [131-2 lb. ft. ; 72 per cent. ; 24-55 miles per hour.] 13. The jib AB of a crane is 20 feet long, it is pivoted at the lower end A, and the lifting chain passes over a pulley at B. From B the chain passes round a pulley at C, 6 feet above A, then round a pulley attached to the jib at D, 5 feet from A, back to a pulley fixed just below C and on to the winding drum just behind A. The drum is 6 inches in diameter, and has a wheel with 119 teeth driven by another of 20 teeth, there being a handle on the latter with a radius of 18 inches. What is the velocity-ratio for this arrangement and what is the efficiency if it takes an effort of 112 pounds to lift 1 ton? Point out why the raising and lowering of the jib is facilitated by the extra pulley at D. [35-7 ; 56 per cent.] 14. The Figure attached represents a block for lifting heavy weights. The spindle A and the toothed wheel B are both secured to the block so that they cannot revolve. The toothed wheel C revolves loosely on A and is bolted to the grooved pulley wheel D, around which is led the chain by which the weight is lifted. E is a grooved pulley wheel turning freely upon A about which the rope for hauling is led. F and G are two toothed wheels keyed to the spindle H. H revolves freely in a bearing fitted through E. B has 30 teeth. C has 31 teeth. F and have the same number of teeth. "The diameter of E is 1 2 inches. The diameter of D is 7 inches. Find how many times E must be rotated to give D one complete revolution. Find the velocity ratio of haul to lift. Given that a weight of 70 Ibs. will just raise a load of 1 ton, find the efficiency of the machine for this arrangement. [31 ; 53-1 ; GO'S.] 15. The Figure below represents the mechanism of a ship's, steering gear. A rotation of the shaft A sciews the piece B to the right and C an equal distance to the left. These motions communicate rotation to the tiller through the connecting rods BD, CE. If the efficiency of this mechanism is 50 per cent., calculate the torque which must be applied to the shaft A to overcome a moment of resistance of R foot-tons in the shaft F for the position shown. One turn of the shaft A displaces B and C each through a distance of -inch. [J* ton ft.] L 226 115 16. In the mechanism shown in the Figure below A and B are fixed centres. C is a block sliding on the arm ED. E is a block sliding on the straight line AE, which is perpendicular to AB. p t -t-- --I 6 The lengths are AC = 8 inches, AB = 25 BD = 36 AC rotates uniformly, making 30 revolutions a minute. Find the time occupied by E in the outstroke and return stroke respectively. Neglect the obliquity of the link DE, and find the greatest forward and backward velocities of E. [1-208 and -792 seconds ; 2-3 and 4 '44 ft./sec.] 17. The Figure attached illustrates an oscillating cylinder. The stroke is 4 feet, the distance AB is 8 feet. The crank makes 120 revolutions per minute. Calculate the angular velocity of the cylinder in revolutions per minute (1) when = 0, (2) when = TT, (3) when B = . If the couple required to rotate the cylinder in its bearings is 20 Ibs./feet, calculate the couple required to rotate the crank shaft for the positions (1) and (2). [40 rev./min., 24 rev./min., 17 ' 15 rev./miu. ; 6 and 4 Ib. ft.] u 5777. 116 18. The mechanism in the Figure below consists of a crank AB, 7 ' 5 inches long ; a link CB and a lever CDE, capable of \ \ oscillating about D. Find the instantaneous centre of the link CB in the given position, the angular velocity of CB, and then deduce the velocity of the point C on the assumption that AB is at the instant turning at the rate of 20 radians per second. You must show that your method of deduction is correct. [9-2ft./sec.] 19. AB is a link connecting two cranks OA, PB turning about the fixed centres 0, P (see Figure below). The crank OA is turning uniformly, making 120 complete revolutions per minute. 117 OA, AB, PB are 1, 3, and 4 feet long respectively. (1) Calculate the acceleration of B in the first position shown (PB perpendicular to OAB). (2) Determine the velocity of B', and the angular velocity of A' B', for the second position shown. [(1) 210- 7 ft./sec. 2 (2) 8-4 ft./sec. 4'2 radians sec.] 20. The point B (see Figure below) is moving backwards and forwards along PQ with simple harmonic motion, the number of complete oscillations being 120 per minute. Con- sider the mechanism for the position when B is 3 inches from Q, determine the velocity of B, the velocity of A, and the horizontal component of A's velocity. [4 8 ft./sec. ; 2-88 ft./sec. ; 1-85 ft./sec.] 21. A, A are two parallel rods (see Figure attached), con- nected together by two transverse bars (B, B). This system slides in fixed guides 0, C, and is operated by the eccentric turning about the fixed centre 0. The throw of the eccentric is 2 inches, the radius of the circle shown in the Figure is 4 inches, and the eccentric makes one revolution per second. Take the eccentric in the position shown Calculate (1) The vertical velocity of the moving parts A, B, (2) The relative sliding velocity at the point where the eccentric touches the slide. i 2 118 Suppose the resistance to motion of the AB system is 10 pounds and the coefficient of friction between the bar B and the eccentric is 0*15, calculate the torque which is required to rotate the eccentric. Examine whether the slider can drive the eccentric in this position of the eccentric. r\ n ^ n [ 76 ft./sec. ; 2-87 ft./sec. ; 1-56 Ib. ft.] 22. Describe, with the aid of a sketch, the gear for changing the speed of the leading screw of a lathe. If the leading screw is right-handed and has two threads to the inch, choose suitable change-wheels for cutting a right- handed screw with 12 threads to the inch, the numbers of teeth in the available change-wheels being 20, 25, 30, . . 100. 23. The Figure below illustrates a friction drive. The shaft A drives the shaft B by means of a cylindrical wheel C, 119 pressing upon the disc wheel D. There is no slipping between the wheels at the radius r c and the pressure between them is uniformly distributed. If a torque of 40 foot/lbs. is applied to the shaft A, calculate what torque is communicated to the shaft B, and determine the efficiency of the transmission. -/ = 3 inches ; r n = 5 inches ; diameter of C Dimensions is 2 inches. [160 lb. ft. ; 80 per cent.] 24. The diameters of a pulley and its rope are 5 inches and 1 inch respectively. The diameter of the pin on which the pulley is mounted is 1 inch and the coefficient of friction between the pin and pulley is ' 26. Find the efficiency of the pulley. Find also over how many such pulleys the rope should pass to make a self-holding tackle. [92 per cent. ; 9.] 25. In a Weston's purchase the large pulley is 6 inches in diameter and the small pulley 5 inches, measured in each case to the chain centre. The coefficient of friction between the compound pulley and its pin is ' 2. What should be the least diameter of the pin in order that the purchase may be self- holding ? [1-274 inches.] 26. In the epicyclic gear shown in the Figure below the wheels B, F, and E are mounted independently on the spindle 0, while G and D rotate together on the spindle P, which is carried by the link OP. The wheel B has .12 teeth, 120 C has 30 teeth, and D has 14 teeth. All teeth have the same pitch. How many teeth are there on E and on F ? If the wheel E is fixed, what is the ratio of the velocity of the wheel F to that of the wheel B ? If the wheel B is fixed, what are the ratios of the velocities of E and of F to the velocity of the link OP ? [72; 56; 35' 6 ' 1 O* 27. A reverted epicyclic train is as shown in the Figure below. A is a fixed annular wheel of 60 teeth. BC is a double intermediate wheel, mounted on an eccentric E which is keyed to the shaft H. B has 55 and C has 59 teeth. B gears with A, and C with an annular wheel L of 64 teeth which is loose on the shaft H . Find the number of revolutions made by H for -f 1 revolu- tion of the wheel L. [-176.] 28. In the Humpage epicyclic gear, sketched, the numbers of teeth in the various wheels are B 14, C 48, D 24, F 44, E 58. If E is fixed, compare the speeds of B and F ; if B is fixed, compare those of E and F ; and if the spindle S is pre- 121 vented from revolving round the main axis of the gear, compare the speeds of B and F, and of E and F. l TO'J 29. The Figure below shows a diagrammatic representation of the Adam's 3-speed and reverse epicyclic gear. First or lowest speed, F is held ; second speed, C is held ; third or highest, F is locked to G ; reverse, G is held. If the speeds are respec- tively 11, 20, 30 and 8 miles per hour, find the values for the ratios r ly r 2 , r 3 and r 4 of the numbers of teeth on the central wheels to the number on the corresponding epicyclic wheels if r lt r 2 = 1. G c. J [8165; 1-2247; 1-7237; 1-1081.] 3D. The Figure below represents diagrammatically a gear- box for a motor-car of the type in which the gear ratio can be 122 gradually varied. The propeller shaft A carries the bevel pinion B, which gears with two crown wheels C and D running loose on the sleeve L. C carries the small satellite pinions E and F of an epicyclic gear, of which the outer, internally toothed, wheel G is fixed to the box of the differential gear H and so drives the back axle. The centre pinion K of the epicyclic gear is fixed to the loose sleeve L which carries the friction surface M, while the other friction surface N is carried by the crown wheel D. P and Q are friction rollers connecting the surfaces M and N, the spindles of the rollers being carried by the gear-box in such a manner that they can be inclined at any angle to the back axle. If the numbers of teeth on the wheels are B 16, C and D 46, E and F 15, G 50, K 20, calculate the gear ratio when the rollers touch N and M at radii in the ratio 4:1, and also when they touch at radii in the ratio unity. and *-/ in opposite directions.] 31. Sketch and describe Hooke's joint for connecting two shafts whose axes intersect. Obtain an expression for the velocity ratio of two inclined shafts connected by such a joint, and say how you could arrange with one or more Hooke's joints to transmit a constant velocity ratio between two inclined shafts. 123 Section 8. THE STEAM AND GAS ENGINES. 1. The Figure attached illustrates the arrangement of a rope brake for taking the brake horse-power of an engine. Calculate the B.H.P. from the following data : Diameter of brake wheel - 20 inches. Speed - 750 revolutions per minute. Spring balance reading - 5 pounds. Dead weight supported Weight of rope Diameter of rope - 25 pounds. pound per foot. 1 inch. VI V ___ -* (ft When the rope is lifted off the top hook the spring balance reads \ pound. [2-56.] 124 2. A steam-engine has the following dimensions : crank, 9 inches long ; connecting rod, 3 feet long. Find, by means of a graphical construction or otherwise, the turning moment on the crank when it makes an angle of 60 degrees with the line of stroke, having given that the total effective pressure on the piston in that position is 1,500 pounds. [1104 Ib. ft.] 3. In a steam-engine the crank is 1 foot long ; the connect- ing rod, 5 feet long ; the crank pin diameter, 3 inches ; the crosshead pin diameter, 2 ' 5 inches ; and the speed, 120 revolu- tions per minute. Find the rubbing velocity at the surface of each pin at the instants when the crank makes 45 degrees with the line of stroke. [21*5 and 2*23 inches per second.] 4. The Figure below is the indicator diagram of a gas- engine. The length of the connecting rod AB is 48 inches. The length of the crank CB is 10^ inches. The internal diameter of the cylinder is 12 inches. 1 a) J- 550 300 Z50 \oo 50 125 Consider the position during an explosion stroke when the angle ACB is 30 degrees and make the following calculations : (1) The total thrust against the end of the piston ; (2) The thrust in the connecting rod ; (3) The thrust of the piston against the cylinder walls ; (4) The turning couple on the crank shaft. Graphical methods are to be employed. Friction and inertia of the moving parts are to be neglected. [26,000 pounds, 26,200 pounds, 3,000 pounds, 13,000 Ib. ft.] 5. The Figure below is the indicator diagram for a gas- engine. The connecting rod is 3 cranks long and the piston is 100 square inches. Neglect the inertia of the reciprocating parts and sketch roughly the form of the crank effort diagram Scale, /' tc WO #*> /n for the explosion stroke. Obtain accurately the greatest crank effort measured in foot-pounds. The length of the crank is 1 foot. [11,800 Ib. ft.] 6. Prove that the acceleration / of the piston of a direct- acting engine may be approximately represented in terms of the velocity F of the crank pin, and the radius R of the crank in the form / = ^ (cos 6 + ] cos 26), H n where 6 is the angle which the crank makes with the line of the stroke and nR is the length of the connecting rod. Plot the values of / on an angle base for half a revolution of the crank when R is 1'6 feet, n is 4, and V is 12 feet per second. Explain how this diagram is used in connection with an indicator diagram to obtain a curve of crank effort. 126 7. Find the forces necessary to accelerate the reciprocating parts of a direct-acting steam-engine at the beginnings of two successive strokes from the following data : Length of crank, 1 foot ; length of connecting rod, 5 fee: ; weight of reciprocating parts, 280 pounds ; and speed, 300 revolutions per minute. [4 -63 and 3 -08 tons.] 8. An engine has a stroke of 18 inches and runs at 400 revolutions per minute. Assuming that the motion of the piston is a simple harmonic motion, find the maximum acceleration of the piston. Draw to scale the curve of acceleration for one complete revolution (1) On a piston base - scale, i full size. (2) On a crank angle base - scale, 100 degrees to inch. Scale for acceleration : 500 feet per second per second to 1 inch. At the beginning of the forward stroke the steam-pressure is 80 pounds, and the pressure remains constant until cut off at half stroke. Draw the crank effort diagram for this portion of the stroke (1) Neglecting the inertia of the moving parts, (2) Taking into account the inertia of piston. [Diam of cylinder, 12 inch. Mass of piston, 100 pounds.] [l,316ft./sec 2 .] 9. The Fig. attached is the indicator diagram for a double- acting steam-engine whose stroke is 2 feet and length of connecting rod 4 feet. The area of the piston is 250 square inches. Consider the case when the crank has turned through Spring 1/50. 45 degrees from the left dead centre A, and determine the turning couple on the crank shaft. If the reciprocating mass is 500 pounds and the engine is making 240 revolutions per minute, find the turning couple for the same position, corrected for inertia. [12,200 Ib. ft. 6,380 Ib. ft.] 127 10. Explain what are the functions of a governor and of a flywheel in regulating the speed of an engine. The revolutions of an engine of 200 h.p. are not to be greater than 400 ' 5, nor less than 399 '5 per minute, and the greatest fluctuation of energy is 32 per cent, of the energy exerted in a revolution. Determine the mass of the flywheel, assuming that it has a mean diameter of 6 feet. [4,280 pounds.] 11. The crank effort diagram for a double-acting steam- engine is shown in the Fig. attached. The two peaks correspond to crank efforts of 1,800 and 1,500 Ib. feet. Calculate the mean value of the crank effort. If the engine is making against a steady resistance equal to the crank effort, calculate the revolutions per minute for the crank positions defined by the letters ABCD. The revolutions are 500 per minute when = 0. The mass of the flywheel is 400 pounds. The mass of the flywheel may be taken as situated 2 feet from the axis of rotation. [825 Ib. ft., 499-3, 502-3, 500, 501 -8.] 12. What would be the fluctuation of speed at full load of a 4-stroke cycle single cylinder 15 h.p. engine at 275 revolutions per minute mean speed ; the weight of the flywheel, supposed concentrated at the rim whose diameter is 6 feet, being 3 tons and the resistance to motion constant? The fluctuation of energy per cycle is 80 per cent. [^ rev./min.] 13. The stroke of an engine is 3 feet, the connecting rod is 4 feet long and the piston is 2 feet in diameter. The mass of the reciprocating parts is 600 pounds and the indicator diagram shown below. Draw the crank effort diagram, corrected for inertia. 128 The mass of the flywheel is \ ton and it may be regarded as concentrated in a ring of 6f feet diameter. Find the fluctuation of speed at 200 revolutions per minute. Spring 1/50. [4 rev./min.] 14. The diameter of the high-pressure cylinder of a cruiser is 43 inches ; stroke, 3 feet 6 inches ; connecting rod, 7 feet long, and revolutions at full power, 135. When the piston has completed of the down stroke, the forward steam-pressure is 180 pounds and the back pressure 110 pounds per square inch. The weight of the reciprocating parts is 2 tons and of the connecting rod, 2 tons. The e.g. of the latter is f the length from the crank pin. Find, by Klein's construction, the acceleration of the piston, and find also the effective twisting moment in foot-tons on the crank shaft in this position. [I75ft./sec. 2 ; 51-75 ton ft.] 15. A shaft, running in bearings 13 feet apart, carries 3 masses of 40, 50 and 60 pounds at distances of 3, 8 and 12 feet from one bearing and at equal radii. The second mass is 130 degrees and the third mass 245 degrees ahead of the first mass. Find magnitudes and angular positions of the two necessary balancing masses, each to be placed outside the bearings and 1 foot from the nearest one. [42 and 19 potmds ; 38 degrees and 215 degrees ahead of the first mass.] 16. A locomotive with two driving wheels has two cranks of 12-inch radius at right angles and 30 inches apart. The rotating masses of the cranks are 450 pounds each, and the reciprocating masses of each cylinder are 500 pounds. Find the position and magnitude of balance masses to be attached to the driving wheels in planes 58 inches apart at a radius of 2 feet to balance the rotating and f of the reciprocating masses. [312 pounds at 162^ degrees to near crank.] 129 17. A simple governor consists of two balls, each of mass 10 pounds, attached to links pivoted on the axis ; a collar on the axis is carried by two similar links, pivoted to the collar and at the centre of the balls, so that the collar rises twice as fast as the balls. The frictional force reduced to the collar is + ^ pound. If the mean position of the balls corresponds to 60 revolutions per minute, find the rise and fall of speed necessary before the collar begins to move from that position. [1^ rev./min.] 18. In a simple governor the links form a parallelogram ABCD, the four links being equal and of uniform section. The length of each link is 9 inches and the mass of each, 1 ' 2 pounds. AC is vertical and balls of mass 5 pounds each are placed at B and D. Find the limits of speed at which the balls will float at a radius of 5 inches, allowing for the weight of the rods, the centrifugal action on them, and a frictional force + 1 pound acting on the sleeve at C. [72 to 82 rev./min.] ' 19. In a loaded governor the mass of each ball is 5 pounds and of the load 50 pounds. The balls rise half as fast as the load. Find the height of the governor due to 120 revolutions per minute and the force necessary to prevent the load from moving up when the speed increases to 125 revolutions per minute. [2-23 ft. ; 4'6 pounds.] 20. In a loaded governor the mass of each ball is 7 pounds and the mass of the load is 150 pounds. The frictional force reduced to the collar is + 2 pounds. The length of each link is 1 ' 2 feet. Find the range of speed to alter the balls from a radius of ' 5 foot to a radius ' 6 foot. [244 to 253 rev./min.] 21. In a crank shaft governor when the radius of the circle described by the balls is 8 inches the controlling force is 450 pounds and when 12 inches it is 750 pounds. The mass of each ball is 30 pounds. Find the radius of the circle which the balls describe when the speed is 240 revolutions per minute, the controlling force curve being a straight line. [5 '84 inches.] 22. A Hartnell governor, when running at a mean speed of 300 revolutions per minute, has two balls, weighing 9 ' 5 Ibs. each, revolving at a radius of 6-f inches. The smallest radius of revolution of the balls is 4 inches, at which the valve is fully open to steam ; the largest radius of revolution is 8 inches, when the steam is entirely cut off. The radii of the bell crank lever are 5 inches and 4 inches as shown in the Figure below. The spring compresses uniformly i inch for every 10 pounds of load upon it. Determine the speed at which the governor balls 130 begin to move outwards and the speed at which steam is cut off, neglecting the weights of the balls and also the effect of friction. Is this governor stable ? [221 and 335 rev./min. Yes.] Section 9. STRESSES IN JOINTED STRUCTURES. 1. In the crane shown in the Figure below the load of 10 tons at the middle point of AB may be regarded as equivalent to Y7///7////77/7/7/9 so >6 - 30 131 5 tons at A and 5 tons at B. Find by graphical construction the stresses in the various members of the framework and indicate the members which are acting as struts. 2. The frame ABCDEF, consisting of light rods freely jointed together, is hung from smooth pins B and C, and supports weights as shown. 20 Tons. 5 Tons. 15 Tons. 10 Tons, Find which of the two dotted bars should be inserted so as to be in tension and, when this has been put in, determine the stresses in all the bars of the frame. rT ,_ -. [BE.J 3. The Figure below illustrates a floating crane. Draw the reciprocal diagram for the framework and determine thence, or otherwise, the stresses in the three lowest members of the frame- work. Indicate which of these members is acting as a strut. --- -so -- [153, 36 and 220 tons.] u 5777. 132 4. The framework shown below is supported at its extremities in the manner indicated. The depth of the framework is 10 feet and the vertical members are spaced at distances of 10 feet apart. Find the reactions at the supports by help of the link polygon and draw the stress diagram. Write down the stress in the member AB and find it independently by moments. '4-Tons 6 Tons 8 Tons * 4- Tons * 3 Tons [13-2 and ITS tons. 25 -5 tons.] 5. The framework shown in the Figure below is fixed at B and carried on rollers at A, so that R A , the reaction at A, is loooLbs vertical. The framework carries a vertical load of 1,000 pounds, and two forces of 400 pounds, due to wind-pressure, also act upon the frame. Determine the reaction at B, and construct to scale the reciprocal figure for the framework. [l,150lbs.] 133 6. An arch, hinged at A, B, and 0, is loaded in the manner shown, and the hinges A and C are fixed to masonry abutments. Ascertain the directions and magnitudes of the reaction at A, B, and 0, and then construct to scale the reciprocal figure for one half of the arch. [Reactions at A and C, 5 '87 tons at 47 to vertical. Reaction at B, 4 '29 tons, horizontal.] 7. The Figure attached shows a gangway extending from the shore to a floating landing-stage. The gangway is hung from a framework by means of three vertical tie-rods, the tension in each tie-rod being one ton. Suppose that the landing-stage is K 2 134 being pushed towards the shore, so that the framework is subjected at A and B to horizontal thrusts amounting to five tons. Determine the vertical reaction at these points. Draw the reciprocal figure for the framework, and determine the greatest tension and compression stress existing in the framework. __*___'/ [At A, 1 ton down, at jB, 4 tons up ; 3 -2 and 8 '9 tons.] 8. In the framework illustrated in the Figure below the bars passing through P and Q are independent at the crossing L. 10 * I I V- 10 points. Show that if pin joints were introduced at P and Q the stresses in the framework would be unaltered. Determine the stresses in the various members of this framework. [Stress in vertical member is 1 ton. 3 135 9. The total span of a Fink truss is 30 feet, the depth is 4 feet, and the four bays are equal. It is loaded with 2 tons per foot run over the whole span and all the joints are pin joints. Draw the reciprocal figure and find the stresses in all the members. 10. Determine the reactions at the bearing A and footstep B in the crane when loaded as shown in the Fig. attached. Draw SO cwts. i .*.___ a stress diagram showing the amount and kind of stress in the members C, D, E, F, and G. 136 The chain is parallel to the adjacent member and one end may be assumed to be fixed to a bracket attached to the vertical centre post. Neglect the effect of friction. The pulleys are of 8 inch radius. [5-5, 74-3 tons ; 48, 80, 92, 49, 22 tons.] 11. The bracing of the crane shown in the accompanying Figure consists of isosceles triangles, having equal bases upon an arc of 30 feet radius. The radius of the inner arc is 28 feet and the bars A and C are 8 feet apart at the ground level. Determine graphically the stresses in the bars A, B and C when the crane is supporting a 5-ton load in the manner shown. [23, 4, and 16 tons.] 137 12. The Fig. attached represents the mechanism of a steam excavator. The operation of excavating produces a force of 3 tons, acting on the bucket along the given line of action GF. Calculate the reaction at the point B, where the bucket is hinged to the jib. Assume that pulley wheels turn without appreciable friction and determine the pull in the rope leading to the winding engines. By considering the forces acting upon the jib, find the tension in the siipporting tie-rod and the reaction at the hinge A. Graphical methods are recommended. The weights of the various parts are to be ignored. Dimensions : AB == 17 feet, AC = 22 feet, AD = 31 feet, EG = 5 feet horizontal, EF = 16 feet vertical. D is vertically above E. EBD is a right angle. The winding rope has one end attached to the point C. It passes round two 3-foot pulleys and is led away to the winding engine in a direction parallel to AD. [2-5, 2-6, 6-5 and 11-5 tons.] 138 13. The Fig. below is a sketch of a small locomotive crane. The jib of the crane is hinged at its lower end, and is supported by a tie-rod in the manner shown. By drawing the force polygon and! funicular polygon determine graphically the tension in the tie-rod and the reaction at the hinge. [8 and 13 tons.] 14. In the travelling crane shown below, AB is the jib, which can swing about A as centre by the circular rack and 2000\lbs pinion. At B is pivoted a beam CE, the end C being secured to the tie-rod CD whose end is secured to the fixed point D. 139 The design is such that at all radii CE is horizontal. A load of 2000 pounds is suspended from E. Find the longitudinal forces in AB and CD and the maximum bending moment in AB. [4,243 and 2,828 pounds ; 70,000 Ib. ft.] 15. Determine by the method of sections the stresses in the bars a, b, c, which are cut by the plane PQ as shown in the Fig. below. Any length required may be obtained graphically. The central load is 6,000 pounds and each of the others 2,000 pounds. [5,830, and 5,000 pounds.] 16. Explain the principle of the method of sections for determining the stresses in a framed structure. A ship's gang- way is loaded in the manner shown in the Fig. attached. The lower end is carried on rollers, and the upper end is hinged 140 to the side of the ship. Determine the stresses in the members of the second panel cut by the plane AB. [7,728, 1,131 and 7,303 pounds.] 17. A symmetrical pin-jointed frame is loaded as shown in the Fig. below. Determine the stresses in the members A, B and C of the frame, using the method of sections. [18,000, 6,200 and 12,500 Ibs.] 18. In the roof truss shown in the Fig. below,^the tie AB is jointed at C but is continuous through D and E. Determine the forces in all the bars in the Figure and sketch the curves of bending moment and shearing force on AC and CB. \ ton [Maximum bending moments in AC and CB, 8^ and 4 ton ft.] 19. In the roof principal ABODE, the dead loads are as shown and the wind-pressures are equivalent to a positive 141 pressure of 30 Ib./ft. 2 normal to AB, zero pressure normal to BC, a negative pressure of 20 Ib./ft. 2 normal to CD, a negative pressure of 40 Ib./ft. 2 normal to DE. The roof principals are UJ 15 feet apart, the end A of each principal "being hinged and the end E on rollers. Draw the polygon of external forces for the roof principal and determine the forces in the bars meeting in F. Span, 55 feet ; radius of circle through upper pin joints, 30 feet. [Reaction at E, 1 ton vertical ; reaction at A, 7 '9 tons at 31 to horizontal. Starting at FA in a clockwise direction 9 '3, 3, 3-5 and 6'8 tons.] 142 20. The legs of a tripod are 6, 7 and 8 feet long, and the points where the legs touch the ground form an equilateral triangle of 8-feet side. If the load suspended from the apex is 10 tons, find the thrust in each leg. [6*7, 5 and 1 5 tons.] 21. A pin-jointed structure in a vertical plane, similar to the Great Wheel (Earl's Court or Blackpool), consists of a regular polygon with an even number of sides, each point of junction carrying a load W and being connected to the centre by a radial tie-rod. If each tie-rod when in the upper vertical position is subjected to a tension T, find the tension in a tie-rod in the lower vertical position. [ T + 4 fT.] 22. A chain, which with its load weighs % ton per foot of horizontal run, is suspended between two points A and B, 40 feet apart horizontally, B being 10 feet below A. Find the greatest tension in the chain when its lowest point is 4 feet below B. [8 -9 tons.] Section 10. APPLICATIONS OF HOOKE'S LAW. 1. Describe the behaviour of a specimen of mild steel when subjected to pull in a testing machine until fracture takes place. A steel specimen, 1 ' 01 inches in diameter, was subjected to a gradually increasing tensile stress in a testing machine until it broke, and the following readings were obtained : Load, Tons. Extension, Inches. Load, Tons. Extension, Inches. 13-6 0-2100 2 0-0014 15-0 0-3200 4 0-0029 16-0 0-5600 6 0-0044 18-5 0-80 8 0-0059 20-4 1-40 10 0-0074 19-6 2-13 12 0-0089 Specimen broke. Original length of specimen between measuring points - 8 inches. Diameter of section at fracture - - -f inch. Draw a load-extension diagram to scale from the above data, and determine the modulus of elasticity of the specimen, the stress at the yield point and at the maximum load, the reduction of area at fracture, and the percentage extension at fracture. [13,500, 16'9, 25*5 tons per square inch; 61 per cent. ; 26 -6 per cent.] 2. An iron and brass wire, of 10 feet lengths and of diameter 0'075 and O'l inch respectively, hang vertically from two 143 points in the same horizontal, and distant 5 inches apart. To the lower ends of the wires is attached a light rod, which supports a weight of 100 pounds mid-way between the wires. Find the angle at which the rod will set itself to the horizontal, due to the stretching of the wires. [The value of E for brass is 5,500, and for iron, 12,500 tons per square inch.] [9'.] 3. Two vertical wires, each 20 feet long and 1 foot apart, support at their lower ends a cross piece on which rests a spirit level. The upper surface of the level tube is a part of a circular arc of 20 feet radius. When a weight of 40 pounds is hung on one of the wires the bubble of the level is displaced \ inch. How much has the wire extended ? If the diameter of the wire is i- inch, deduce the value of E. [025 inches ; 31,278,500 lbs./in 2 .] 4. The wire leading from the signal box to the most distant signal is 2,100 feet. It requires a pull of 300 pounds to work the signal, and, owing to resistances along the wire, the pull at the signal box must then amount to 500 pounds. The wire is !%- inch in diameter, and the signal end of the wire must move through a distance of 6 inches. Assuming that the resistance along the wire is uniformly distributed, find what movement must be given to the signal-box end of the wire. [E = 30,000,000 pounds per square inch.] [18 -6 inches.] 5. The two pieces A and B, shown in the Figure below, fit freely into the ends of a straight tube and are drawn together 144 by a bolt and nut. The section of the bolt is 1 square inch, the section of the tube is 1^ square inches, and bolt and tube are made of the same steel. If the ends A and B are pulled apart so that an extra pull of 2 tons is caused on the bolt, find how much the thrust in the tube has been decreased ; it may be taken that there is no change of shape in the pieces A and B. [4 tons, nearly.] 6. A foundation bolt with a square end is secured by means of a cotter as shown ; determine the dimensions marked " D," " 6," and " ," in terms of the diameter " d" in order that the shearing stress on the cotter may be three-fourths and the intensity of the pressure of the bolt on the cotter may be twice the tensional stress in the bolt. Deduce the limit of: deviation from the centre line of the bolt of the force transmitted so as not to cause reversal of stress in the cylindrical portion (d) of the bolt. 7. A bar of steel, 1 inch in diameter, has a gun-metal sleeve round it, of the same diameter internally, and \\ inches external diameter. If the bar and sleeve are firmly fixed together at each end and a load of 5 tons is suspended from one end, find the intensities of stress set up in the steel and gun-metal. \E for steel = 30 X 10 6 and for gun-metal, 10 X 10 6 pounds per square inch.] [4-5 and 1'5 ton/in 2 .] 8. A steel rod, 1 inch in diameter, is placed inside a copper tube, the internal and external diameters of which are 2 and 3 inches. The rod is screwed at the ends, and fitted with thick washers and nuts which are just screwed down on the ends of the tube at 60 F. Find the intensities of stress set up in the steel and copper when the rod and tube are heated to 160 F. 145 la \E for steel = 30 x 10 6 and for copper, 16 x 10 6 pounds per square inch. The coefficient of expansion for steel is 6'67 x l(r 6 and for copper 10 x 10~ 6 .] [7,260 and 1,450 Ib./inA] 9. The accompanying sketch shows a steel rod fitted with a collar and passing through a boss which projects from a fixed casting. It is secured by a nut, which is tightened up sufficiently to give an initial tension of 4 tons. A downward pull of 2 tons is then applied to the free end of the rod. Taking Young's modulus for the cast iron as 15,000,000 pounds per square inch, and for the steel as 30,000,000 pounds per square inch, find the resulting tension in the part of the rod between the collar and the nut. r [4-8 tons.] 10. A flanged pipe-joint is made with 6 steel bolts and elastic packing. The bolts are first screwed up so that the initial load on each is 2 tons. The compression in the packing is found to be twice as great as the extension in the bolts. When the water-pressure comes on the pipes the flanges are pulled apart with a force of 10 tons. Find the load on each belt. [3-11 tons.] 11. In the framework shown in the Fig. attached the bars AC, BC are freely hinged at C and hinged freely to the points A and B, which are to be regarded as absolutely fixed. Assuming that E for the material of the bars is 30,000,000 pounds per square inch, and that the bars remain perfectly 146 straight, calculate the approximate vertical and horizontal displacements of the point C. [1 inch; -019 inch.] 12. A crane frame is shown in the Fig. below, AD being a rigid wall and the joints being regarded as pin joints. 147 The lengths in feet of the members are AD, 10 ; AB, 7 ' 5 ; BC, 7 ' 5 ; CD, 7 ' 5 ; and AC, 3 ' 5. The cross-sectional areas in square inches are as follows : AB, ' 6 ; BC, 2 ' ; CD, 3 ' ; AC, 1'5. Find the vertical deflection of B when a load of 2 tons is applied there. [E - 30 X 10 6 lbs./in 2 .] [1 inch.] 13. The framework of a small wall-crane has the form and dimensions shown in the accompanying sketch. Find the stresses in each of the members. If the rods are designed for a stress of 6 tons per square inch, and the struts for a stress of 3 tons per square inch, calculate by the principle of work the depth through which the outer part of the crane is depressed by the load, assuming that the parts of attachment A and B are absolutely rigid. IE is 12,000 tons per square inch.] [-354 inch.] 14. A ram weighing 400 pounds falls from a height of 2 feet on to a crosshead which is supported by two steel rods, each 5 feet long and 2 inches in diameter. Assuming the energy of the blow to be all absorbed in the longitudinal extension of the rods, which have a modulus of elasticity 30 X 10 6 pounds per square inch, find the intensity of stress induced in the rods. u 5777. [39,100 lb./in 2 .] L 148 15. A crane rope, of sectional area 1 square inch, carries a load of 1 ton, which is being lowered at a uniform rate of 2 feet per second. When the length of the rope unwound is 40 feet the load is suddenly brought up. Find the intensity of stress induced in the rope if E = 30 X 10 6 pounds per square inch. Find also the extension. [6-7 ton/in 2 . ; -^ inch.] Section 11. THEORY OF BENDING. 1 . A band-saw is made of an endless strip of steel, \ inch wide and J-Q inch thick, passing over two pulley wheels each 12 inches in diameter. The tension in the straight part of the band is 100 pounds. Calculate the maximum tensile stress in the portion lying found the pulley, assuming E = 30,000,000 pounds per square inch. [60,000 pounds per square inch.] 2. A steel cable consists of 6 strands, with 7 wires in each strand (so that there are 42 wires), and the diameter of each wire is T V inch. It is wound round a drum 14 feet in diameter and carries a cage at its extremity. If the maximum intensity of stress has not to exceed 11^ tons per square inch and E for the wires is 13,000 tons per square inch, find the weight of the cage. [858 tons.] 3. A beam, 40 feet long, carries a load of 1 ton per foot run, uniformly distributed along its whole length. The beam is carried on two supports, each 8 feet from an end. Draw to scale the bending moment and shearing force diagrams. Find the maximum bending moment and the positions of the points where it is zero. Find also the position of the supports which will reduce the bending moment to a minimum. [40 ton ft. 11 ft. from ends ; 8-28 ft. from ends.] k- 2.0 ^ 2.0' ^k 2.0' - -^ I I ' A i Sootbiafrer ffrorrov? cit&tHbufed FromAVoB 6 "~ 4. The Fig. above represents a beam, strengthened by tie- bars and vertical struts. By tightening the tie-bars, and 149 consequently increasing the thrusts exerted by the struts, it is possible to arrange that the beam shall be free from bending moment at the sections over the struts. Draw carefully to scale the bending moment diagram for this state of affairs and scale off the greatest bending moment to which the beam is subjected. Calculate also the tension in the horizontal tie-bar. [25,000 lb. ft, ; 25,000 pounds.] 5. Case (1). Abeam, 20 feet long, rests upon two supports, A and B, as shown in the Figure below. A load of 20 tons is uniformly distributed along the beam. A terminal couple of Q B 25 foot-tons is applied at A, and a terminal couple of 75 foot- tons is applied at B. Determine the reactions at A and B, and draw the bending moment and shearing force diagrams for the beam. [7'5 and 12-5 ton ft.] Case (2). The same beam, cariying the same uniformly distributed load, is subjected to a couple of 50 foot-tons, applied at P in the manner shown in the Figure above. Determine the reactions at A and B, and draw the bending moment and shearing force diagrams for the beam. [12-5 and 7 '5 ton ft.] L 2 150 6. A bent rod ABCD is fixed horizontally at A in a vertical wall and loaded in the manner shown in the Figure below. The weight of the rod is 3 pounds per foot run. Draw to scale the bending moment diagrams for the length DC, GB and BA. 1 <_ 12 .' _ ^ f A B f C 1. \ , (^ 10 -> f 7. A beam of 60 feet span carries a load of 20 tons, uniformly distributed along its length. The beam is trussed in the manner shown, the central strut being screwed up until it exerts a thrust of 10 tons. Draw to scale the bending moment and shearing force diagram for the beam so loaded, and compute the thrust in the central strut which would reduce the bending moment in the beam to a minimum. 60'$pan 1 [11-7 tons.] 8. A cast-iron piston ring works in a cylinder 4 inches in diameter. The ring is inch deep and the pressure between the ring and the walls of the cylinder is 20 pounds per square inch. The ring is split by a single cut parallel to the axis of the cylinder. Show, by means of a diagram, the magnitude of the bending moment at all sections taken at different points round the ring, and find the maximum bending moment. [40 lb. in.] 151 9. In the roof frame shown in the accompanying Figure ABCD is a continuous beam. The vertical EB and GO are jointed to this continuous member at B and C. IO ^ All the other bars of the frame, AE, EF, FG, GD, are freely jointed at their extremities. Construct the bending moment and shearing force diagram of the beam ABCD. 10. Fig. 1 represents the front axle of a motor-car. *k 14- - >! V^L Fig. 1. - 152 Fig. 2 shows the section of the straight part of the axle. ^1) Find the moment of inertia of this cross-section about a horizontal axis tlirough e.g. of the section. (2) Draw the bending moment diagram for the straight portion of the axle. (3) Calculate the greatest stress set up in the material. fc- - Fig. 2. CM i [T^ i n - 4 '> 10*8 tons per square inch.] 11. An I beam, with flanges 6 inch by | inch and a web lOi- inches by f inch, is supported at both ends and has a span of 16 feet. The allowable stress on the metal is 12,000 pounds per square inch. Determine the load which may be applied at the centre, and also the load which may be applied when uniformly distributed along the beam. [14,396 pounds ; 1,800 pounds per ft. run.] 12. A circular steel tube, 1 feet in diameter and i-inch thick, is freely supported at the two ends. The span is 100 feet and the total distributed load is 30 tons. The tube is closed at its ends and used as a gasholder. Find the pressure of the gas so that the longitudinal stress is a tensile one at every point. [87 lb./in 2 .] 13. Find the maximum uniformly distributed load which can be supported by a beam of T section over a span of 12 feet, so that the maximum intensity of stress does not exceed 6,000 pounds per square inch. The width of the flange is 4 inches, the total depth 4 inches, and the thickness of the metal \ inch. [70 lb./ft.] 14. A spar, 10 feet long and of uniform diameter 6 inches, is planed down on the opposite sides so that it has two parallel plane faces, each 2 inches from the centre. It is placed with these faces horizontal, on two supports at its ends, and carries a load of 500 pounds at its middle. Neglect the weight of the spar and find the maximum intensity of stress due to bending. [1,098 lb./in 2 .] 153 15. The skin and plate deck of a ship have at the midship section the dimensions shown in the diagram. Find the position ,<- -50 -i S /Q PLATING. T^ i i , i o / r-o > > ' L X^ 1 \ 1 of the neutral axis and calculate the sagging bending moment which will set up a compressive stress of 4 tons per square inch in the deck plates. [Note. The distance of the e.g. of a semicircular arc from 2 the centre is - times the radius.] /i [13-45 feet below deck ; 51,000 ton-feet.] 16. The arrangement of a ship's davit is shown in the Fig. below. The vertical load carried by the davit is 2 tons. - SECTION AB- Calculate the greatest compression and tension stresses set up at the cross-section AB. [11 '41 and 11 '15 tons per square inch.] 154 17. In the hydraulic crane shown below, find the greatest intensity of compressive stress set up at the section AB. 30 B <-30%| |c2e-- j WMHUHUV [11-3 ton/in. 2 ] 18. The coupling rod of a locomotive is of uniform I section and has the following dimensions : Depth = 6 inches, Width = 3 inches, Thickness of web = li inches, Thickness of flanges = 1 inch, Length between centres = 7 ft. 6 ins. The driving wheels have a diameter of 7 feet, and the cranks operating the coupling rod have a throw of 14 inches. The weight of the rod is ' 3 pounds per cubic inch. Calculate the maximum longitudinal stress due to bending when the engine runs at 70 miles per hour. [3 -44 ton/in. 2 ] 19. A solid cylinder, 12 inches long and weighing 100 pounds, is supported by being laid across the middle points of three beams, placed side by side and having a common span of 10 feet. The centre beam is circular and 4 inches in diameter, and the two outer ones have a square section of 4 inch side. The material is the same for each beam. Find the load carried by each beam. [38-6, 22-8, 38 -6 pounds.] 155 20. A smooth plank, 12 inches wide and 2 inches deep, is laid across a 20-foot gap. Another smooth plank, 12 inches wide and 1 inch deep and of the same length, is placed on it, and a central load of 300 pounds is applied. Find the greatest fibre stress in each plank. [ 1,000 and 2,000 lb./in 2 .] 21. A spiral spring, made of rectangular section steel rod, is fixed at one end and connected at the other to a circular shaft coaxial with the spring and carrying a flywheel, the weight being taken by bearings. Rotation of the wheel pro- duces pure bending of the spring, the length of which is 20 feet, width of rectangular section 1 inch, depth radially f inch. The mass of the shaft and flywheel is 2 tons and radius of gyration, 15 inches. E = 30,000,000 pounds per square inch. Find the period of free oscillation. [6 '4 seconds.] Section 12. THEORY OF TORSION. 1. A steel rod, 1 inch in diameter, is subjected to an axial torque of 100 ft. Ib. and is found to twist through an angle of 27 ' 6 minutes, measured over a length of 8 inches. Find C. [5,430 pounds per square inch.] 2. In designing solid shafts of circular section to transmit power, a working rule which is often adopted is to limit the angular twist of the shaft to one degree for every twenty diameters of length of the shaft. If the value of C is 12,000,000 pounds per square inch, calculate the maximum shearing stress in the shaft consequent on adopting this working rule. [5,238 pounds per square inch.] 3. The shaft of a steam turbine is transmitting 10 H.P. at 30,000 revolutions per minute. If the shearing stress is not to exceed 6 tons per square inch, calculate the least possible diameter of the shaft. [2 inch.] 4. A steel shaft, 1 inch in diameter, is provided with enlarged portions, P and Q, 1 inches in diameter. The shaft is held twisted by an axial torque of 40 ft. Ib. While in this condition a steel tube, -sV-iflch thick, is shrunk on to the enlarged portion as shown in the Fig. above. When thfl tube 156 has firmly gripped the shaft the applied torque is removed. Calculate what twisting couple remains in the shaft, assuming that the shaft and tube are made of the same material. [23'91b. ft.] 5. The shafting of the turbines at Niagara Falls consists of a steel tube, 38 inches in diameter and |-inch thick. Find what H.P. can be transmitted at 250 revolutions per minute, when the working intensity of stress is limited to 9,000 pounds per square inch. Find also the diameter of a solid shaft which would be equivalent to the above. [57,224 ; 20 -14 inches.] 6. Power is transmitted at 150 revolutions per minute through a hollow shaft, whose external and internal diameters are 10 inches and 5 inches respectively. It is found that on a length of 100 feet the angle of twist is 9 degrees. Find the H.P. transmitted and the maximum intensity of shearing stress in the material of the shaft. [C = 12,000,000 pounds per square inch.] [3440 ; 7,850 lb./in 2 .] 7. A hollow steel propeller shaft is to transmit 10,000 H.P. at 120 revolutions per minute. The maximum twisting moment is 1 ' 2 times the mean, and the maximum intensity of shearing stress is 3 tons per square inch. If the inside diameter is ' 6 times the outside diameter, find the external diameter of the shaft. If the bolt circle is 25 inches in diameter, and the number of bolts is eight, find their diameter. [17 '64 inches ; 3 '5 inches.] 8. Design a spiral spring to give a deflection of ' 1 inch for a load of 1 pound, and to stand a safe deflection of 1 inch. The maximum intensity of stress allowed is 15,000 pounds per square inch and C for the material of the spring is 13,000,000 pounds per square inch. The radius of the coils is to be 10 times the radius of the wire. [Number of coils, 21 17 ; diameter, 1 '3 inches.] 9. A spiral spring made of wire of circular section is required to absorb 20 ft. Ib. of energy, the deflection being 5 inches and the intensity of stress not exceeding 12,000 pounds per square inch. Find a suitable diameter and length of wire, the mean diameter of the coils being 6' inches. C = 12,000,000 pounds per square inch. [496 inch ; 34 '47 feet.] 10. A spiral spring carries a weight of 25 pounds. The mean diameter of the coils is 2 inches, of the wire T V inch and there are 120 coils. If C = 10 7 pounds per square inch, find the number of vibrations per minute when the weight is set into a state of vibration. [47-6.] 157 11. A flywheel, whose moment of inertia is 2 ton ft 2 ., is mounted on the end of a shaft 10 feet long and 4 inches in diameter, the other end of the shaft being fixed. If C for the material of the shaft is 13,000,000 pounds per square inch, find the period of torsional oscillation of the flywheel. [156 sec.] 12. A gas-engine has a flywheel of moment of inertia 2 ton ft. 2 on one end of the crank shaft, and on the other an equal flywheel with the armature of a dynamo of moment of inertia '8 ton ft 2 , bolted to it, the length of the shaft between the flywheels being 6 feet and the diameter of the shaft 3 inches. If C for the shaft = 5,500 tons per square inch, find the engine speed at which dangerous torsional oscillations would occur. [356 rev./min.] Section 13. FLUID PRESSURE. 1. The Fig. below illustrates an automatic arrangement for maintaining a constant level of liquid in the vessels B and C. The float A, which is a semi-circular cylinder, turns freely about a horizontal axis passing through the centre of the semicircle. Prove that, in order to keep the liquid at the level of this axis, the specific gravity of the float must be exactly half the specific gravity of the liquid. 2. The water-line sectional area of a ship is 9,000 square feet, and her displacement is 4,000 tons. Find her alteration in draught consequent on going from fresh water weighing 62 pounds per cubic foot to salt water weighing 64 pounds per cubic foot. [4-48 inches.] 3. A log of wood, 20 feet long, having a square cross-section 12 inches X 12 inches, is floating freely in salt water. A load is applied at one extremity of the log so that it draws a depth 158 of 10 inches at one end and 5 inches at the other. Deduce from this the weight of the log and also the magnitude of the applied load. [711 and 89 pounds.] 4. Assuming for purposes of calculation that a submarine is a cylinder 6 feet long and 8 feet in diameter, calculate the reserve of buoyancy when she is floating half immersed, and when she is floating awash with only six inches appearing above the surface. Draw a curve showing the variation in buoyancy as she sinks from the former to the latter draught. [Note. The area of the segment of a circle which subtends an angle 6 at the centre of a circle of radius r is^r 2 (0 sin #).] [29-6 and '103 tons.] 5. A cube of wood having an edge of one foot floats half immersed in a cistern of water. The sectional area of this cistern is 200 square inches. Calculate the work which must be done to depress the cube until it is just completely immersed. A cubic foot of water weighs 62 pounds. [2 -2 ft. lb.] 6. A rectangular sluice-gate is 6 feet broad and 8 feet deep. The water-level on one side is 7 feet above the bottom edge, and on the other side the level is 3 feet above the bottom edge. Calculate the magnitude of the resultant horizontal thrust against the sluice. [7,500 lb.] 7. ABCD is a vertical section of a vertical dam, stayed against water-pressure by the rod BE, which is inclined at 45 degrees to the vertical and freely hinged at B and E. It is desired that the dam shall tip over automatically when the water under flood rises to A. At what point should B be placed ? What is the stress in BE just before the water reaches A if AD = 9 feet and the dam is 4 feet wide ? [3 feet above D ; 6* 4. tons.] 8. (a) Find the depth of the centre of pressure of a circular plate 1 foot in diameter immersed with its plane vertical and its centre at a depth 3 feet below the surface. (6) Find the whole pressure on and centre of pressure of a rectangular plate 6 feet by 4 feet whose two longer sides are respectively 2 feet and 3 feet below the surface of still sea- water. [(a) 3 feet 0-25 inches ; (ft) 3,840 pounds 2-13 feet from upper edge.] 9. The delivery end of a pipe is closed by a flat plate, and this plate is kept in position by a weight acting through a bent lever pivoted at A in the manner shown in Fig. below. Calcu- late the greatest difference in water-levels which can be maintained in this manner. [10-2 inches.] 10. A horizontal circular boiler, of internal diameter 6 feet, has plane vertical ends, above which is steam at atmospheric pressure. Find the point of action of the total thrust on the inside of one of its ends. [4 inch below centre.] 11. A bridge spans a stream in a single masonry arch, which is a semicircle of 15 feet radius. The breadth of the bridge measured in the direction of the stream is 20 feet. During a flood the water rises to a level of 2 feet above the crown of the arch. The arch itself remains watertight, and the 160 roadway of the bridge is not flooded. Show that there is a con- siderable force in existence tending to lift the bridge off its foundations, and calculate its magnitude in tons. [87 tons.] 12. A deep-sea cable, [for purposes of repair, has been fished up from a depth of 400 fathoms and fastened to a buoy. The buoy is spherical in form, 7 feet in diameter, built up of -f-inch steel plates. If by miscalculation this buoy is dragged to the bottom, calculate the stress in tons per square inch which the metal will have to stand in compression, provided that the buoy maintains its spherical form. [13-3 ton/in. 2 ] 13. In a hydraulic ram the water is prevented from escaping through the gland by means of a leather collar, which is forced by the water-pressure into close contact with the ram. If R is the radius of the ram and h is the height of the collar, show that the efficiency of the ram is .. _ 2ph R' \JL is the coefficient of friction between the leather and the ram.] 14. An Armstrong's cylinder, used for hoisting out boats, has a stroke of 6 feet ; the piston and piston rod diameters are respectively 6 and 3 inches. Find the tension of the chain when working at steady speed (1; with light loads, (2) with heavy loads. The water-pressure is 2,000 pounds per square inch. What is the horse-power of the arrangement if the rate of lift is 4 ft./sec. ? [21,200 pounds ; 28,290 pounds ; 206.] 15. Describe briefly the object and mode of action of an hydraulic accumulator. The stroke of an accumulator is 15 feet, the diameter of the ram is 15 inches and the working pressure when the ram is falling is 600 pounds per square inch. What is the amount of useful energy which can be stored in the accumulator expressed in foot-pounds and in horse-power hours ? [1,591,000; -8.] 16. The displacement of a ship at 25 feet draught is 10,000 tons and the tons per inch immersion curve is a parabola having its axis horizontal and at the 25-foot draught line. Find the tons per inch at 20 feet and 25 feet. [44 -7 and 50.] 161 Section 14. HEEL, CHANGE OF TRIM AND OSCILLATIONS OF SHIPS. 1. A collier, of displacement 6,000 tons, lias a metacentric height of 3 feet. 200 tons of coal on board has its centre of gravity shifted 20 feet transversely and 10 feet vertically upwards. What heel does the ship take owing to this ? [14 degrees.] 2. The metacentric height of a ship for alteration of trim is 400 feet, and the length of the ship is 450 feet. The dis- placement of the ship is 8,000 tons, and a mass of 100 tons is shifted from the bow to the stern through a distance of 300 feet. Calculate the alteration in the trim produced thereby. [4 -2 feet.] 3. To enter a certain dock it is required to decrease the draught of a ship 6 inches by the bow and 12 inches by the stern. To do this, cargo is removed at two points, distant 100 and 300 feet respectively from the bows of the ship. Calculate the number of tons which must be removed from the two points respectively. Length of the ship is 400 feet and displacement, 14,000 tons. Metacentric height for alteration of trim, 400 feet. The horizontal section of the ship at water-level, 30,000 square feet. The e.g. of this section is distant from the bow 200 feet. [286^ and 3561.] 4. The cross-section of a pontoon is shown in the Fig. below. The portion has plane vertical ends and is afloat in sea- water with a draught of 10 feet. Its length is 120 feet. Calculate its weight and the position of its centre of buoyancy. 162 Find also the height of its metacentre ahove the centre of buoyancy. 4-T [The e.g. of a semicircular area is distant -^ from the OTT centre, and a cubic foot of sea-water weighs 64 pounds.] [633 tons ; 4 ft. 8 3 in. deep ; 3 ft. 7 3 in.] 5. A single-screw vessel of 200 tons is being propelled by an engine of 250 horse-power at 72 revolutions per minute. If the metacentric height is 30 inches, find the angle of heel due to the turning of the screw. [56'.] 6. Prove that the centre of area of a uniform semicircle of 4r radius r is - - distant from its centre. 7T Deduce the metacentric height of a log of semicircular section, radius 6 inches, floating in water, and show that it is in stable equilibrium with its flat side horizontal and uppermost. [2 54 inches.] 7. The metacentric height of a battleship when at deep draught is 3 ' 7 feet, and when light it is 3 ' 3 feet. What effect has this alteration on the period of rolling of the ship ? [Increased in ratio 1'03 : 1.] 8. A vessel has 8 guns capable of firing on the broadside, the mean height being 30 feet above the centre of gravity. The weight of the projectile is 850 pounds, and of the charge 260 pounds. The muzzle velocity is 2,850 feet per second ; dis- placement of ship, 18,000 tons, and metacentric height, 5 feet. Find the angle of heel caused by simultaneously firing the 8 guns on the broadside, omitting any resistance to heel. The period of the ship is 18 seconds. [2 5'.] Section 15. RESISTANCE OF SHIPS. 1. Find the frictional resistance at 20 knots of a ship in which the under-water surface is 6,000 square feet. [5*8 tons.] 2. The under-water surface of a ship is 25,000 square feet and she is steaming 12 knots. If the E.H.P. for this speed is 1,200, how much do the residuary resistances amount to? [5 tons.] 163 3. A model for a ship, 400 feet long and of under-water surface 26,000 square feet, is constructed of length 10 feet and when towed at 1 ' 9 knots the residuary resistances amount to ' 9 pounds. Find the corresponding speed of the ship, and the E.H.P. to drive her at this speed. [12 knots ; 2,935.] 4. Two ships are built on similar lines, of 5,000 and 10,000 tons displacement respectively ; the former is 360 feet long on the water-line and her under-water surface is 25,000 square feet. Find the corresponding dimensions of the other ship and her skin friction at 10 knots. [454 ft. ; 39,685 ft. 2 . ; 10 "6 tons.] 5. A ship is to be built having a length of 390 feet, under- water surface 30,000 square feet, and maximum speed, 20 knots. A proportionately sized model is constructed on a scale of f inch to the foot, and when towed at the corresponding speed the total resistance of the model is 4 ' 6 pounds. The coefficient of friction for the model is '01131 and for the ship '00886. The index of velocity is 1 ' 825. The mechanical efficiency of the engines will be about 85 per cent, and the efficiency of the propellers 65 per cent. What should be the indicated horse- power of the engines ? [11,430.] 6. A war- vessel has a bunker capacity of 1,080 tons of coal. When steaming at 15 knots, she burns 170 tons per day. Find approximately the quantity of coal burnt per day at 10 knots ; and, if the maximum speed of the ship is 19 knots, find the distance she can steam at full speed, starting with bunkers full. [50 tons ; 1,425 miles.] U 5777. 164 PART III NAVIGATION AND NAUTICAL ASTRONOMY. Section 1. Time problems. 2. Sunrise, Sunset, &c. 3. Great circle sailing. Composite sailing. 4. Small errors. , , 5. Deviation analysed. 6. Tide questions. 7. Course and distance : Instruments. ,, 8. Current questions. 9. Scouting problems, &c. 10. Observations to determine deviation. ,, 11. -Position lines by observations of sun, stars, &c. 12. Error and rate of chronometer. 13. Day's work and observations, for use with Charts A., B., C., D. Section I. TIME PROBLEMS. 1. Why is the R.A.M.S. the same as the Sidereal Time at G.M. Noon? 2. Distinguish between a Mean Solar and a Sidereal year, and find the number of days (mean solar) in the latter, given that the former consists of 365 ' 2422 days. [365-2563.] 3. Explain how the table for converting an interval of Mean Solar Time into Sidereal Time is constructed, and calculate the equivalent in Sidereal Time of 7 h Mean Solar Time. 4. Determine the Sidereal Time at Washington (77 3' W.) on June 3rd. 1912, at 10 P.M. Local Apparent Time. [14 h 46 m 45 s ] 5. Find the sidereal time of sunrise at Berehaven, lat. 51 40' N., long. 9 46' W., on April 20th, 1912. [18 h ol m 40'.] 165 6. Given R.A.M.S. at G.M. Noon to be ll h 5 m 49'2 s , find S.M.T. of passage of the First Point of Aries over the meridian of a place in long. 115 40' E. [12 h 53 m 19 -7 s .] 7. In lat. 35 S. on a certain day two stars, whose declina- tions are and 16 S. respectively, rise at the same instant, Calculate the sidereal interval between the instants at which they set. [l u 32 m 40 s .] 8. Given that the R.A.M.S. is 2 !l 36 m 39 s and the Sun's Apparent Long, is 40 50' 15", determine the Equation of Time. [3 ul + to M.T.] 9. If the " Sidereal Time " given in the Nautical Almanac is 14 h 46 m 2 s , what is the " Transit of the First Point of Aries " given for the same day. [9 h 12 m 27 s .] 10. If the mean time of transit of the First Point of Aries at Greenwich is 5 h l m 32' 06 s , find the R.A.M.S. at the follow- ing mean noon at a place in long. 22 35' E. [19 U l m 20- 12 s .] 11. The mean time of Transit of the First Point of Aries given in the Nautical Almanac for a certain day is 8 h 33 m 8 ' 13 8 . Find the Equation of Time at G.M. Noon on the same day, having given the Sun's Apparent Right Ascension 15 h 9 m 42 -68 s . [15 m 44-9 s to A.T.] 12. The declinations of two stars, X and F, are 16 19' N. and 28 15' N. respectively, and the angular distance between them is 45 1'. If a chronometer, with no daily rate, shows 3 h 12 m 10 s when X, the more westerly star, is on the meridian, what will it show at the time of meridian passage of Y ? [6 h 20 m 38 s .] 13. If the sidereal time at Greenwich is 8 h 15 m 12 s , the S.M.T., 6 h 10 m 3O, and the R.A.M.S. at the preceding G.M. Noon was 18 h 56 m 40 ' 8 s , find the longitude. [106 27' 30" W.] 14. The Sun's Apparent Right Ascension at Mean Noon was 18 h 4 m 25 ' 89 9 and at Mean Noon on the following day it was 18 h 8 m 52 '48 s . Determine the variation in the equation of time in one hour. [1 '25 seconds.] 15. The mean time of the transit of the First Point of Aries at Greenwich being 20 h 54 m 56 '94 s . find the R.A.M.S. at O h 15 m (S.M.T.) at a place in long. 25 oo' W. [3 h a m 2-i-3 s .] M 2 166 16. Prove that at mean noon R.A.M.S. = Sidereal Time. On a certain day the Nautical Almanac gives Sidereal Time as 12 h 39 m 15 '69 s . What is the mean time of the transit of the first point of Aries on the same day ? [ll h 18 U1 52 -79 s .] 17. What do you understand by the Equation of Time ? Explain the causes of it and draw a diagram showing its value throughout the year. If a watch keeping accurate mean time showed 2 h ll m 10 s when the Sun was on a certain meridian on January 1st, what time did it show when the Sun was on the same meridian on January 10th? [2 b 15 m 12 s .] 18. Prove the formula R.A.M.S. + S.M.T. == R.A. of x + W.H.A. of x, x being any heavenly body, and 24 hours being added to either side when necessary. [You need only consider one case.] What will be the time shown by a deck watch (3 m 20 s fast on G.M.T.) when the star Pollux is on the meridian of 60 W. on April 1st, 1912 ? [Il u 3 m 34 s .] 19. At what local mean time will Capella be on the Prime Vertical west of the meridian in 56 0' N., 108 15' E. on the night of April 7th, 1912 ? What are the Meridian altitudes of Capella above and below Pole in the same latitude ? [7 h 12 m 6 s P.M. ; 79 54-9'; 11 54-9'.] 20. On a certain date the Sun's declination was 15 20' 15" S., the obliquity of the ecliptic 23 27', and R.A.M.S., 21 h 8 m 49 s . Find the Equation of Time, and explain clearly whether it is -f or to Mean Time. [14 ni 18- 3 s to M.T.] 21. Prove that Star's W.H.A. = S.M.T. + R.A.M.S - Star's R.A., and determine as accurately as possible the G.M.T. of passage of Aldebaran over the meridian of 105 W. on December 29th, 1912. [16 U 57 m 52 s .] 22. Prove that S.M.T. - Star's W.H.A. + Star's R.A. -- R.A.M.S. At what time by a deck watch, fast l h 10 m on G.M.T., did Aldebaran pass the meridian of 150 W. on January 15th, 1912 ? [8 h 3 m 27 s .] On June 20th, at 9 h 15 m S.M.T., the hour angle of an unknown bright star was found, by means of the observed altitude and azimuth, to be approximately 5 h . By using the above formula determine the approximate R.A and thence the star. \_RegulusJ] 167 23. State the reasons why Apparent Time cannot be kept accurately by a clock, and explain now the difficulty is got over in practice. At G.M. Noon a deck watch showed l h 12 m 37 s and a sidereal clock 5 h 5 m 43 s . If the sidereal clock was l h 2 m 48 s fast, and neither the clock nor the watch were gaining or losing, determine the day in 1912 when this happened, and the times shown by the clock and the watch at 9 h 41 m 50 s G.A.T. on the same day. [May 23rd ; H h 45 m 43 s ; 10 h 51 m 2 s .] 24. Prove. S.M.T. - Star's W.H.A. + Star's R.A. -- R.A.M.S. S. Sid. T. = S.M.T. + R.A.M.S. Hence show how to find the time of a star's Meridian Passage. Find the mean time of the Meridian Passage of Aldebaranat Greenwich on September 1st, 1912, and its altitude when on the Meridian. [5 h 50 m 47 s A.M. ; 54 51 -6'.] 25. Draw a diagram on the plane of the horizon, radius 2' 5 inches, representing the celestial concave at 8 h P.M. on May 25th in latitude 30 S., showing the Equinoctial, Ecliptic, Canopus, Procyon, Regulus, a Crucis, Arcturus, and the Moon. 26. Draw a diagram on the plane of the horizon, radius 2 ' 5 inches, representing the celestial concave at midnight on December 27th in latitude 40 S., showing the Equinoctial, Ecliptic, and the six stars given on pages 190 and 191 of the Nautical Almanac. Section 2. SUNRISE, SUNSET, &c. 1. If the obliquity of the ecliptic be 23 27', what is the length of the longest day in lat. 60 N. ? [18 U 29 m 38 s .] 2. What time will the Sun set in lat. 61 S. if his declina- tion is 20 N. ? [3 h 15 m 50 s P.M.] 3. Find the S.A.T. of sunrise and sunset and the length of the day at a place in lat. 25 10' N., when the Sun's declina- tion is 10 15' N. [5 U 40" 1 30 s A.M. ; 6 1 ' 19 m 30 s P.M. ; 12" 39 m .] 168 4. Find the mean times of sunrise and sunset in lat. 52 10' N., when the Sun's declination is 14 20' N., the equation of time being 10 m + to A.T. [4 h 53 m 10 s A.M. ; 7 b 26"' 50 s P.M.] 5. Find the duration of the longest day in lat. 45 N. [15 h 25 m 40 s .] 6. Find the declination of the Sun when in lat. 52 N. he rises at 4 h 3 m A.M. S.A.T. [20 53' 45'' N.] 7. In lat, 43 N., when the Sun's declination is 12 N., find the hour angle when rising. [17 h 14 m 16 8 .] 8. If on the longest day in the year the Sun rises at a certain place at 4.30 A.M., at what time will he rise on the shortest day, and what will be his bearing at rising? [7 h 30 m A.M. ; S. 58 E.] 9. At Greenwich, lat. 51 28' N., the Sun sets at 7 h 15 m P.M. apparent time on a certain day. Find the local apparent time of sunset at Edinburgh, lat. 55 55' N., on the same day, neglecting the change of declination. [7 h 28 m 55 8 .] 10. Determine the time of sunrise and duration of morning twilight (not using the special Table in Inman) at Cherbourg, lat. 49 39' N., on March 2nd, using the declination for Greenwich Apparent Noon of that day. [6 h 34 m 20 s A.M. ; l h 51 m 51 s .] 11. On what days of 1912 does the Sun set at Plymouth most nearly to 5 h P.M., local apparent time, and how long does twilight last on either of those days ? [N.B. Use declination for G.M. Noon, without correction.] [February 18tb and October 25th ; l h 54 m 39 s on October 25tb.] 12. In 36 N. 3 E. the Sun rose at 4 h 47 m by D.W., its compass bearing being N. 81 E. ; and set at 6 h 37 m by D.W., its compass bearing being N. 53 W. The D.W. was 2 m 13 s slow on G.M.T. Find the Equation of Time, Variation of the Compass, and Declination of the Sun. [3 m 47 s + to M.T. ; 14 W. ; 18 7' N.] 13. What is the obliquity of the ecliptic? From the information given for Thursday, May 16th, on p. 51 of the Nautical Almanac for 1912, find the obliquity of the ecliptic. [23 27 -2'.] 169 14. Find the time of the beginning of morning and the end of evening twilight at a place in lat. 54 36' N., the Sun's declination being 8 30' N. (supposing the declination constant, and twilight beginning and ending when the Sun is 18 below the horizon). [2 h 45 m 44 s A.M. : 9 h 14 m 16 s P.M.] ^^J---"To. Required the duration of morning twilight in lat. 49 N., the Sun's declination being 17 27' N. [2 h 30 m 24 s .] 16. Find the duration of twilight in lat. 36 N. on the shortest day, and also the lowest latitude in which twilight will last all night when the declination is 14 S. [l h 36 m 42 s ; 86 N. or 58 S.] 17. At what time will the Sun be on the twilight circle before rising, in lat. 50 N., when his declination is 0? [4 h 5 m 4 s A.M.] 18. What are the causes of twilight? How long does twilight last -at a place on the Equator when the Sun's declina- tion is 12 30' N. ? [l h 13 m 49 s .] 19. At a place in North latitude, on the longest day, the Sun rose at 3 h 55 m A.M. Find latitude of place and duration of twilight. [50 5' N. ; all night.] 20. Find the duration of morning twilight in lat. 44 N., the Sun's declination being 20 50' N. What is the lowest latitude in which twilight will last all night, with the given declination ? [2 h 16 m 58 s ; 51 10' N.] 21. At a certain place, when the declination of the Sun was 22 30' N., the Sun set at 6 h 45 m 30 s apparent time. At what time did twilight end on the same evening? [8 h 18 m 22 s P.M.] 22. Find the duration of evening twilight at a place in 42 N. when the Sun's declination is 12 51' N. [l h 49 m 21 s .] 23. If the mean times of sunrise and sunset be 5 h 56 m and 6 h 12 m , find the equation of time, showing clearly whether it is + or - to A.T. [4 minutes + to A.T.] 24. On a certain day the mean times of sunrise and sunset were 8 h 3 m A.M. and 4 h 14 m P.M. What was the equation of time ? Was it additive to mean time or subtractive from it ? 8n so** _ to M>T> 170 25. At a place in long. 45 W. the Sun rose at 6 h 40 m 9 8 by chronometer and set at 7 h 44 m 9 B . Determine the equation of time and whether it is + or to M.T., the chronometer being l h 42 m 40 s slow on G.M.T. [5 m 11' + ioM.T.] 26. At a certain place the Sun rose bearing by compass E. 16 N., and set bearing W. 34 S., the deviation being 5 30' E. Find the variation. [19 30' E.] 27. In lat. 27 S. the Sun rose by compass N. 84 E. and set by compass N. 58 W. The variation was 10 W. Determine the deviation, and the Sun's declination. [3 W. ; 16 51' 45" N.] 28. The Sun rose at 5 h 10 m S.M.T., bearing by compass S. 79 E., and set at 6 h 55 m P.M., bearing by compass S. 55 W. Find the compass error and the latitude of the place. [12 E. ; 33 19' S.] 29. In lat. 55 N., long. 40 W., the sun rose at 9 h 12 m 52" by Deck Watch, bearing S. 82 30' E. by compass, and set at 12 h 54 m 2 s by Deck Watch. The error of the watch was 2 h 17 m 19 s fast on G.M.T. Find the Sun's declination, the equation of time, the compass bearing of the Sun at setting, and the compass error. [18 N. ; 6 m 8 s + to A.T. ; N. 17 18' W. ; ~ 40 6' W.] 30. Find the length of the longest day at Oban (lat. 56 25' N.). At Oban (lat. 56 25' N., long. 5 31' W.) the Sun rose on a certain day at 12 h 3 m 5 s by Deck Watch, bearing S. 27 30' E. by compass, and set at 7 h 3 m 15 s by Deck Watch. The error of the watch was 3 h 21 m 16 s fast on G.M.T. Find the Sun's declination, the equation of time, the compass error, and the compass bearing of the Sun at setting. [I7 h 26 m 30 s ; 22 S. ; 10 m 10 s to A.T. ; 19 52' W. ; S. 67 14' W.] 31. Calculate to the nearest minute the S.M.T. of moonrise at the following places on the given dates. Would the Moon be visible at these calculated times ? Give reasons : (a) Lat. 50 21' N., long. 3 35' W. on February 4th, 1912. Lat. 35 50' N., long. 14 50' E. on June 3rd, 1912. (c) Lat. 12 S., long. 178 W. on December 27th, 1912. (d) Lat. 43 30' S., long. 176 E. on July 3rd, 1912. [(a) 7 h 2 m P.M. ; (b) 10 h 32 m P.M. ; (c) 10 h 15 m P.M. ; (d) 7 h 48 m P.M.] 171 32. Calculate to the nearest minute the interval between sunset and moonrise at the following places on the given dates : Lat. 50 51' N., long. 5 E. on January 6th, 1912. (6) Lat. 55 50' N., long. 1 W. on September 28th, 1912. (c) Lat. 33 56' S., long. 18 30' E. on March 5th, 1912. Lat. 30 S., long. 75 30' W. on August 4th, 1912. [(a) 2 h 29 m ; (6) 25 m ; (c) l h 23 m ; 33. Find, approximately, the time elapsing between the end of twilight and moonrise on March 6th, 1912, in 50 20' N., 4 9' W. [The amplitude tables may be used.] [2 h 20 m .] Section 3. GREAT CIRCLE SAILING. COMPOSITE SAILING. 1. Find the initial course and the distance on the arc of the great circle from lat. 32 N., long. 160 W. to lat. 46 N., long. 141 W. [N. 40 49' E.; 1215'.] 2. Find the distance saved by sailing on the great circle instead of on the rhumb line from Easter I. (27 10' S., 109 26' W.) to Otago harbour (45 47' S., 170 45' E.). [130'.] 3. Find the distance saved by steaming on the arc of the great circle, instead of on the rhumb line, from lat. 51 20' N., long. 9 40' W., off the Fastnet, to lat. 46 40' N., long. 53 W. off Cape Race. [26'.] 4. Find the distance by great circle, and the highest latitude reached, in ? proceeding from lat. 35 24' N., long. 139 50' E. to lat. 48 25' N., long. 124 W. [4077-8'; 54 31' 45" N.] 5. Determine the distance on a great circle from 41 20' S., 174 50' E. to 33 1' S., 71 40' W. Find also the latitude of the vertex. [5025 ; 54 29' S.] 172 6. Find the difference in distance in steaming from Bermuda (32 30' N., 64 40' W.) to Madeira (32 30' N., 17 0' W.) on the parallel and great circle tracks ; and determine the initial course and highest latitude reached on the great circle. [21'; N. 7639'E. ; 34 51V N.] 7. Find the initial and final courses and the distance in steaming on a great circle course from Port Otago, New Zealand (45 47' S., 170 45' E.) to Callao (12 4' S., 77 14' W.). [S. 65 45' E. ; N. 40 33' E. ; 5765'.] 8. A ship steams on a great circle from lat. 31 20' N., long. 62 30' W. ; her initial course is N. 54 E. ; find her latitude and longitude and her course when she has steamed 200 miles. [33 15' N. ; 59 16' W. ; N. 55 44' W.] 9. How long will it take a ship to steam at 14 ' 5 knots on the parallel from lat. 11 22' N., long.. 60 27' 30" W. to lat. 11 22' N., long. 19 30' W. ? How much time would she save by employing the great circle track ? [166*16 hours ; about 9 minutes.] 10. What is the distance and the highest latitude reached on the great circle track joining CaUao (12 4' S., 77 8' W.) and Yokohama (35 26' N., 139 39' E.) [8364 -5 miles; 42 50|' N.] 11. Determine the initial course, distance, and the latitude of the vertex in steaming on a great circle from Port Otago (45 47' S., 170 45' E.) to Callao (12 4' S., 77 14' W.) [S. 65| E. ; 5765 miles ; 50 32' S.] 12. Find the initial course and distance on the great circle track joining Honolulu (21 18' N., 157 52' W.) and Hong Kong (22 16' N., 114 10' E.) [N. 69| W., 4819 miles.] 13. At noon on April 15th a ship was in lat. 38 23' S., long. 177 55' W. and wished to reach Otago (45 47' S., 170 45' E.) by about 6 P.M. on April 16th. Find her least average speed. [21-8 knots.] 14. Determine the initial course and distance on a great circle from Otago (45 47' [S., 170 50' E.) to Coquimbo (29 56' S., 71 24' W.) and also the course and distance on the rhumb line. [S. 50 E., 5138' ; N. 80 E., 5619'.] If two ships left Otago for Coquimbo on January 10th at local mean noon and steamed, one on the great circle and the other on the rhumb track, each averaging 12 knots on the passage, what were the dates and local times of their arrivals ? [4 P.M. Jan. 27th ; 8 A.M. Jan. 29th.] 173 15. Find the distance on the composite track from Durban (29 52' S., 31 4' E.) to Port Philip (38 18' S., 144 37' E.), the maximum latitude reached being 42 S. [5381'.] 16. A ship steams on a composite track from lat. 38 N., long. 145 E. to lat. 44 N., long. 125 30' W., the highest latitude reached being 48 N. ; find her initial course, and the distance on the parallel. [N. 58 T E. ; 586'.] ^ 17. A ship is to steam on a composite track from lat. 35 30' S., long. 57 3' W., off Piedras Point, to Port Adelaide, lat. 34 46' S., long. 138 30' E. Her initial course is S. 67 E. Find the maximum latitude and the distance on the parallel. [41 27' 45" S. ; 5446'] 18. Find the initial course, and the distance, on the composite track from lat. 37 50' N., long. 122 30' W., off San Francisco to C. Lopatka, lat. 50 50' N., long. 156 40' E., the maximum latitude reached being 50 50' N. [N. 53 W. ; 3402'.] X19. A ship steams on a composite track from lat. 49 50' N., )ng. 6 25' W. to St. John's, Newfoundland, lat. 47 34' N., long. 52 40' W., the maximum latitude being 50 30' N. Find her initial course and the distance on the parallel. [N. 80 27' W. ; 314'.] 20. Find the initial course and the distance on the parallel in sailing on a composite track from lat. 29 55' S., long. 31 10' E., off Durban, to lat. 35 12' S., long. 117 40' E., off West Cape Howe, the maximum latitude reached being 40 S. [S. 62 E. ; 322'.] "21. Find the distance by composite track from lat. 51 20' S., long. 76 30' W. to lat. 41 22' S., long. 175 30' E, the highest latitude reached being 53 S. [4219'.] 22. A ship sails from Wellington, New Zealand, for Cape Horn (56 S., 68 W.) on a composite track, not going further south than lat. 60 S. Taking 41 40' S., 175 E. as her departure point, determine the initial course and the distance she will run on the parallel. [S. 42 1'E. ; 803-5.] 23. A ship steams at 12 ' 5 knots on a composite course from lat. 43 44' S., long. 146 E., to lat. 48 38' S., long. 69 12' E., the maximum latitude being 52 S. Find her first course, and the number of hours she will be steaming due west. [S. 58 26' W. ; 22-7 hours.] 174 24. Determine the distance from 34 50' S., 20 0' E., off Cape Agulhas, to Adelaide (35 0' S., 135 0' E.) on a composite track, not going further south than the parallel of 45. [5313 '5 miles.] Section 4. SMALL ERRORS. 1. In practice we consider a position line found by deck watch time and altitude of a heavenly "body to be independent of any error in the assumed position. Give reasons for this and illustrate with a diagram. In what circumstances will this assumption be unjustifiable ? 2. Define " Geographical Position of a Heavenly Body." Find the Geographical Position of the*Sun when its centre is in the horizon of an observer at Greenwich (51 28' 38" N.) on the evening of September 29th, 1912. [2 27 -6' S., 86 54' 15" W.] 3. What is the " geographical " position of a heavenly body ? Find the geographical position of Jupiter on April 6th, 1912, at 9 P.M. M.T.P. in longitude 120 E. [21 50-6' S., 135 14' 45" W.] 4. On May 1st, 1912, an observer has Antares in his zenith. What will he find to be the true altitude and true bearing of a 2 Centauri ? [50 56 -4'; S. 21JW.] 5. From a ship a lighthouse in lat, 51 N. bears N. 26 E. ; the ship then runsN. 58 E., 10 miles, when the lighthouse bears N. 47 W. The bearings and course are by compass, the error being 6 E. In plotting, the error is applied W. instead of E. Find the errors in latitude and longitude of the position obtained at the second bearing. ['8'; 1-3'.] 6. The run (N. 40 E. true, 52 5') between the D.R. positions of a ship at 8 A.M. and 11.30 A.M. was correctly worked by Traverse Table, and the result obtained from observations of the Sun at those times were 4 '4' nearer, Sun's T.B N. 74 E., and 2 '6' nearer, Sun's T.B. N. 14 E., respectively. In plotting, the course was laid off N. 50 E. true by mistake and the resulting position at 11.30 A.M. was lat. 34 50' S., long. 113 46' E. Find the correct position, and the D.R. position at 11.30 A.M. [34 43-5' S., 11338-1'E. ; 34 45-4' S., 113 33' E.] 7. Two observations of the Sun gave intercepts of 1*7' away, Sun's T.B. S. 75 E., and 2' nearer, Sun's T.B. S. 12 E. ; and both were plotted from the second D.R. position, giving a result lat. 35 20' N.. long. 17 30' E. The first sight was 175 carelessly plotted with Sun's T.B. S. 75 W. Find the correct position at the time of the second sight. [35 19' N., 17 24- 4' E.] 8. Two nearly simultaneous observations of stars gave intercepts 2 '5' nearer, Star's T.B. S. 60 E., and 2' away, Star's T.B. S. 20 \V., giving a result lat. 42 45' N., long. 176 10' E. The second intercept had been plotted nearer, instead of away. Find the correct position. [42 48- 5' N., 176 12-8'E.] 9. Nearly simultaneous observations of Rigel and Regulus gave respectively intercepts of 3 '5' away, Star's T.B. S. 15 W., and 2 '5' nearer, Star's T.B. S. 59 E., with a resulting position lat. 48 10' N., long. 20 15' W. It was afterwards discovered that the T.B. of Regulus had been taken from the Azimuth Tables. " DECLINATION contrary name to LATITUDE." Obtain the correct bearing of Regulus, the D.R. position, and the correct observed position. [S. 75^ E. ; 48 7-7' N., 20 21 -4' W. ; 4810'3'N. ; 20 16-6' W.] 10. Two stars were observed nearly simultaneously and the observations worked out by using the height of eye (19 feet) for the upper deck, instead of the upper bridge (38 feet). The true bearings of the stars were N. 84 E. and S. 40 E. If the position thus obtained was lat. 35 30' N., long. 26 15' W., find the errors in latitude and longitude due to the mistake. [45" S., 2' 20" E.] 11. A careless observer, in lat. 42 30' N., took the Sun's altitude at about 8.45 A.M. with the sextant inclined at an angle of 4 to the vertical. After applying the usual corrections the true altitude used was 21 23' and the Sun's declination was 10 15' S. What error was caused in the calculated longitude by the faulty observation, and in which direction ? [4' 45" too far E.] 12. On a certain evening in D.R., lat. 40 3' N., long. 5 16' E. nearly simultaneous observations of stars were made by an experienced observer, which, when correctly worked, gave the following results : (1) 4'!' nearer, Star's T.B. S. 86i E. (2) 3-9' North. (3) 4-6' S.82W. (4) 3-6' S. 16FW. (5) 4' _ S. 5i E. Suggest an explanation (other than instrumental errors), and obtain the probable position of the ship. What instrumental errors, and what error in observing, would produce results similar in direction ? [40 3' N., 515-7'E.] 176 13. The Sun's declination being 14 10' 30" N., an observa- tion, taken in the artificial horizon in lat. 22 17' N., long. 114 9' 15" E., gave the S.A.T. to be 8 h 15 m 36 s A.M., and the error of the chronometer, 5 L 40 m 11 '4 s slow on mean time at place. The index error (!' 30" ) was applied the wrong way. Find the correct error of the chronometer on M.T.P. and on G.M.T. [5 h 40 m 5 s slow ; l h 56 m 32 s fast.] 14. Two stars were observed on bearings N. 80 E. and S. 15 E., and gave intercepts 2' away, and 1*5' nearer, respectively. The position obtained was lat. 35 40' N., long. 15 49' E. It was found afterwards that the D.W. error used in each case had been O h O m 8 s fast, instead of O h O m 8 s slow. Find the correct position. [35 40' N., 15 45' E.] 15. The position lat. 47 25' N., long. 17 28' W. was obtained from two observations of the Sun on bearings S. 75 E. and S. 10 E., and the intercepts were 4' away, and 2' away respectively. In working the first observation the D.W. error had been read O h 5 m 34 s fast, instead of O h 5 m 24 8 fast. Find the correct position at the time of the second observation. [47 24- T N., 17 30- 7' W.] 16. The position obtained at 11.50 A.M. by observations of the Sun was lat. 25 19' S., long. 65 16' E. The intercepts were 3' nearer, Sun's T.B. S. 86 E., and 1' away, Sun's T.B. N. 15 E. The latitude not agreeing with that by meridian altitude it was suspected that the D.W. had been read (at the second sight) 8 h 4 m 58 s , instead of 8 h 3 m 58 8 . Assuming this mistake to have been made, find the correct position at 11.50 A.M. [25 15-5' S., 65 16-2' E.] 17. In measuring a distance on a chart in latitude 53 N. an error of 17 ' 5' was made through using the scale of longitude, instead of that of latitude. Required the correct distance. [26-5 miles.] 18. The obs. alts, of Sun were read off 10' too great at two distinct observations. Find the error in magnitude and direction in position of ship. The intercepts were both zero and the bearings S. 86 E. and S. 42 E. [10 '9 miles S. 65 E. from true position.] 19. Find an expression for the error in calculated Z.D. resulting from a small error in the Hour Angle. Is the resulting error of greater or less magnitude than the original error ? In latitude 40 N. an error of 15 s is made in the H.A. ; the Sun's true bearing is S. 60 E. Find the error in Z.D. by the Traverse Table or otherwise. [2 -5 miles.] 177 20. The position of a ship determined by the observation of two stars (bearing N. 40 E. and S. 60 E.) was 50 K, 15 W. But it was found that the correction for dip, 5' 30", had in each case been added. Find the latitude and longitude of the ship. [49 58- 5' N., 15 21' W.] Section 5. DEVIATION ANALYSED. 1. An iron bar A, B, is held horizontally in the magnetic meridian (A towards the North) and tapped with a mallet. Describe the behaviour of a compass needle when A is brought near its N. end. 2. A pole of a weak magnet is gradually brought near the N. end of a powerful compass needle. At first it repels and then attracts the compass needle. Explain this. What difference would there be in the behaviour of the compass if a piece of soft iron were gradually brought near ? 3. A 12-inch gun fires several rounds when pointed ahead, and is forward of the compass in the fore and aft central line of the ship. If the ship steams in the direction of the magnetic meridian the whole time, what effects, if any, upon the compass would you expect as the gun is afterwards rotated through 180 so as to point aft ? 4. A ship is built in England, head N. Sketch roughly the deviation you would expect to find on various compass courses. If when the course is E. the deviation is 30 W., what deviation would you expect to find on course S.W. ? [21 -2 E.] 5. A ship is built in England, head S.E. On what courses would you expect to find no deviation due to permanent magnetism ? [S.E. and N.W.] 6. The upper end of a funnel (no permanent magnetism) is abaft the standard compass and nearly on a level with it. Sketch roughly the deviations you would expect to be caused in England. 7. If in Question 6 the deviation on course N.E. is 6|-, find the magnitude and direction of the deviation on N.W. 178 8. The deck below a standard compass is supported by two fore and aft mild-steel girders, one on each side. Sketch generally the nature of the deviation produced in England. 9. If the girders in Question 8 are athwartship, give the general nature of the deviations. 10. If the deviations are small, the deviation on any compass course is given by S = A + B sin + G cos + D sin 2 + E cos 2 . The following deviations were observed : N. - 10 S. + 8 N.E. 6 S.W. + 9 E. - 3 W. + 5 S.E. + 2 N.W. - 5 Find the coefficients A, B, C, D, and E, and determine the deviation on N. 60 E. [0, 3-4, 8-4, + 1-5, 1; 5 20' W.] 11. Which of the above coefficients would you expect to be reduced after correction by (1) horizontal permanent magnets ? (2) soft iron spheres ? What error is corrected by permanent magnets placed vertically below the centre of the compass ? 12. A compass needle is deflected 10 degrees by a magnet placed end on, due E. of the centre of the needle. The compass needle is then surrounded by a hollow cylinder of iron, the magnet remaining outside. What is the deflection ? [10 degrees.] 13. A single wire conveying a current of electricity from starboard to port runs horizontally below the centre of the compass needle. Sketch roughly the deviations produced on various courses. 14. Of what nature is the magnetic field produced by a current in a straight conductor ? An electric-light lead, conveying a current from starboard to port, passes above a compass needle : what will be the direction of the error produced when the ship's head is E. ? [Westerly.] 15. A current of electricity flows up a vertical conductor. How will a compass needle placed, first, due North, secondly, due East, of the wire be affected ? 16. An electric current is flowing along a wire. Given a small compass needle, how will you find the direction of the current-flow if the wire be (a) horizontal, (6) vertical, (c) coiled in a horizontal circular hank ? 170 17. Give examples of magnetic screening. Will the compass error produced by a ship's magnetism be affected by surrounding the needle with an iron bowl ? Will a copper conductor conveying a given current have the same effect on the compass if the conductor is placed in an iron tube ? Give reasons for your answers. 18. Define : " Induction," " Lines of Force," "Line of Total Force." What part of the ship's magnetism do the spheres correct ? 19. If the period of a card was 22 at Glasgow (hor. force ' 9), what would you expect it to be at the Equator in long. 5 E. (hor. force 1 ' 6) ? How do you find a period of a card ? Why is it different at two places which differ in magnetic latitude ? [16 '5 sees.] 20. How do you stow compass cards ? How do you know in the dark if the Thomson's cards are stowed correctly ? 21. You take the bearing of an object by the Pelorus and find it is N. 60 W. The ship's head was N. 60 E. (magn.) as set on the Pelorus. The helmsman was 5 to Starb. of his course. What was the correct bearing ? [N. 55 W.] 22. What are the chief causes of Heeling Error ? Describe the observations made with the H.E.I, on board, supposing the number of scale divisions on shore was 29 and the ship's multiplier at that compass position was '8. 23. In a ship built head N.E. in England, the maximum effect due to this was 10, also the maximum effect due to vert. soft iron is 10. What will be the deviation on N.W. at Bombay due to these causes ? England Hor. Force, 1 0. Vert. Force, 2 33. Bombay 2'0. I'OO. 24. Analyse the following deviation table, and state the correctors yovi would use to correct the coefficients : - - 8 E. - 4 E. - - 8 W. - 15 W. -35, 7-35, -f 4, 0.] 25. The Sun's true bearing was N. 79 E. and comp. bearing N. 81 Q E. The variation was 6 W. What is the Deviation? [4 E.} u 5777. N. - 8 W. S. N.E. - - 4 E. s.w. E. - 8 E. W. S.E. - - 7 E. N.W. [0, + 7 26. A. ship is properly and accurately corrected for semi- circular deviation in England. Would you expect any alteration on change of magnetic lat. ? Explain your answer. 27. What is the effect of sub-permanent magnetism if not allowed for ? 28. Why is it that, in a modern ship which does not usually have a steady heel, it is necessary to correct the heeling error ? 29. Explain what you understand by induction in soft iron. Show by diagrams the effect of induction in soft iron spheres (a) at a place on the magn. equator ; (6) at a place in N. Hemi- sphere where dip is 45. 30. Describe, with aid of a figure, the construction of a Liquid Compass. What are the different sizes of Liquid Com- passes supplied ? 31. What do you understand by the " Period " of a Card ? Why are Kelvin's cards made with a period of 30 sees, at Glasgow ? What would be the period of a card at Malta ? [Hor. force at Glasgow, ' 9 ; hor. force at Malta, 1 ' 5.] [23 -24 sees.] 32. What precautions are necessary as regards the stowage of compass gear ? 33. Why is it so important to keep the correcting magnets as low down as possible in the binnacle ? 34. What would a real coefficient A and E be caused by ? 35. Describe how you would correct your Heeling Error. 36. What is semicircular deviation ? what is it caused by ? and how does it vary ? 37. How would your deviation change on change of magnetic latitude, if due to (a) hard iron ? (6) vert, soft iron ? (c) hor. soft iron ? 38. If the combined effects of B and C cause a maximum deviation on N.N.E. of 10 E. in a ship built in the N. Hemi- sphere, what was the direction of ship's head when building ? [E.S.E.] 39. Define : " Natural magnet," " Artificial magnet." Explain the process of making an artificial magnet by the " divided touch " method. 181 40. State exactly how a freely suspended magnetised needle would lie in a place where H = 1 ' 5, Z = 1 ' 75, and find the value of T. 41. Give a short description of the deviation caused by induction in vertical soft iron, with the ship upright, stating on what points it is a maximum, how it varies, &c. 42. You leave England with a + B. of 3. When swinging on the Magnetic Equator you find B. = 0. Is any correction desirable ? Give reasons. 43. Explain what is meant by Sub-Permanent Magnetism. Can you estimate in any way what deviation it will cause in either sign or amount, and when would you particularly be on the look-out for it ? 44. What are the causes of Quadrantal Deviation, and why is it always positive ? You have a + D. of 5 in England, what deviation will it cause on N.N.E. at Bombay ? [3 5 E. nearly.] 45. Given B = + 4, C = - 3, D = + 2, find the deviation on S.E. Would you expect the deviation on N.W. to be the same in amount ? [2-9E.] 46. What are the principal causes of heeling error and how does the deviation caused by each vary on change of geographical position ? 47. Define: " The magnetic poles," " Poles of a magnet," " Line of total force." At Portsmouth, if an iron rod be held vertically and hammered, do you expect it to become magnetised? Give reasons. 48. By sketches, show the distribution of permanent magnetism (a) By a stern view, ship built head East. (6) By an upper deck view, ship built head North. Both ships were built at Cape Town. Dip, 60. 49. Why is a 6" compass used in the following positions : (a) Conning tower ? (&) "Lower steering position ? 50. A ship has been steaming for some days on a Westerly course in the Southern Hemisphere. On altering course to North, would you expect any deviation ? Give an illustration, and explain your answer. 51. What points must be considered in selecting the position for a Standard Compass ? N 2 182 52. At a steering compass, the following deviations were observed : N. 5E. E. 10 E. S. 4E. W, Nil. N.E. 8 E. S.E. 8 E. S.W. 2 E. N.W. 2 E. Find the approximate coefficients. How would you allow for or correct each ? [+4-87, +4-6, + -25, 0, -25.] 53. Explain why the spheres help to correct that portion of the " Heeling Error" which is caused by vertical induction in horizontal soft iron. 54. If the period of an 8" compass card is 26 sees, at Green- wich, what will it be at Hongkong, where H. is 2 ' 0, and at South Georgia (55 S.), where H. is T4 ? [18-38 sees. ; 21-97 sees.] 55. How should compass cards be stowed ? Give the two most likely causes which would result in the needles losing their magnetism when stowed away. 56. In recently built battleships, the compass in the lower control position is near to, and immediately under, the bottom of the armoured tube. What correctors do you have to place on this account, say, at Portsmouth, and why ? 57. Describe, with sketches, what magnetism a horizontal soft iron bar lying in the direction of the magnetic meridian will have (a) N. Magnetic Pole ; (6) Equator ; (c) 50 S. lat, 58. What is the use of the Pelorus, and what are the essential conditions to be observed when using it ? 59. You wish to place the ship's head N.E. magnetic. Describe how you will do so by means of the azimuth mirror at a Standard Compass, having found that the true bearing of Sun was S. 70 E. Var., 18 W. 60. A ship was built S.E. in England, and her maximum deviation due to permanent magnetism was 10 in England. What deviation will she have due to permanent magnetism on N.E. at Gibraltar ? [7 35' E.] 61. What are the causes of heeling error? How are they corrected ? If closely corrected in England, would you expect it to change on change of geographical position ? Give reasons. A ship being swung for deviation with a constant heel, the heeling error being uncorrected, what effect will it have on her deviation table ? 183 62. State the order of placing correctors at a Compass. Give reasons. 63. What is Quadrantal Deviation caused by ? What is the usual sign, and why ? If you had a quadrantal deviation of 1 + in England, would it change on change of geographical position ? Give reasons. 64. Give all the methods of finding the magnetic bearing of a distant object you wish to swing by. State what distance it should be away from you to give a satisfactory result. 65. Define : " Magnetic Meridian," " Magnetic Equator," "Line of Total Force," " Period of a Card." 66. Show by sketches the distribution of magnetism in (a) A ship built head East at Portsmouth. (6) A ship built head West at Cape Town. 67. If the period of a compass card at Glasgow is 30s, where H. is ' 9, what will be the period of the same card at Singapore where H. is 2 ' 1 ; and at a place where the dip is 60 and Z. = - 1-75? [19-64 sees.; 28 '31 sees.] 68. The following observations were made when swinging ship by distant object : - Ship's Head. C.B. of Object. Ship's Head. C.B. of Object. N. N. 75 GO' W. S. N. 75 00' W. N.E. N. 67 20 W. S.W. N. 72 10 W. E. N. 69 10 W. W. N. 61 10 W. S.E. N. 67 20 W. N.W. N. 67 30 W. Find the deviation on the 8 points. How would each be corrected ? 69. State the order for applying the correctors to a compass, and give fully the reasons for adopting this order. What rules must be . adhered to when applying correctors ? 70. What would be the effect on a compass under the following conditions : (a) If heading N. for a length of time, then suddenly altering co. to E. ? (fe) If heading E. for a length of time, then suddenly altering co. to S. ? What steps would be taken in each case ? 184 71. It has been found necessary to steer by the compass in the after lower steering position. After close correction on the cardinal points, and the application of 12" spheres to this compass, there still remains + 4 of quadrantal deviation. The ship's head being N.E. by compass, you alter course to S.E. by compass (passing through East). Through what arc of the horizon will the ship's head have passed ? [82.] 72. A compass was placed in a binnacle on shore where the Earth's H.F. was 1 ' 0. A horizontal magnet, fixed 20 feet below the compass, caused 3 degrees of deflection of the card. A pair of soft iron spheres, secured on the brackets, caused 6 degrees of deflection. Without moving either magnet or spheres, the compass and binnacle were carried on board ship, where the H.F. was only 0'3. How much deflection of the card would the magnet now cause, and how much the spheres? [10, 6.] 73. Define : " Natural magnets," " Artificial magnets." Explain how the latter are made. You wish to magnetise a bar AB by the method known as divided touch, so as to make A a red pole. How will you proceed ? 74. What are the principles of the mechanical correction of a compass ? In what order are the correctors applied, and why? 75. What do you understand by quadrantal error, what causes it, and how is it corrected ? 76. What are the principal causes of heeling error ? How does the effect of each change on change of geographical position ? What noticeable effect would there be at a compass where the H.E. had not been corrected ? 77. Having found "B" at Spithead = -14 and "D" = + 4, the ship then goes to Capetown, the correctors not being moved, and "B" is found to be -4. How will "B" be corrected ? X = ' 9. 78. Given : A = 0, B= - 4, C = + 8, D= + 4, what will be your deviation on N.N.E. ? [8-7E.] 79. How would you find out if there was any permanent magnetism in your Flinder's bar ? If you had any, how would you remove it ? 80. Describe how you would find the Variation at sea. 81. How would you proceed to test an azimuth mirror? 185 82. Make a table of deviations from the following observa- tions, and find the approximate coefficients : Ship's Hd. . True Bg. Sun. Comp. Bg. Sun. N. N. 82 30'-W. N. 81 00' W. N.E. 82 50 81 15 E. 83 30 81 15 S.E.- 80 00 77 15 S. 80 30 77 30 S.W. 81 00 78 00 W. 81 20 78 45 N.W. 82 05 79 45 [0,+ 16', +42', + 7V, + 5'.] 83. Make a deviation table from the following observations. Analyse the table, and state how you would correct your compass : Ship's Head. C.B. of Sun. T.B. of Sun. N. S. 71 50' E. S. 82 10' E. N.E. S. 78 00 E. S. 80 00 E. E. S. 66 40 E. S. 78 40 E. S.E. S. 59 30. E. S. 77 30 E. S. S. 54 30 E. S. 76 10 E. S.W. S. 52 40 E. . S. 74 40 E. W. S. 53 20 E. S. 73 20 E. N.W. S. 49 00 E. S. 71 00 E. [0, + 6-3, + 5-65, + 4, 0.] 84. Form a deviation table from the following observations, analyse it, and state what correctors are necessary. A pair of 1" spheres being already on, set out 2". Ship's Head. T.B. of Sun. C.B. of Sun. N. S. 71 17' E. S. 46 30' E. N.E. S. 72 47 E. S. 54 00 E. E. S. 73 47 E. S. 63 00 E. S.E. S. 67 17 E. S. 60 30 E. S. " S. 68 17 E. S. 57 30 E. S.W. S. 68 47 E. S. 50 00 E. W. S. 69 47 E. S. 45 00 E. N.W. S. 70 17 E. S. 43 30 E. Var. 18 17' W. State, approximately, in what direction ship was built. [+ , + 7, 7.-l,0 ; head S.W.] 186 85. Make a table of deviations from the following observa- tions with the Standard Compass, and find the approximate coefficients : Ship's Head. Comp. Brg. Sun. True Brg. Sun. S. 89 E. N. 85 30' E. N. 85 E. N. 87 30' E. N. 89 E. S. 87 30' E. S. 81 E. S. 85 30' E. N. 78 E. N. 79 E. N. 80 E. N. 81 E. N. 82 E. N. 83 E. N. 84 E. N. 77 E. [0, + 4-9, -2-9, + 2, 0.] 86. What is the object in analysing a deviation table, and in what order should the correctors be placed ? Give reasons. 87. Give a short description of the principal differences between the old Standard compass (Thomson's 10") and the new standard now adopted for use in H.M. Navy (the Chetwynd- Clark). 88. By means of the following Deviation Table calculate the values of the coefficients A, B, C, D, and E : Direction of Ship's Head. Deviation. N. N.E. E. S.E. 2 45' E. 10 0' E. 8 50' E. 1 50' E. Direction of Ship's Head Deviation. S. s.w. w. N.W. 3 O'W. 7 0' W. 8 50' W. 4 50' W. [0, + 8'5, + 3-2, + 1-5, 0.] 89. The following deviations were taken of a compass with spheres in place. Construct a curve. Horizontal scale, \" = 1. Vertical Scale, 1" = 4 points. N. 2 OE. N.E. SOW. E. 2 W. S.E. 1 35 W. S. 2 W. S.W. 30 W. w. 2 OE. N.W. 2 40 E. 90. (a) Deduce the approximate coefficients from the deviation table in preceding question. (6) State briefly how you would correct the deviation due to each coefficient. [0,- 1-75, + 1-75, --5, 0.] 187 91. (a) What are the causes of heeling error? (6) What type of deviation does it cause and on what points has it a maximum and minimum effect ? 92. What is the arrangement in a Lord Kelvin compass to obviate the effect of vibration ? Section 6. TIDE QUESTIONS. 1. Low water at 8 A.M. - - sounding, 6 fathoms. High water at 2 P.M. - 8 ,, Springs rise 20 feet. Reduce these to L.W.O.S., and compute the reduction to be applied to a sounding taken at 9A.M. [2^ feet, 17^ feet, 3^ feet are the reductions.] 2. A tide-pole registered 4 feet at 10 A.M. (L.W.) and 20 feet at 4 P.M. (H.W.). Springs rise 22 feet. Find the correc- tions to be applied to soundings of 4| fathoms and 6 fathoms taken at 11.30 A.M. and 2 P.M. respectively. [ 5 ft. 4 ius. ; 15 feet.] 3. A tide-pole shows 4 feet at 9 A.M. (L.W.) and 20 feet at 3.12 P.M. (H.W.). Springs rise 26 feet. Find the reading of the tide-pole corresponding to L.W.O.S., and the reduction to be applied to a sounding of 3 fathoms taken at 11 A.M. [ _ 1 foot ; 8 ft. 10 ins.] 4. A tide-pole at L.W. (7 A.M.) shows 6 feet and 20 feet at H.W. (1 P.M.). Springs rise 20 feet. How will the following soundings appear on the chart : 8 A.M. - 6 fathoms. 10.30A.M. - - - 4 Noon - - - - 1 [5 fms. ; 2 fms. ; dries 7 feet.] 5. At ship it was L.W. at 8 A.M., with a depth alongside of 10 fathoms ; and H.W. at 2 P.M., with a depth of 12 fathoms. Springs rise 16 feet. Find the reduction to be applied to soundings taken in the neighbourhood at 9 A.M. and Noon. [2ft. 10 ins. ; 11 feet,] 6. At St. Helier the half mean spring range is 15 ft. 9 ins., and a certain tide rises 25 ft. 6 ins. Find the height of the tide li hours before H.W. [22 ft. 8 ins.] 188- 7. At 10 A.M. (H.W.) a tide-pole showed 26 feet, and' at 4 P.M. (L.W.) 10 feet. Springs rise 22 feet. Construct a table showing the reduction to be applied to soundings for each half-hour from 10 A.M. to 4 P.M. 8. The depth of water on the bar at the entrance to a harbour at low water springs is 10 ft. 6 ins. The spring range is 22 feet, and a certain tide rises 17 ft. 6 ins. If the time of high water is 10 h 51 m A.M., find the depth of water on the bar at 9 A.M. [25 ft. 3 ins.] 9. At Douglas, springs rise 23 feet above the level to which Soundings are reduced on the Chart. Spring range 21 feet. What water will there be over a 3-fathom patch at L.W.O.S. ? [20 feet.] 10. With a regular tide rising 18 feet, how much (roughly) will it rise during the 2nd hour of the flood ? [3 feet.] 11. Construct a diagram for a tide that rises 4|- hours, whose \ Mean Spring Range is 8 ft. 6 ins., and from it find the depth of water above L.W.O.S. at \\ hours from Higli Water on a day when the tide rises 13 ft. 6 ins. [11 ft. 10 ins.] 12. Explain the terms: "Vulgar Establishment of the Port," " Priming and Lagging of the Tides," " Semi-mensual Inequality." What is the usual approximate relation between the heights of spring and neap tides at a given place ? 13. Draw a diagram, on the scale of 1 inch = 5 feet, to show the depth of water at any time for a six-hour tide whose range is 22 feet ; spring rise, 28 feet ; depth of Avater at L.W.S., 5 feet. Show that for a six-hour tide the correction to be applied to mean tide level is \ range of tide X cosine (twice time interval from H.W.). 14. H. W.F. and C. at Auckland, New Zealand (long. 175 E.) is 7 h 32 m . Determine approximately the time of H.W. on the evening of December 12th, 1912. [9 h 50 m P.M.] 15. Define the terms : " Lunitidal Interval," " H.W.F. and C." The least water marked on the chart in a certain channel is 1| fathoms. It is high water at 2 h 16 m P.M., and the height of the tide by the tables is 13 feet, and half spring range 7 ft. 9 ins. Find the least depth of water at 5 P.M. [16| feet.] 16. At 6 A.M. the tide-pole at L.W. showed 10 feet, and at noon, at H.W., 20 feet. Springs rise 19 feet. Calculate the reductions to be made to soundings at each intermediate half- hour for insertion on a chart. 189 17. Define the terms : " Age," " Rise and Range of a tide," " L.W.O.S." ; and explain briefly, with the aid of diagrams, the cause of Neap and Spring tides. 18. The depth of water over a submerged rock is shown on the chart as f of a fathom. It is high water at 8 h A.M., the height of the tide by the tables is 34 ft. 8 in., and half mean spring range, 15 ft. 9 ins. Find the depth of water over the rock at 9 h 30 m A.M. [33^ feet.] 19. At 9 h A.M. the tide-pole at L.W. showed 12 feet, and at 3 h P.M. at H.W. 26 feet. Springs rise 19 feet. Calculate, graphically or otherwise, the reduction to be made to soundings at each intermediate half-hour for insertion on a chart. Section 7. COURSE AND DISTANCE : INSTRUMENTS. 1. Ship left 49 30' N., 8 10' W., and steamed as follows :- Compass course. Deviation. Variation. Distance. S. 27 W. 2E. 19 W. 29' S. 030' E. 19 W. 31' S.E. 3 W. 18 W. 42' E. 4W. 18 W. 112' Find her position "in." [48 57' N., 4 26' W.] 2. Find the D.R. position of the ship at noon from the following : Time. Course. Knots. Deviation. Remarks. ll h P.M. S. 87 W. 13-5 1 40' W. Var., 17 W. Lundy Isle Light (51 10' N., 4 40' W.) was dead astern, distant 14'. 5 h 45 m A.M. S. 34 W. 15 9 W. Var., 18 20' W. 8 h 45 m A.M. N. 80 W. 15 10' E. 18 40' W. 10 h 15 m A.M. S. 12 E. 16 7 50' W. 18 40' W. Estimated a 3-knot current setting N. 58 E. (magn.) between 9 A.M. and noon. [49 28^ ^. 7 20V W.] 3. Work up the D.R. position at 9.30 A.M. from the following notes : Variation, 25 W. 6.0 P.M. Achill Head [53 58' N., 10 15' W.] bore by com- pass N. 22| E., distant 8 miles ; Deviation, 3 W. Set course by compass N. 28 W., 15j knots ; Deviation, 3 W. 8.0 P.M. Altered course East, 15 knots ; Deviation, 5 E. 1.0 A.M. Altered course N. 8 W., 14 knots ; Deviation, 3iW. 5.20 A.M. Altered course S. 67 E., 14 knots ; Deviation, 5 E. Allow for a current setting E.S.E. (Magnetic) at 1^ knots after 5.30 A.M. [55 18' N., 8 9' W.] 4. On November 15th, at 5 p.M.,Longships Light (50 4' N., 5 45' W.) bore by compass S. 30 E., distant 6 miles. Ship's Head, S. 40 W. by compass. Nov. 15th : 5 P.M. Set C* S. 40 W. ; Variation, 18 W. ; Deviation, 2 W. ; Speed, 13 knots. 7 P.M. Altered C* N. 89 W. ; Variation, 18 W. ; Deviation, 3 W. ; Speed, 13 knots. 11 P.M. Altered C 11 S. 3 E. ; Variation, 18 W. ; Deviation, 1 W. ; Speed, 12 knots. Nov. 16th : 3 A.M. Altered C^ S. 73 E. ; Variation, 17f W. ; Deviation, H E. ; Speed, 13 knots. 6A.M. Altered O N. 81 E. ; Variation, 17 W. ; Deviation, 2 E. ; Speed, 14 knots. [From 11 P.M. to 3 A.M. there was a 2 knot current to the N.N.E. (Magnetic)]. Work up the D.R. Position at 9 A.M. on November 16th. [49 8^' N., 4 50' W.] 5. At noon a ship was steaming S. 53 E., 20 knots, and a point of land (8 31' N., 76 57' E.) was on the port beam, distant 10'. She then steamed as follows : Time. Compass Course. Deviation. Variation. Speed. Rema] Noon Midnight. S. 53 E. N. 81 E. 5 W. 1 W. 2 E. 1 W. 20 knots. 20 knots. Alter course. Altered 6A.M. N. 1E. 1W. 21 knots. <> c speed. Find by traverse table the D.R. position at noon, allowing for a current setting East (true) 1 knot from noon till 6 A.M. [8 41^' N., 82 37f E.] 191 6. Work up D.R. position at 5.30 P.M. from the following notes : 8.0 A.M. Set C Q by compass S. 60 W., 10 knots ; Deviation, 3 50' E. ; Variation, 5 15' W. 8.30 A.M. Casuarina Point Light, 33 19' S., 115 39' E. bore by compass S. 30 E., distant 8'. 10.0 A.M. a/c N.W., 12 knots ; Deviation, 1 40' E. ; Var- iation, 5 10' W. Noon. a/c W., 12 knots ; Deviation, 3 E. ; Variation, 5 10' W. 1.30 P.M. a/c S., 12 knots ; Deviation, 30' E. ; Variation, 5 5' W. Allow for a current 2% knots S.E. (Magnetic) the whole time. [347'S., 115 1^'E.] L Extracts from Log. Variation, 17 W. [Start Point is in 50 13' N, 3 38' W.] 7.0 P.M. Set C^ W., 12 knots ; Deviation, 3 E. 9.0 P.M. Sighted Start Pt. Light 3 points on Starboard Bow, distant 20'. Midnight. Altered G 3 - W.S.W., 15 knots ; Deviation, 2 E. 4.30 A.M. Reduced speed to 12 knots. 6.0 A.M. Altered C^ South, 15 knots ; Deviation, 1 E. 8.30 A.M. Took Sights. Estimate a current setting E. (Magnetic) at 2 knots from 6 P.M. to Midnight, and at 1 knot from Midnight to 10 A.M. Work up D.R, for Sights. [48 331' y M 5 10' w.] Find the Course and Distance in the following cases : 8. From the Fastnet (51 23' N., 9 36' W.) to Cape Race (46 39' N., 53 4' W.). [S. 801 W. ; 1,732'] 9. From Bishop Rock (49 53' N., 6 25' W.) to Barbados (13 10' N., 59 25' W.). [S. 50 W. ; 3,428'] 10. From the Eddystone (50 10' N., 4 16' W.) to Madeira (32 43' N., 16 39' W.). [S. 27| W. ; 1,183'] 11. From Malta (35 54' N., 14 32' E.) to Alexandria (31 12' N., 29 51' E.). [S. 69| E. ; 816'] 12. From Ascension I. (7 55 S., 14 25' W.) to Cape Town (33 56' S., 18 28' E.). [S. 49^ E. ; 2,399'] 192 13. From C. Guardafui (11 50' N., 51 16' E.) to Colombo (6 56' N., 79 51' E.). [S. 80 E. ; 1,716'] 14. From Honolulu (21 18' N., 157 39' W.) to San Francisco (37 48' N., 122 25' W.). [N. 61 E. ; 2,079'] 15. From Callao (12 4' S, 77 9' W.) to Coquimbo (29 55' S., 71 21' W.). [S. 16|E. ; 1,119'] 16. From Acapulco (16 50' N., 99 57' W.) to Payta (5 5' S., 81 6' W.). [S. 40E. ; 1,726'] 1 7. The divisions on the scale of a barometer represent ' 05 of an inch. Twenty-five divisions on the vernier are equal to twenty-four on the scale. What is the accuracy of reading ? [002 inch.] 18. The arc of a sextant is divided to 10', and the length of the vernier, which is divided into 40 equal parts, is equal to 79 arc divisions. Find the accuracy of reading. [15"] 19. If the perpendicular distance from the centre of the index glass of a sextant to the line of collimation is 1 ' 5 inches, what is the shortest permissible distance of an object (observed for index error) so that the index error may be correct within 10"? [860 yards, nearly.] 20. Nearly simultaneous meridian altitudes of Markdb (dec. 14 41' 36" N.) and Dubhe (dec. 62 15' 40" N.) were observed in the same artificial horizon, and were 77 7' 50" and 76 56' 0" respectively, Dubhe being below the N. pole. Find the latitude and the error of the sextant. [66 10' N.; 2' 13"] 21. The line of collimation of a sextant is inclined at an angle a' to the plane of the instrument. Prove that when the sextant reads x the error due to collimation is given in minutes by a 2 sin 1' tan x. Is this additive or subtractive to the sextant reading? If a =T 10" and x = 120, find the error of collimation. [1-55"] 22. If with the above notation a = 40' and x = 126 30', find the true angle. [126 29' 5"] 23. If the angle read off is 104 15' 20" and the true angle is 104 12' 52" find the inclination of the line of collimation to the plane of the sextant. 193 24. Two stars, observed when in the same vertical circle, had altitudes 61 43' 20" and 48 50' 0", and the observed distance between them was 69 35' 10". Height of eye, 20 feet. Index error + 1' 20". Find the centring error. [-37"] Section 8. CURRENT QUESTIONS. 1. A headland bears S.E. (true) from a ship, distant 20 miles. On what compass course and at what speed must a ship steam, through a current setting East (true) 2 ' 5 knots, so as to pass 10 miles to the West (true) of the headland after an interval of two hours ? Variation, 20 W. Deviation, 4 E. [S. 19 W. ; 7-1 knots.] 2. Find the compass course to be steered by a vessel which can steam 12 knots, in order to reach a port bearing N. 79 E. (true) from her, distant 106 miles, and the time in which she will reach it. The compass error is 19 E., and a current is running N. 27 E. (true), 2 miles per hour. [N. 67^E. ; 8-1 hours.] 3. A ship steaming 16 knots through a 3 knot current running S. by E. is making for a port 64 miles S.W. of her. What course must she steer, and how long will it take her to reach the port ? [S. 55i W. ; 3 -6 hours.] 4. What course should a cruiser, steaming at 12 knots, steer in order to reach a port 38' N.W. by W. of her in a current setting N.E. at a rate of 2 knots ? [N. 68 W.] 5. A port bore from a ship S. 34 E., distant 88 miles. After steering that course for 7 hours at 11 knots, the port bore S. 10 E. and was 22 miles distant. Find the set and drift of the current. [N. 10E. ; 1-8 knots.] 6. A cruiser steams S. 70 W. at 12 knots for 3 hours through a 2 knot current setting S.W. Find approximately how long it would take her to get back to her starting point, supposing the speed of ship and set and rate of current remain constant. Find also the course she should steer. [4-4 hours; N. 62 E.] 7. A ship is 12 '5 miles distant from a port which bears from her N. 38 E. (true), and is steaming directly towards it at the rate of 10 knots. After 30 minutes a lightship over a sandbank, 4 miles N. 79 W. (true) from the port, is observed 194 to bear N. 16 E. magnetic, and to be 5 miles distant from the ship. Find the set and drift of the current experienced ; also the ship's magnetic course to the port. Variation, 16 W. [N. 6 E. ; 3-6 knots ; N. 85 E.] 8. A ship runs a measured mile (6,080 feet) with the tide in 2 m 51 ' 5 8 , and against the tide in 3 m 45 8 . Find the speed of the ship and the rate of the tide. knots ; 1\ knots.] 9. A ship ran a measured mile in 3 m 50 s with the tide, and in 4 m 35* against the tide. Find speed of ship and rate of tide. [14-4 knots ; 1 '3 knots.] 10. Find the latitude and longitude arrived at after steaming N.W. for 24 hours at 11 knots from 41 7' N., 61 12' W., through a current setting E.N.E. 2 knots. [4437'N. ; 6411'W.] 11. From a ship steaming N. by E. at 12 '2 knots a light- ship bore N.N.E., distant 6 miles. Half an hour later it bore 'E. by S., distant 2 '25 miles. Find the set and drift of the tide, and the course and distance made good. [W., 2-2 knots; N. 1 W., 5-88 miles.] 12. Two ships, A and B, in line abreast, 5 cables apart, are steering north at 13 knots. A headland bears simultaneously N. 35 E. from A and N. 24 E. from 5. Fifteen minutes later it bears S. 61 E. from A, and S. 52 E. from B. Find -by protraction the set and drift of the current. [8.47 W. ; 1-7 knot.] 13. From a beacon on shore a lighthouse bears S. 85 E., 4 miles. At ll h A.M. the beacon bore from the ship S. 60 W. and the lighthouse S. 25 E. After steaming S. 26 W. at 12 knots until ll h 30 m the beacon bore N. 23 W. and the lighthouse N. 35 E. Determine the set and rate of the tide. All bearings and the course are true. [S. 85 E. ; 2-5 knots.] 14. On June 1st, 1912, a ship was in lat. 5 N., and there was a current setting E.N.E., but of unknown strength. On what bearing would you take the Sun in the afternoon so that the fix obtained from a later observation would be prac- tically unaffected by the current ? Illustrate your reasons by a sketch. [N.N.W.] .5. A 12-knot Picket Boat is ordered to proceed to a pier 5 miles from the ship, and bearing N. 60 E. (true). A 4-knot current is setting S.S.E. What courses must the boat steer to reach the pier and then to return to the ship ? [N. 41 E. ; S. 80 W.] 195 16. From a ship at anchor a pier bears E. by N., 3 miles. A 4-knot tide is running S.E. What course must a 12-knot pinnace steer in order to go from the ship to the pier, and how long will the journey take ? What course must she steer in order to return to the ship from the pier ? [N. 63 E., 13 minutes ; N. 8o^ W.] 17. A ship, steering N. 73 E. (true) at 20 knots, observes Wolf Rock (49 57' N., 5 48' W.) due North (true). Half-an- hour later the Light bore N. 52 W. (true). The tide sets East (true) 4 knots. Find the latitude and longitude of the ship at the tin*e the second bearing was taken. [4948'N. ; 5 29 -9' W.] 18. A ship steamed from 50 10' N., 4 30' W. on the following courses : S. 55 W., 35 miles ; N. 66 W., 29 miles ; N. 15 E., 49 miles, and was then found by observation to be in 50 55' N., 5 49' W. Determine the current experienced. [N. 52 W. ; 9-8 miles.] Section 9. SCOUTING PROBLEMS, &c. 1. Find the course to steer and time taken to get into station : Flag Course and Speed. First Station. Second Station. Own S }eed. 1. North 7 k be. Stb. Beam 8' Astern - 3' 12-5 kt s. 2. S. 10 E. 10 Astern - 1' Stb. Bow - 6' 18 3. N.W. 8 Pt. Bow - 2' Pt. Bow - 10' 16 4. S. 40 W. 9 Ahead - 2' Pt. Beam - 5' 9 5. West 11 Pt. Beam - 10' Pt. Bow - 3' 20 6. East 6 S.S.W. - 60' Join Flag. 17 ^7. South 7* Pt. Qtr. - 8' Astern - 2' 16 8. N. 30 E. 11 Stb. Beam - 3' Stb. Beam - 10' 14 9. N. 20W.10 West - - 42' Join Flag. 18 10. S.E. 8 S. 10 W. - 56' Within - 2' 16 11. N.E. 12 Pt. Beam - !' Pt. Bow - %' 15 [1. N. 79 W., 39 min. 2. S. 9 W., 44 miu. 3. N. 70 W., 51 miu. 4. S. 4 E., 47 min. - 5. N. 47i W., 33 ' 7 min. 6. N. 41* E., 4 h 14. 7. S. 33 W., 36 min. 8. N. 67i E., 50 min. 9. N. 58 s E., 2 h 15 m . 10. N. 33 E., 2 h 49 m . 11. N. 68 E., 11 miu.] 2. A cruiser A sights a merchant ship B bearing N. 10 E., 10' distant. Suppose B is steaming N. 85 E. at 9 knots and u 5777. O 196 that A can steam 16 knots : what course should A steer in order to come up with B as quickly as possible, and how long will A take to overhaul B ? [N. 43 E. ; 55 miu.] 3. A squadron is steaming N. 58 E. at 10 knots ; a cruiser, which bears from the flagship S. 32 E. 5', is ordered to take up a position 4 cables on the starboard beam of the flagship. Supposing that the cruiser can steam 16 knots, what course should she steer, and how long will it take her to reach the required position ? [N. 7 E. ; 22 min.] 4. A detached scout with a speed of 19 knots is in lat. 40 13' N., long. 21 11' W., and at the same time the flagship with the main squadron is in lat. 37 35' N., long. 22 45' W., and steaming N.W. at the rate of 11 '5 knots. Find the shortest time in which the scout can reach a point N. by W., 5' of the flagship, and the course that will enable her to do this. [8'8 hours ; S. 60 W.] 5. A squadron is steaming N. 50 E. at 10 knots ; three cruisers, B, C and D, are posted on a line N. 15 E. from the flagship F, distant respectively 4', 8', and 12'. The cruisers are ordered to take up new positions on a line bearing N. 30 W. from flagship, their distances being 7', 14' and 21' respectively from the flagship. If B can steam 18 knots and C and D can steam 20 knots, find the course each cruiser should steer to take up the required position, and the approximate time she will occupy in doing so. , N. 33 W., 15 min. (about); C, N. 36 W., min. (about) ; D, N. 36 W., 4l min. (about).] 6. The fleet is steaming N. 30 E. at 10 knots. A cruiser A, in station 2 points on the port bow of the flagship, distant 10 miles, is told that the course will be altered at 10 P.M. 8 points to starboard, and that she is to alter course in just sufficient time to take up the same relative position to the flagship at that time, using her available speed of 18 knots. When should she alter course, and what course must she steer ? [At 9 h 16 m P.M., S. 68 E.] 7. A battle squadron is steering N.E. at 10 knots. Two cruisers, C and R, are stationed N.N.E., 20 miles, and E. by N., 25 miles, respectively, from the flagship. They are ordered to exchange stations in 3 hours. What will be their courses and speeds ? Show that in executing this manoeuvre the cruiser shifting from the port to the starboard bow of the flagship will always, theoretically, have to alter course to avoid collision with the other cruiser. [C, N. 79 E., 13 knots ; R, N, 7 E., H ' knots.] 197 8. A squadron is steaming N. 25 E. at 10 knots, and a cruiser is detached with orders to communicate with a Coast Guard Station and rejoin the squadron at a speed of 16 knots. When the cruiser proceeds in execution of her orders the Coast Guard Station bears N. 70 E., distant 32 '5 miles, and the cruiser is \ mile from the Coast Guard Station when she alters course to rejoin the squadron ; find the course she should then steer, and the approximate time that she will take to rejoin her squadron, supposing that she made her signal without reducing speed. [N. 331 W. ; l h 40 m .] 9. A cruiser is ordered to rejoin her fleet at 13 knots, the fleet bearing N. 16 W., 25 miles, and proceeding N. 70 E. at 10 knots. The cruiser has to pass to the westward of a head- land N. 17 E., 21 miles from her, giving it a clear berth of one mile. Find her course after rounding the headland and the whole time required to rejoin. [N. 54 E. ; 3 h 45 m .] 10. A cruiser near the flagship is ordered at 4 A.M. to proceed at 17 knots, preserving a bearing N. 50 W. from flag- ship, and to rejoin squadron at 6 P.M. Squadron is steaming N. 10 W. at 10 knots. At what time should cruiser alter course, and what should her course then be ? [2 h 30> P.M. ; S. 72 E.] 11; The flagship, steering E.N.E., bears from you N.N.E., 30 miles. You are ordered to close on present bearing. What course would you steer if your speed is 14 knots and that of the flagship 10 knots ? [N. 53 E.] 12. Fleet steaming E. by S., 12 knots. You are on a bearing S. by W. from flag, 8 cables. You are ordered to look out on present bearing 30 miles. If you increase speed to 15 knots, how much will you have to alter course ? [3|- points to starboard.] V 13. The fleet is steaming N. 15 W. at 10 knots. A cruiser, close to the flagship, with a speed of 16 knots, is ordered to look out 45 miles on a bearing S. 65 E., and, having done so, to rejoin the fleet. She leaves at 8 h A.M. ; what courses must she steer, and when should she be back again ? [N. 86 E. ; N. 36 W. ; 8 h 5 m .] 14. A cruiser is ordered at 5 A.M. to scout N.E. at 18 knots, and to rejoin the squadron, which is steaming North at 10 knots, at 4 P.M. When should the cruiser alter course, and what should the course be ? [Il h 20 ni A.M. ; N. 70 W.] 2 15. A cruiser is ordered to proceed N. 55 E. at 15 knots, and to rejoin the squadron in 4 hours' time at a position N. 20 E., 40 miles from the place where she was detached. Find how far she should steer on the given course, and what her course should be when she starts to rejoin. [36-7 miles; N.W.] 16. A scout accompanying a squadron is ordered at noon to proceed at 18 '5 knots on a N.E. course and to rejoin by 7.30 A.M. next day, arriving at a point 5 miles E. by N. of the flagship. The squadron is steaming N. by W. at 11 knots. At what time must the scout alter course, and what will her new course be ? [9 h 34i' n P.M. ; N. 62 W.] x^ 17. The fleet is steaming S. 20 E. at 10 knots, when, at 2 P.M., a cruiser C is ordered to steer S. 80 E. at 16 knots and to rejoin the flag at 6 A.M. next day. When must G alter course and what course must she then steer ? [9 U 5 m P.M. ; S. 23 W.] 18. On August 5th, 1912, at 6 A.M. (S.A.T.) a fleet in lat. 49 N. is steaming West at 12 knots, and a scout S (speed 18 knots) is ordered to steer N.W. by W. and to rejoin the fleet at sunset. When must S alter course, and what will her new course be ? Neglect the change of longitude. [2 h 20 m P.M. ; S. 23 W.] 19. The fleet is steaming S. 30 W. at 12|- knots. At 8 A.M. a cruiser C (speed 19 knots) is ordered to steer N. 75 W. for two hours, then S. 60 W., and to rejoin the fleet at 6 A.M. next day. When must G alter course from S. 60 W., and what will her new course be ? [9 h 22 m P.M. ; S. 31 E.] 20. A mail steamer, whose speed is 21 knots, sights a cruiser, whose maximum speed is believed to be 15 knots, bearing S. 81 W , distant 8 miles. What is the closest the mail steamer can keep to her course, which is N. 45 W., in order to prevent the cruiser getting within 4 miles of her ? [N. 24 W.] 21. A battleship with a speed of 14 knots discovers an enemy's transport 6' off, bearing S. 70 W., in a bay open from N. to E.S.E. The transport has steam for 17 knots, and the battleship's effective range does not exceed 3,800 yards. What course should the battleship steer to intercept the transport, and in what time will the battleship be within the required range ? [N. 35 W. ; 30 ' 5 min.] 199 22. A cruiser C, stationed 25 miles N. 25 W. from the fleet, which is steaming N.E. at 16 knots, wishes, to approach to a distance of 10 miles as quickly as possible at 14 knots. What course should she steer, and how long will it take her ? [S. 65 E. ; l h 4 m .] 23. A mail steamer, speed 18 knots, sights a cruiser, whose maximum speed is believed to be 15 knots, bearing N. 15 E., distant 8 miles. What is the closest the mail steamer can keep to her course (N. 60 W.) in order to prevent the cruiser getting within 4 miles of her ? [N. 71 W.] M 4. A cruiser, with speed 18 knots, is steaming N.N.E. along a coast-line lying to the westward. She is sighted at a distance of 7 miles bearing W. by N. from a battleship with speed 16 knots. Find what course the battleship must. steer to endeavour to intercept the cruiser, and how long she will be within a range of 5,000 yards. [N. 4 20' W. ; 26 min. (nearly).] 25. A battleship B, with an available speed of 12 knots, discovers a cruiser A, bearing N. 36 W., at a distance of 7 miles, in a bay which is only open to the South east. A's available speed is supposed to be 18 knots and she cannot steer to the Northward of East. What course must B steer to approach A as closely as possible if A endeavours to escape to the Eastward, and what will the shortest distance between them be ? [N. 42 E. ; 3000 yards.] 26. A cruiser with a speed of 17 knots is sighted near a shore lying to the Northward by two of the enemy's ships, bearing respectively from her E. by S., 6 miles, and S. 75 W., 7 miles. If their respective speeds are 12^ knots and 13^ knots and their effective gun-range 5000 yards, find within what limits the cruiser must steer so as to avoid an action. [S. 3^ W. and S. 8^ E.] 27. A cruiser A, with steam for 15 knots, sights a cruiser B 5 minutes due S. of her and steaming E.N.E. at 18 knots. What course must A steer to get as close as possible to B, and t will this distance be ? [S. 78 E. ; 1 mile.J 28. A fleet is steaming N. 10 E. at 10 knots. At 4 A.M. a scout is detached and ordered to proceed N. 30 W. at 15 knots, and rejoin the flag at 7 P.M. At what time must she alter course, and what will her new course be ? [12.30 P.M.; N. 68 E.] 29. The fleet is steaming N. 70 E. at 12 knots when a destroyer, speed 20 knots, is detached at 6 h A.M. to fetch the mails from a station bearing N. 20 E., 10 miles. She is detained 1 hour. What course should she steer to rejoin, and when will she do so ? [N. 84 E. ; 9 U 4- A.M.] 200 30. A squadron is steaming N. 10 E. at 15 knots. A cruiser, with an available speed of 20 knots, bearing S. 67 E., 25' from the flagship, is ordered to close. What course must she steer? After steaming on this course for f hour a break- down causes her to stop for one hour. What will be her new course and what will be the total time taken to join the flagship ? [N. 19 W. ; N. 2 W. ; 5 h 49 m .] 31. Two ships, X and Y, are steaming East, 14 knots, and S. 5 E., 18 knots, respectively ; at a certain time Y bears N. 50 E., 8 miles from X. Find by graphical methods (a) how near X will pass to Y ; (6) for how long Y will be within 3' of X. [2 1 5 miles ; 11*4 minutes.] 32. A fleet is steaming N. at 12 knots. A cruiser, with an available speed of 19 knots, is ordered to proceed N. 75 W., 50 miles, and then rejoin the fleet. She leaves at 5 A.M. What return course must she steer and at what time will she rejoin the fleet ? [N. 33 E. ; 12 h 20 m P.M.] The fleet is steaming S. 20 W. at 12 knots, when a cruiser, bearing N. 65 E., 2 miles from the flagship, is ordered to take station 5 cables on her starboard beam at 18 knots. What course must she steer, and how long will she take to get into station ? [S. 41 W. ; 18 minutes.] 34. From a cruiser, A, steaming N. 20 W. at 10 knots, another cruiser, B, steering a steady course, is observed bearing N. 80 E., distant 4' and after an interval of half an hour, B bears N. 65 E., distant 5 '5'. Find, by a graphical con- struction, the course and speed of B. [N. 6 W. ; 12-8 knots.] 35. A battleship B, speed 10 knots, discovers a cruiser C bearing S. 70 E., at a distance of 10 miles, in a bay which is only open to the Westward. C's available speed is 15 knots and she cannot safely steer a more southerly course than S. 60 W. What course must B steer to approach C as close as possible, and what will the distance be ? [S. 12 W. ; 1-4 miles.] 36. A is steaming N.E., at 20 knots. is E.S.E., 6 miles from A, and her speed is 16 knots. C wishes to get within 4,000 yards of A as soon as possible. What course should she steer, and how long will she be in getting into position ? [N. 14 W. ; 16 minutes.] 37. A cruiser, with steam for 18| knots, is lying to the northward of a shore running E. and W. when sighted by two of the enemy's battleships, whose speed is 12 knots. One 201 bears W. by N., 5 miles, and the other E., 5 miles. If the effective range of the enemy's guns is 3,500 yards, find within what limits the cruiser may shape her course so as to avoid an action ? [N. 18 W. and N. 27 E.] 38. Of two ships, C bears from A N. 25 W., 15 miles, and is steaming N. 80 E., 18 knots. A wishes to close to 7 miles at 14 knots as quickly as possible. What course should she steer ? [N.N.E.] 39. A cruiser is steaming N. 20 E. at 18| knots, and is under orders not to alter course unless compelled to do so. An enemy's ship is sighted bearing E. by S., distant 7| miles ; her speed is 16^ knots and her effective gun-range, 5,000 yards. Will the cruiser be able to avoid an action without altering course ? [No ; limiting course is N. 19 20' E.] 40. A cruiser is 30 miles N. 20 W. of the flagship, which is steaming N. 80 E. at 10 knots. The cruiser is ordered to approach within 5 miles of the flagship at a speed of 17 knots. What course should she steer, and what time will be necessary to carry out the order ? [S. 50 E. ; 2 hours.] 41. A cruiser bears from the flagship N. 80 W., distant 18 miles, and has steam for 15 knots. The flagship is steaming S.W. at 7 knots, and orders the cruiser to approach to 4 miles as quickly as possible. What course must the cruiser steer and when will she be at the assigned distance ? [S. 64 E. ; 46| minutes.] 42. A cruiser C, stationed 30 miles S. 30 W. from the fleet, which is steaming N. 70 W. at 12 knots, wishes to close to a distance of 5 miles as quickly as possible at 16 knots. What course should she steer, and how long will it take her ? [N. 9 W. ; l h 49 m .] 43. A cruiser A, with steam for 20 knots, wishes to approach as quickly as possible to a distance of 2 miles from another cruiser, distant 8 miles, S. 50 W., and steaming S. 30 E. at 18 knots. What course should A steer, and how long will she be in getting to the required distance ? [S. 1 E. ; 43 miu.] 44. A cruiser B, steaming at a constant speed, sights a vessel C which is steering a steady course at a constant speed. B alters course to overhaul C, and finds that, on a certain course she keeps C on the same bearing ; show that on this course B is approaching C by the shortest route. 45. A scout sights an enemy's ship in the offing, apparently steaming N. at a constant speed, and bearing N. 66 W. at the time of observation. After the scout has proceeded N. for 202 20 minutes at 10|- knots the enemy's bearing is found to bo N. 48 W. The scout then stops engines and after another 20 minutes a third bearing is found to be N. 27 W. Find the enemy's speed, and distance when first sighted. [ 1 6 6 knots ; 5 * 87 miles.] 46. A scout, G, sights the enemy at 10' 1 A.M., bearing S. 80 E. C remains stationary, and bearings are again taken at 10 h 10* (N. 8b' 15' E.) and at 10 h 20 m (N. 70 E.). After ascertaining the course of the enemy from these bearings C proceeds on a parallel course, and at 10 h 24 m the enemy bears N. 64 30' E. At 10 h 34 m C comes to a standstill, the enemy bearing N. 53 30' E. and C having gone two-thirds of a mile in the ten minutes. At 10 h 44 m the enemy bears N. 39 E. Determine the enemy's course and speed. Do you consider this method reliable ? Give reasons for your answer. [X. 22 W. ; 12 knots.] 47. Two scouts, A and B, in line abreast, N.W. and S.E., respectively, from each other, and 3 miles apart, are steering N.E. at 20 knots. At daylight they sight a battleship, F, bearing simultaneously right ahead of A, and N. 25 E. from B. Ten minutes later F bears N. 27 E. from A and due North of B. Determine approximately the course and speed of F. [N. 63 W. ; 9-1 knots.] 48. From a cruiser a lighthouse bears due N., distant 12 miles. The cruiser is ordered to steam in a North-easterly direction till the lighthouse bears due West, and thereafter to proceed to a rendezvous which bears from the lighthouse S. 50 E., 44 miles. Find the shortest route by which the cruiser can execute her orders, and obtain the courses she should steei and her distance from the lighthouse when she alters course. [N. 40 E. ; S. 40 E. ; 10-04 miles.] 49. A scout observes an enemy's ship in the offing at a distance of Similes, bearing S. 10 E. The engines are stopped, and a second bearing, taken 12 minutes later, is S. 19 W., while a third bearing at the end of another 12 minutes is S. 44 W. Find the course and speed of the enemy, assuming that both remain unvaried. [N. 79^ W. ; 13^ knots.] 50. A scout, C, sights the enemy at 4 P.M., bearing S. 17 W. C remains stationary and takes bearings at 4' 6 P.M. (S. 32 W.) and 4.12 P.M. (S. 46 W.). After ascertaining the course of the enemy from these bearings C proceeds on a parallel course and at 4 '15 P.M. the enemy bears S. 52^ W. At 4 "21 P.M. C comes to a standstill, the enemy bearing S. 59f W. and C having advanced half a mile in the six minutes. At 4 '27 P.M. the enemy bears S. 68f W. Determine the enemy's course and speed. [N. 65 W. ; 15 knots. 203 Section 10. OBSERVATIONS TO DETERMINE DEVIATION. 1. June 28th, 1912, at about 3.20 P.M., in latitude and longitude by account 19 15' S., 169 45' E., the Sun bore by compass N. 28 W., when the Deck Watch showed 7 h 9 m 46 s . The watch was 3 h 5 m 53 9 fast on G.M.T., and the magnetic variation was 20 19' W. Required the deviation. [2 45' W.] 2. April 18th, 1912, at about 3.20 P.M., in latitude and longitude by account 24 40' S., ?5 51' W., the Sun bore by compass N. 75 W., when the Deck Watch showed 3 h 51 m 35 s . The watch was l h 13 m 57 8 slow on G.M.T., and the magnetic variation was 17 0' E. Required the deviation. [2 25' W.] 3. June 4th, 1912, at about 9 A.M., in D.R. 16 47' S., 62 10' E., the Sun bore by compass N. 35 30' E., when the Deck Watch showed 6 h 7 m 43 8 . The watch was l h 18 m 51 s fast on G.M.T., and the magnetic variation was 12 20' E. Required the deviation. [2 E.] 4. October 15th, 1912, at about 7.50 A.M., in D.R. 27 34' S., 68 37' E., the Sun bore by compass S. 70 E., when the Deck Watch showed 4 h 15 m 43 s . The watch was l h 15 m 53 s fast on G.M.T., and the magnetic variation was 21 W. Required the deviation. [4 10' W.] 5. November 30th, 1912, at about 6.15 P.M., in D.R. 50 10' S., 112 43' W., the Sun bore by compass due West, when the Deck Watch showed 2 h 21 m 13 s . The watch was Qh 4gm 333 f ast 011 Q.M.T., and the magnetic variation was 20 36' W. Required the deviation. [4 E.] 6. September 24th, 1912, at about 7.30 A.M., in D.R. 4 12' N., 63 19' E., the Sun bore by compass S. 70 E., when the Deck Watch showed l h 55 m 29 s . ' The watch was l h 12 m 5 8 slow on G.M.T., and the magnetic variation was 21 W. Required the deviation. [3 E.] 7. April 27th, 1912, at about 7.30 A.M., in D.R. 53 18' N., 5 20' W., the Sun bore by compass S. 65 E., when the Deck Watch showed 7 h 23 m 23 8 . The watch was O h 26 m 48 8 slow on G.M.T., and the magnetic variation was 17 19' W. Required the deviation. [2 25' E.] 204 8. On March 20th, 1912, at 7 h lb' m 37* A.M. (S.M.T.\ in 41 30' N., 126 40' W., the Sun bore E. by compass. The variation was 18 10' E. Find the deviation. [6 25' W.] 9. April 13th, 1912, about 5 h 15 m A.M. S.A.T., in lat. 52 20' N., long. 11 15' W., the Sun rose by compass E. f S. Variation, 24 W. Find the deviation. [50' E.] 10. December 2nd, 1912, about 6 h 20 m A.M. S.A.T., in lat. 12 5' N., long. 62 50' E., the Sun rose by compass S. 63 30' E. Variation, 1 10' W. Find the deviation. [2 55' W.] 11. April 30th, 1912, about 5 h 5 m P.M. S.A.T., in lat. 41 35' S., long. 100 15' E., the Sun set by compass N.W. by W., variation being 18 W. Required the deviation. [4 5' E.] 12. April 4th, 1912, about 6 h 20 m P.M. S.A.T., in lat. 41 41' N., long. 35 20' W., the Sun set by compass N. 60 20' W. Variation, 24 W. Find deviation. [2 5' E.] 13. January 21st, 1912, about 1.30 A.M., lat. 35 S., long. 116 E., Spica bore by compass N. 82 E. Deck Watch showed 4 h 45 m 29 8 and was l h 12 m 44 s slow on G.M.T. Variation, 5 W. Find the deviation. [125'E.] 14. October 5th, 1912, about 8.15 P.M., lat. 34 16' S., long. 113 30' E., a Cygni bore by compass N. 3 E. Deck Watch showed 10 h 43 m 8 s and was l h 42 m 47 s slow on G.M.T. Variation, 5 W. Find the deviation. [1 30' W.] 15. June 21st, 1912, about 9 P.M., lat. 48 40' N., long. 7 30' W., Arcturus bore by compass S. 38 W. Deck Watch showed 9 h 42 m 12 s and was O h 13 m 42 s fast on G.M.T. Variation, 17 W . Find the deviation. [1 5' E.] 16. March 12th, 1912, about 8 P.M., lat. 42 41' N., long. 11 W., Capella bore by compass N. 56 W. Deck Watch showed 7 h 16 m 20" and was l h 29 m 37 8 slow on G.M.T. Variation, 16 W. Find the deviation. [1 15' E.] 17. August 24th, 1912, about 7 P.M., lat. 43 49' S., long. 174 35' E., Jupiter bore by compass N. 32 W. Deck Watch showed 7 h 25 m 24 s and was O h 2 m 31 8 fast on G.M.T. Variation, 15 E. Find the deviation. [1 10' W.] 205 18. July 15th, 1912, about 4 A.M., lat. 50 50' N., long. 6 10' W., Saturn bore by compass S. 65 E. Deck Watch showed 4 h 38 m 27 8 and was O h 3 m 46 s fast on G.M.T. Variation, 17 W. Find the deviation. [20' E.] 19. February 2nd, 1912, about 5 A.M., lat. 20 30' S., long. 15 30' W., Venus bore by compass S. 54 E. Deck Watch showed 5 h 49 m 41 s and was O h 12 m 17 s slow on G.M.T. Variation, 23 W. Find the deviation. [2 E.] 20. March 21st, 1912, about 7 P.M., lat. 47 50' N., long. 9 40' W., the Moon's centre bore by compass N. 73 W. Deck Watch showed 7' 1 51 m 17 s and was O h 3 m 20 s fast on G.M.T. Variation, ] 8| W. Find the deviation. [50' W.] 21. February llth, 1912, about 6 A.M., lat. 38 40' S., long. 174 10' E., the Moon's centre bore North by compass. Deck Watch showed 6 h 27 m 44 s and was O h 3 m 24 s slow on G.M.T. Variation, 14 E. Find the deviation. [1 15' E.] 22. March 8th, 1912, in 37 10' N., 19 15' E., at 7.45 A.M. the observed altitude j21 was 17 1' 0" and its compass bearing N. 80 E. Index error, 3' 30" -. Height of eye, 20 feet, Variation, 27 19' E. Required the deviation. [3 E.] 23. January 21st, 1912, at 4.30 P.M., in. 40 53' S., 81 25' W., the observed altitude _0 was 29 22' 0" and its compass bearing N. 81 W. Index error, 3' 10"+. Height of eye, 17 feet. Variation, 17 38' W. Required the deviation. [7 E.] 24. October 28th, 1912, at 9.10 A.M., in 53 30' N., 141 40' W., the obs. alt. G) was 14 16' 50" and its com- pass bearing, S. 28 E. Index error, 3' 10" + . Height of eye, 16 feet. Magnetic variation, 21 33' W. Required the deviation. [7 30' E.] 25. May 26th, 1912, at 7.10 A.M., in 53 25' N., 174 12' E., the obs. alt. _0 was 26 44' 0" and its compass bearing S. 55 E. Index error, 3' 50" -. Height of eye, 18 feet. Magnetic variation, 30 W. Required the deviation. [4 30' W.] 26. November 12th, 1912, at 6 P.M., in 36 20' S., 173 35' E., the obs. alt. _0 was 10 12' 10" and its compass bearing was West. Index error, 2' 20" +. Height of eye, 31 feet. Magnetic variation, 20 3' W. Required the deviation. ., [5 40' E.] 206 27. August llth, 1912, at 9 A.M., in 43 12' S., 107 44' W., the obs. alt. Q was 18 6' 20" and its compass bearing, N. 25 E. Index error was 3' 15" -f . Height of eye, 23 feet. Magnetic variation, 17 30' E. Required the deviation. [3 45' E.] 28. June 1st, 1912, at 10 P.M., in 70 45' N., 90 17' W., the obs. alt. _Q was 5 8' 10" and its compass bearing, N. 30 W. Index error was 3' 10" -. Height of eye, 19 feet. Magnetic variation, 9 E. Required the deviation. [6 W.] 29. December 10th, 1912, at 8.30 A.M., in 27 20' N., 117 12' W., the obs. alt. .0 was 18 25' 50", and its compass bearing, S. 59 E. Index error, 2' 20" +. Height of eye, 27 feet Magnetic variation, 12 7' E. Required the deviation. [3 30' W.] 30. January_6th, 1912, at 3.45 P.M., in 52 45' S., 56 56' E., the obs. alt. O was 39 0' 40" and its compass bearing, N. 64 W. Index error, 3' 40" -f. Height of eye, 20 feet. Variation, 35 50' W. Find deviation. [23 40' E.] 31. MarchJOth, 1912, at 7.15 A.M., in 41 30' N., 126 40' W., the obs. alt. O was 13 0' 30", and its compass bearing, East. Index error, 10' 30" +. Height of eye, 27 feet. Variation, 18 10' E. Find deviation. [6 25' W.] 32. March 9th, 1912, about 8.30 P.M. (S.M,T.), lat. 50 10' N., long. 6 40' W., Denebola bore by compass S. 57 E., and the observed altitude was 29 48'. Index error, 2' 20" - ; Eye, 24 feet ; Variation, 19 W. Find the deviation. [H w.] 33. April 16th, 1912, about 7.30 P.M. (S.M.T.), lat. 35 20' S., long. 115 30' E., Aehemar bore by compass S. 35 W., and the observed altitude was 18 41'. Index error, 2' 30" - ; Eye, 40 feet ; Variation, 5 W. Find the deviation. [If E.] 34. January 6th, 1912, about 8 P.M. (S.M.T.), lat. 33 10' S., long. 114 20' E., Sir ius bore East by compass, and the observed altitude was 37 37'. Index error, 3' + ; Eye, 30 feet ; Deviation, 50' E. Find the variation. [5 W.] 35. July 9th, 1912, about 8.40 P.M. (S.M.T.), lat. 50 48' N., long. 9 20' W., Spica bore by compass S. 60 W., and the observed altitude was 20 41'. Index error, 2' 10" - ; Eye, 40 feet ; Deviation, 1 W. Find the variation. W.] 207 36. January 10th, 1912, about 5 A.M. (S.M.T.), lat. 10 20' S., long. 55 20' E., Venus bore by compass S. 72 E., and the observed altitude was 30 10'. Index error, nil ; Eye, 41 feet ; Deviation, 2 E. Find the variation. [5 W.] 37. May 7th, 1912, about 8.30 P.M. (S.M.T.), lat. 25 20' N, long. 17 30' W., Mars bore by compass N. 65 W., and the observed altitude was 33 23'. Index error, 2' ; Eye, 28 feet ; Variation, 15 W. Find the deviation. [3 E.] 38. September 29th, 1912, about 7 P.M. (S.M.T.), lat. 35 20' N., long. 45 W., Jupiter bore by compass S. 61 W., and the observed altitude was 19 21'. Index error, 3' ; Eye, 39 feet ; Variation, 20 W. Find the deviation. [2 40' E.] 39. March 3rd, 1912, about 10 P.M. (S.M.T.), lat. 37 30' N., long. 18 45' E., the Moon's centre bore by compass S. 53 E., and the observed altitude Moon's U.L. was 42 51'. Index error, 1' 30" - ; Eye, 45 feet ; Variation, 6| W. Find the deviation. [20' W.] 40. August 28th, 1912, about 4 A.M. (S.M.T.), lat. 12 20' S., long. 101 20' E., the Moon's centre bore by compass S. 85 W., when the observed altitude Moon's U.L. was 33 11'. Index error, 2' 30"+; Eye, 40 feet; Variation, 2 W. Find the deviation. CF w.] Section 11. POSITION LINES BY OBSERVATIONS OF SUN, STARS, &c. POSITION LINES BY OBSERVATION OF THE SUN. 1. October 9th, 1912, in D.R. 25 C 14' S., 87 30' E., observed meridian altitude Sun's L.L. was 70 40' 20". Index error, - 2' 30". Height of eye, 16 feet. [4-5' away, Sun's T.B. North ; or lat. 25 18^' S.] 2. May 10th, 1912, in D.R. 3 53' S., 11 16' W., observed meridian altitude Sun's L.L. was 68 20' 40". Index error, + 1' 50". Height of eye, 18 feet [3-3' nearer, Sun's T.B. North ; or lat. 3 49-7' S.] 3. March 8th, 1912, in D.R. 33 5' N., 139 48' E., observed meridian altitude Sun's L.L. was 51 49' 30". Index error, - 3' 15". Height of eye, 15 feet. [6-7' nearer, Sun's T.B. South ; or lat. 32 58 -3' N.] 208 4. April 3rd, 1912, in D.R. 27 36' S., 102 52' E., at about ll h 30 m A.M. the observed altitude 0. was 56 28' 10". The Deck Watch showed ll h 53 m 32 9 , and was 4 h 49 m 18 s slow on G.M.T. Index error, + 2' 20". Height of eye, 24 feet. [10-2' away, or lat. 27 46'3' S. ; Sun's T.B. N. 14 E.] 5. November 15th, 1912, in D.R. 40 12' N., 10 17' W., at about ll h 55 m A.M. the observed altitude O. was 31 8' 30", when a chronometer showed l h 25 m 27 s . On October 27th, at G.M. Noon, the chronometer was l h 5 m 40 8 fast on G.M.T. , losing 3' 2 seconds daily. Index error, 3' 30". Height of eye, 40 feet. [5- T away, or lat. 40 17' N. ; Sun's T.B. S. 1$ E.] 6. November 1st, 1912, at G.M. Noon, the chronometer was 2 U 34 m 56 s fast on G.M.T., gaining daily 4' 6 seconds. November 16th, at about ll h 30 m A.M., Comparison : Chronometer - 2 h 15 m s Deck Watch - ll h 36 m 10 8 At ll h 45 m A.M., in lat. 49 17' N., long. 4 W. by account, observed altitude Sun's L.L. 21 48' 40". Deck Watch, ll h 51 m 21 s . Index error, + 3' 20". Height of eye, 28 feet. [3-4' nearer, or lat. 49 13 '7' N. ; Sun's T.B. S. 2 E.] 7. October 2nd, 1912, at G.M. Noon, the chronometer was 3 h 3 m 33 s slow on G.M.T., losing daily 3' 6 seconds. October 15th, at about 11 h A.M., Comparison : Chronometer - 12 h 3 m s Deck Watch 3 h 5 m 7 s At ll h 30 m A.M., in lat. 35 Q 21' S., long. 115 51' E. by account, observed altitude Sun's L.L. 62 6' 50". Deck Watch, 3 h 33 m 52 s . Index error, - 1' 20". Height of eye, 38 feet. [5 1' away, or lat. 36 26' S. Sun's T.B. N. 14 E.] 8. January 26th, 1912, at G.M. Noon, the chronometer was 2 h 47 m 19 8 slow on G.M.T., gaining daily 2 ' 8 seconds. February 19th, about ll h A.M., Comparison : Chronometer 9 h O m s Deck Watch, - ll h 45 m 37 s At ll h 30 m A.M., in lat. 50 47' N., long. 9 3' W. by account, observed altitude Sun's L.L. 27 11' 30". Deck Watch 12 h 20 m 17". Index error, 1' 10". Height of eye, 45 feet. [3-7' nearer, or lat. 50 43-2' N. ; Sun's T.B. S. 8 E.] 9. April 18th, 1912, in D.R. 40 16' N., 35 35' W. at about 7 h 10 m A.M. (S.A.T.) the observed altitude 0. was 19 44' 30", when a chronometer showed ll h 18 m 19 s . On April J&th at 209 G.M. Noon, the chronometer was l h 47 m 15 s fast on G.M.T., and its daily rate 4 ' 3 seconds, gaining. Index error, - 2' 30". Height of eye, 25 feet. [8-5' away, or long. 35 46' 30" W. ; Sun's T.B. S. 87f E.] 10. December 10th, 1912, in D.R. 32 40' S, 81 10' W., at about 7 h 30 m A.M., the observed altitude 0. was 32 12' 40" when a chronometer showed 10 h 57 m 25 s . On November 9th, at G.M. Noon, the chronometer was l h 51 m 17 s slow on G.M.T., and its daily rate was 3 ' 3 seconds, gaining. Index error, + 1' 20". Height of eye, 22 feet. [3' nearer, or long. 81 6' 15" W. ; Sun's T.B. S. 81 E.] 11. September 24th, 1912, at about 7 h 30 m A.M., in D.R. 4 12' N., 63 15' E., the observed altitude 0. was 21 55' 0", when a chronometer showed l h 55 m 29 s . On August 27th, at G.M. Noon, the chronometer was l h 14 m 9 s slow on G.M.T., and its daily rate was 4 ' 5 seconds, gaining. Index error, - 2' 30". Height of eye, 23 feet. [9' nearer, or long. 63 16' E. ; Sun's T.B. S. 88 E.] 12. November 30th, 1912, in position by account 50 10' S., 112 40' W., at about 6 h 15 m P.M., the observed altitude Sun's L.L. was 14 18' 0", when a chronometer showed 2 h 21 m "l3 8 . On October 29th, at G.M. Noon, the chronometer was O h 51 m 14 s fast on G.M.T., and its daily rate was 4 ' 8 seconds, losing. Index error, 2' 50". Height of eye, 15 feet. [6-6' away, or long. 112 29' 15" W. ; Sun's T.B. S. 73 W.] 13. September 25th, 1912, in D.R. 0', 47 12' E., at about 4 h 30 m P.M. the observed altitude of the Sun's L.L. was 22 30' 0", when a chronometer showed ll h 2 m 25 s . On August 20th, at G.M. Noon, the chronometer was slow on G.M.T. 2 h 7 m 31 s , and its daily rate 4' 6 seconds, losing. Index error, 1' 10". Height of eye, 29 feet. [4-T nearer, or long. 47 7' 45" E. ; Sun's T.B. S. 89 W.] 14. Augustllth, 1912, in D.R. 43 12' S., 107 50' W., at about 9 A.M. the observed altitude 0^ was 18 6' 20", when a chrono- meter showed 5 h 14 m 48 8 . On July 3rd, at G.M. Noon the chronometer was O h 56 m 11 s fast on G.M.T., and its daily rate 3'6 seconds, gaming. Index error, 3' 15". Height of eye, 23 feet. [16-6' away, or long. 108 21' 30" W. ; Sun's T.B. N. 46 E.] 210 POSITION LINES BY OBSERVATIONS OF STARS. 1. September loth, 1912, observed meridian altitude of Aldebaran (Z.S.) 75 20' 40". Index error, 2' . Height of eye, 40 feet. [Lat. l32-6' N.] 2. March 10, 1912, observed meridian altitude of Sirius (Z.S.) 59 17' 30". Index error, 1' 30" +. Height of eye, 30 feet. [Lat. 47 22 -8' $.] 3. September 21st, 1912, observed altitude Capella (on the meridian below Pole) 9 47' 20". Index error, 1' 10" -K Height of eye, 35 feet. [Lat. 53 42 '7' N.] 4. July 7th, 1912, observed altitude a Crucis (on the meridian below Pole) 11 38' 10". Index error, 1' 40" +. Height of eye, 45 feet. [Lat. 38 51 -5' S.] 5. On April 14th, 1912, about 2 A.M. ship apparent time, in D.R. 48 55' N., 5 10' W., observed altitude of Altair was 23 6' 10". Time by Deck Watch, 2 h 32 m 32". Error of Deck Watch, ll m 34 8 fast on G.M.T. Index error, 1' 10" -. Height of eye, 30 feet. [4-3' away, or Long. 5 16f W. Star's T.B., S. 76 E.] 6. October 17th, 1912, about 6.50 P.M., in D.R. 9 27' S., 85 54' W., the observed altitude Vega 35 37' 50". Deck Watch, 12 h 17 m 5 s . Index error, 30" +. Height of eye, 50 feet. Deck Watch, O h 4 m 57 s slow. [4-4' away, or Long. 85 43|' W. Star's T.B., N. 26 W.] 7. February 16th, 1912, about 11 P.M., in D.R. 33 0' S., 114 40' E., the observed altitude of Betelguese was 29 50' 30". Deck Watch, 12 h 32 m 30 8 . Index error, + 1' 10". Height of eye, 31 feet. Chrono- meter, 3 h 3 m 39 s slow on G.M.T. Comparison Chronometer, 12 h 25 m s . Deck Watch, 12 h 22 m 37 8 . [5-7' away, or Long. 1 M 48' E. Star's T.B., N. 56 W.] 8. February 24th, 1912, about 7.30 P.M., in D.R. 34 45' S., 115 10' E., the observed altitude of Proeyon was 43 42'. Deck Watch, 12 h 7 m 10 s . Index error, 2' 20" +. Height of eye, 19 feet. Error of Deck Watch, 3 m 58 8 fast on G.M.T. [6-2' nearer, or Long. 115 23|' E. Star's T.B., N. 35 E.] 211 9. May 15th, 1912, about 9 P.M., in D.R. 49 10' N., 6 15' W., the observed altitude of Gapella was 23 12' 40". Deck Watch, 9 h 17 m 53 s . Index error, 2' 10" +. Height of eye, 26 feet. Error of Deck Watch, O h 2 m 24" slow. [4-6' nearer, or Long. 6 24f W. Star's T.B., N. 45^ W.] 10. September 27th, 1912, about 6.30 P.M., lat. 27 24' N., long. 115 4' W., observed altitude Areturus, 24 25' 30". Deck Watch, 2 h 10 m 29". Index error, 1' 10"-. Eye, 30 feet. Deck Watch, O h 4 m 10 8 slow. [2' nearer, or Long. 115 6|' W. Star's T.B., N. 79| W.J 11. February 21st, 1912, about 7.20 P.M., lat. 42 17' N., long. 37 30' W., observed altitude Eigel 39 30'. Deck Watch, 10 h 27 m 39 s , and was O h 38 m 20 s fast on G.M.T. Index error, 2' 10" . Height of eye, 50 feet. [1-6' away, ot Lat. 42 18-6' N. Star's T.B., S. 3^ W.] 12. August 20th, 1912, about 6.25 P.M., lat. 39 56' S., long. 65 26' E., observed altitude Antares 76 25'. Deck Watch, 2 h l m 17 s and was O h 2 m 3 8 slow on G.M.T. Index error, 1' 30" +. Height of eye, 32 feet. [4-7' nearer, or Lat. 39 51 -6' S. Star's T.B., N. 4^ E.] 13. June 18th, 1912, about 7 P.M., lat. 21 53' N, long. 125 47' W., observed altitude Spica 56 33'. Deck Watch, 3 h 10 m 27 s and was O h 12 m 49 s slow on G.M.T. Index error, 30" . Height of eye, 35 feet. [8' away, or Lat. 21 54' N. Star's T.B., S. 14^ E.] 14. November llth, 1912, about 3.45 A.M., lat. 44 19' S., long. 176 25' E., observed altitude Sirius 62 7' 40". Deck Watch, 4 h 17 m 29 s and was O h 18 m 37 s fast on G.M.T. Index error, 3' 20" -. Height of eye, 37 feet. [2-4' nearer, or Lat. 44 16 '6' S. Star's T.B., N. 10 W.] POSITION LINES BY OBSERVATIONS OF PLANETS"! 1. January 6th, 1912, about 5.10 P.M., lat. 47 55' N., long. 6 44' W., observed altitude of Mars, 42 28' 0". Deck Watch 5 h 31 m 20 8 and was O h 4 m 47 s slow on G.M.T. Index error, 2' 30" -. Height of eye, 28 feet. [2' away; T.B. S. 73^ E.] 2. February 8th, 1912, about 5.50 A.M., lat. 9 36' N., long. 137 19' W., observed altitude of Venus, 25 33' 30". Deck Watch 3 h 4 m 11 B and was O h 3 m 57 8 fast on G.M.T. Index error, 1' 10" +. Height of eye, 40 feet. [3-9' nearer ; T.B. S. 60 E.] u 5777. P 212 3. January 12th, 1912, about 3.40 A.M., lat. 46 17' S., long. 84 20' W., observed altitude of Jupiter 22 0' 30". Deck Watch, 8 h 58 m 37 s and was O h 18 m 15 8 slow on G.M.T. Index error, 30" . Height of eye, 26 feet. [3 -4' nearer. T.B., S. 82 E.] 4. January loth, 1912, about 4.50 A.M., lat. 24 40' S., long. 73 47' E., observed altitude of Mercury 13 57' 20". Deck Watch, 11 h 50 m 19 8 and was O h 4 m 2 9 slow on G.M.T. Index error, 1' + . Height of eye, 24 feet. [3 -4' away. T.B., S. 72 E.] 5. December 1st, 1912, about 7.15 P.M., lat. 23 25' S., long. 4 17' E., observed altitude of Venus 26 0' 0". Deck Watch, 7 h 13 m 25 8 and was O h 14 m 48 8 fast on G.M.T. Index error, 2' 30" + . Height of eye, 45 feet. [-6' nearer. T.B., S. 73 W.] 6. February 15th, 1912, about 7 P.M., lat. 13 42' S., long. 65 17' E., observed altitude of Mars 52 38' 20". Deck Watch, 2 h 44 m 7 s and was O h 4 m 59 s fast on G.M.T. Index error, 1' 30" - . Height of eye, 30 feet. [4-9' nearer. T.B., N. 9 J W.] 7. April 16th, 1912, about 5.10 A.M., lat. 12 54' N., long. 56 55' E., observed altitude of Jupiter 46 8' 0". Deck Watch, l h 10 m 47 3 and was O h ll m 58 s slow on G.M.T. Index error, 2' . Height of eye, 35 feet. [1-7' away. T.B., S. 38 W.] 8. February 17, 1912, about 5.40 P.M., lat. 51 40' N., long. 15 25' W., observed altitude of Saturn 51 50' 20". Deck Watch, 7 h 19 m 24 8 and was O h 37 m 56 8 fast on G.M.T. Index error, 1' 40" + . Height of eye, 32 feet. [1-6' nearer. T.B., S. 14 W.] 9. January 10th, 1912, about 6 A.M., lat. 17 47' N., long. 46 25' W., observed altitude of Venus 31 12' 0". Deck Watch, 9 h l m 37 s and was O h 3 m 56 8 slow on G.M.T. Index error, 40" . Height of eye, 34 feet. [5 -9' away. T.B., S. 53^ E.] 10. April 15th, 1912, about 7.20 P.M., lat. 36 20' N., long. 10 37' W., observed altitude of Mars 56 17' 30". Deck Watch, 7 h 57 m 35 s and was O h 5 m 14 8 slow on G.M.T. Index error, 1' 20" +. Height of eye, 42 feet. [2' nearer. T.B., S. 82 W.] 11. August 20th, 1912, about 5.50 A.M., lat. 32 30' S., long. 147 20' E., observed altitude of Saturn 38 23' 50". Deck Watch, 8 h O m 56 s and was O h l m 6 8 fast on G.M.T. Index error, 3' 20" - . Height of eye, 27 feet. [1'away. T.B., N. 8 E.] 213 12. February 10th, 1912, about 6.15 A.M., lat. 35 20' N., long. 20 40' E., observed altitude of Jupiter 31 ()' 0". Deck Watch, 4 h 54 m 42 s and was O h 2 m 37 s fast on G.M.T. Index error, 50" + . Height of eye, 39 feet. [1-7' nearer. T.B., S. 19 E.] 13. January 30th, 1912, in long. 129 20' W., observed meridian altitude of Mars 38 45' 40" (Z.S.). Index error, 2' - . Height of eye, 30 feet. [Lat, 28 59 -8' S.] 14. August 5th, 1912, in long. 146 40' E., observed meridian altitude of Jupiter 34 36' 30" (Z.N.). Index error, 1' 30" -K Height of eye, 45 feet. [Lat. 34 49-8' K] 15. September 1st, 1912, in long. 107 57' E., observed meridian altitude of Saturn 54 14' 0" (Z.S.). Index error, 30" +. Height of eye, 32 feet. [Lat. 16 58' S.] POSITION LINES BY OBSERVATIONS OF THE MOON. 1. January 23rd, 1912, about 5 P.M., lat. 51 47' N., long. 17 29 r W., obs. alt. Moon's L.L. 26 52' 30". Deck Watch, 6 h 14 m 7 s and was O h 5 111 29 s fast on G.M.T. Index error, V 40" -f . Height of eye, 40 feet. [2-3' away. T.B., S. 27^ W.] 2. October 6th, 1912, about 5.4O A.M., lat. 35 42' N., long. 19 36' E., obs. alt. Moon's L.L. 53 10' 20". Deck Watch, 4 h 18 m 56 s and was O h 2 m 11 s slow on G.M.T. Index error, 30" +. Height of eye, 26 feet. [2 -2' away. T.B., S. 76 E.] 3. April 28th, 1912, about 6 P.M., lat. 34 17' S., long. 93 42' W., obs. alt. Moon's U.L. 24 25' 30". Deck Watch, 12 h 19 m 36 8 and was O h 2 m 57 s fast on G.M.T. Index error, 1' 30" -. Height of eye, 37 feet. [4-2' nearer. T.B., N. 75 E.] 4. December 29th, 1912, about 7.15 A.M., lat. 17 45' S., long. 69 21' E., obs. alt. Moon's U.L. 43 16' 20". Deck Watch, 2 h 31 m 14 s and was O h 5 m 2 8 slow on G.M.T. Index error, 2' . Height of eye, 24 feet. [2 -4' away. T.B., N. 59 W.] 5. May 23rd, 1912, about 5.30 P.M., lat. 21 19' N, long. 143 55' E., obs. alt. Moon's U.L. 79 55' 0''. Deck Watch, 7 h 31 m 24 s and was O h 22 m 3 s slow on G.M.T. Index error, 50" +. Height of eye, 30 feet. [1'away. T.B., S. 59fc E.] p 2 214 6. February 22nd, 1912, about 7 P.M., lat. 45 27' S., long. 176 25' E., obs. alt. Moon's L.L. 14 55' 10". Deck Watch, 7 h 15 m 43" and was O h O m 16 8 fast on G.M.T. Index error, 20" +. Height of eye, 35 feet. [3 T away. T.B., N. 62 W.] 7. August 8th, 1912, about 5.30 A.M., lat. 8 42' S., long. 16 37' E., obs. alt. Moon's L.L. 44 21' 50". Deck Watch, 4 h 39 m 47" and was L 14 m 36 s fast on G.M.T. Index error, 1' . Height of eye, 45 feet. [5-5' nearer. T.B., N. 38 E.] 8. June 18th, 1912, about 7.30 p.m., lat. 36 10' N., long. 6 56' W., obs. alt. Moon's L.L. 36 5' 40". Deck Watch, 8 h l m 29 8 and was O h 3 m 57 s fast on G.M.T. Index error, 2' 30". Height of eye, 40 feet. [I 1 away. T.B., S. 89 W.] 9. July 21st, 1912, in long. 176 40' E., obs. mer. alt. of Moon's U.L. (Z.N.) 32 40' 20". Index error, 2' 40" - . Height of eye, 35 feet. [Lat. 45 0' N.] 10. September 4th, 1912, in long. 37 50' W., obs. mer. alt. of Moon's U.L. (Z.S.) 29 47' 30". Index error, 1' 10" +. Height of eye, 30 feet. [Lat. 33 43-1' S.] 11. November 1st, 1912, in long. 66 25' E., obs. mer. alt. of Moon's U.L. (Z.S.) 78 27' 30". Index error, 30" +. Height of eye, 28 feet. [Lat. 14 46- 4' N.] 12. March 25th, 1912, in long. 146 19' W., obs. mer. alt. of Moon's U.L. (Z.S.) 42 13' 50". Index error, 1' 10" -. Height of eye, 45 feet. [Lat. 19 4 -2' S.] CHOICE OF BODIES FOR OBSERVATION. Draw diagrams on the plane of the horizon, showing the heavenly bodies available for observation at the following places and times, selecting in each case the pairs most suitable for nearly simultaneous observation. Specify also, in each case, which pair you would take if you were particularly anxious about (a) your latitude, (b) your longitude. [N.B. No star of less magnitude than 2' 1 to be included.] 1. Lat. 50 N., long. 5 W. ; 45 m after sunset on January 25th, 1912. [S. Sid. T. is l h 35 m .] 215 2. Lat. 32 S., long. 84 W. ; 35 m before sunrise on July 9th, 1912. [S. Sid. T. is l h 38 m .] 3. Lat. 15 N., long. 150 E. ; 30 m after sunset on Decem- ber 14th, 1912. [S. Sid. T. is 23 U 29 U \] 4. Lat. 45 S., long. 126 W. ; 40 m before sunrise on April llth, 1912. [S. Sid. T. is 19 h 1 l m .] 5. Lat. 21 S., long. 65 E. ; 45 m before sunrise on August 5th, 1912. [S. Sid. T. is 2 h 41".] 6. Lat. 36 N., long. 10 W. ; 40 m after sunset on October 29th, 1912. [S. Sid. T. is 20 h 15 m .] 7. Lat. 9 S., long. 78 E. ; 30 m before sunrise on February , 1912. [S. Sid. T. is 14 h 53 m .] 8. Lat. 26 N., long. 40 W. ; 40 m after sunset on Novem- ber 21st, 1912. [S. Sid. T. is 21 h 47 m .] MERIDIAN PASSAGES. 1. Find the S.M.T. of the meridian passage of Aldebaran on January 9th, 1912, in longitude 70 W. [9 h 18 m P.M.] 2. Find the S.M.T. of the meridian passage of Arcturus on February 15th, 1912, in longitude 125 40' W. [4 h 35 m A.M.] 3. Find the S.A.T. of the meridian passage of Procyon on April 3rd, 1912, in longitude 60 E. [6 h 45 m P.M.] , --4. Find the S.A.T. of the meridian passage of Fomalhaut on June 6th, 1912, in longitude 130 E. [5 h 59 m A.M.] 5. Find the Deck Watch time of the meridian passage of Rigel on March 2nd, 1912, in longitude 17 45' E., the Deck Watch being O h 4 m 47 s fast on G.M.T. [5" 23i">.] 6. Find the Deck Watch time of the meridian passage of Sirius on March 24th, 1912, in long. 12 25' W., the Deck Watch being l h 4 m 19 s slow on G.M.T. [6 h 19 U1 .] 216 7. Find the Deck Watch time of the meridian passage of Saturn on January 19th, 1912, in long. 97 20' E., the Deck Watch being O h 2 m 17 s slow on G.M.T. [I2 h 24 m .] / 8. Find the Deck Watch time of the meridian passage of Jupiter on March 17th, 1912, in long. 105 26' W., the Deck Watch being O h 4 m 36 s fast on G.M.T. [12 h 23 m .] 9. Find the S.M.T. and the G.M.T. of the Moon's meridian passage in long. 44 20' W. on October 18th, 1912. [6 h 42 m P.M. ; 9 h 39| m .] 10. Find the S.M.T. and the G.M.T. of the Moon's meridian passage in long. 135 50' E. on June 7th, 1912. [5 h 21 m A.M. ; 8 h I7^ m .] 11. Find the time by Deck Watch when the Moon is on the meridian of long. 67 30' W. on April 23rd, 1912, the Deck Watch being O h 5 m 49 s slow on G.M.T. [10 h 21 m .] 12. Find the time by Deck Watch when the Moon is on the meridian of long. 83 20' E. on February 9th, 1912, the Deck Watch being O h 17 m 48 s fast on G.M.T. [Il h 50 m .] 13. What heavenly bodies suitable for observation pass the meridian within one hour after sunset at the given places and dates : (a) Lat. 50 20' N., long. 12 30' W., March 1st, 1912? (6) Lat. 20 45' N.,long. 72 30' E., December 20th, 1912? (c) Lat. 40 10' S., long. 76 30' W., August 23rd, 1912? (d) Lat. 10 20' S., long. 124 W., January 27th, 1912 ? [(a) Aldebaran, Mars, Capella, Rigel. (6) a. Andromeda, fl Cassiopeia, y Pcgasi. (c) Jupiter, Antares. (c?) 'Moon,?a Ceti, a Persei.~\ 14. What heavenly bodies suitable for observation pass the meridian within one hour before sunrise at the given places and dates : (a) Lat. 40 40' N., long. 17 30' W., March 10th, 1912? (6) Lat. 15 15' N., long. 75 20' W., February 17th, 1912? (c) Lat. 45 50' S, long. 176 30' E., June 7th, 1912? (d) Lat. 25 19' S., long. 20 45' W., September 1st, 1912? [(a) Moon, Jupiter, A Scorpii. (b) a. Corona. (c) a. Andromeda, y Pegasi, a. Phanicis, a. Crucis (below Pole), (e?) Saturn, Aldebaran.~\ 217 LATITUDE BY OBSERVATION OF THE POLESTAR. 1. Prove that the altitude of the Pole is equal to the latitude of the observer. February 8th, 1912, at 9 P.M. S.A.T., in long. 24 30' W, the observed altitude of Polaris was 44 37'. Index error, 3' 40" - . Height of eye, 28 feet. [44 10-2' N.] 2. On April 15th, 1912, at 9.45 P.M. S.A.T., in long. D.R. 38 39' W., the observed altitude of the Pole Star was 47 23' 20". Index error, 1' 10" - . Height of eye, 30 feet. [48 15-4' N.] 3. On January 18th, 1912, at about 9.15 P.M. ship apparent time, in long. D.R. 54 30' W., the observed altitude of Polaris was 39 56'. Index error, 4' 30" - . Height of eye, 27 feet. [397-7'K] 4. September 19th, 1912, about 3 A.M. apparent time, in long D.R. 27 50' W., the observed altitude of Polaris was 54 26'. Index error, 2' 30" +. Height of eye, 25 feet. [53 17' N.] 5. April 3rd, 4.30 A.M., in D.R. long. 25 W., the observed altitude of the Pole Star was 41 20' 30". Index error, 1' 30" +. Height of eye, 27 feet. Deck Watch showed 8 h 35 m 2 s . Error, fast, 2 h 21 m 27 8 . [4l53'N.] 6. April 26th, 4.55 A.M., in D.R. long. 27 E., observed altitude of Polaris 33 20' 50". Index Error, 1' 40" -. Height of eye, 27 feet. Deck Watch showed ll h 37 m 31 s . Error, slow, 3 h 27 m 23 s . [33 18-8' N.] 7. September 27th, 1912, about 6.20 P.M., long. 115 4' W., observed altitude Polaris 27 20'. Deck Watch, l h 53 28 s and was O h 4 m 10 s slow. Index error, 1' 10" - . Height of eye, 30 feet. [27 25 -5' N.] 8. January 17th, 1912, about 6.30 A.M., long. 17 20' E., observed altitude Polaris 36 20'. Deck Watch, 5 U 17 m 20 s and was O h 3 m 19 s slow. Index error, 2' . Height of eye, 20 feet. [37 20 -6' N.] 9. December 10th, 1912, about 5.15 P.M., long. 37 20' W T ., observed altitude Polaris 57 20' 10". Deck Watch, 7 h 50 m 29 s and was O h 4 m 46 s fast. Index error, 1' 30" +. Height of eye, 45 feet. [56 25-1' N.] 218 10. March 9th, 1912, about 5.30 A.M., long. 46 33' W., observed altitude Polaris 24 45'. Deck Watch, 8 h 53 m 37 8 and was O h 18 m 19 8 fast. Index error, 1' +. Height of eye, 30 feet. [25 25 -9' N.] 11. September 1st, 1912, about 6.30 P.M., long. 142 27' E., observed altitude Polaris 37 49' 20". Deck Watch 8 h 56 m 47 B and was O h 2 m -39 8 slow. Index error,!' 30"-. Height of eye, 35 feet. [38 34- 3' N.] 12. July 23rd, 1912, about 3.30 A.M., long. 9 42' W., observed altitude Polaris 47 35' 30". Deck Watch, 3 h 27 m 53" and wasO h 41 m 16 9 slow. Index error, 3' 20"-. Height of eye, 28 feet. [46 25' N.] To FIX THE SHIP'S POSITION BY Two OBSERVATIONS OF THE SUN, the run in the interval being known. [N.B. In the following Questions the second observation is supposed to be worked with the position obtained by running on from the first position point.] 1. At 8 A.M. the D.R. position of a ship was 36 40' N., 74 10' W. An observation taken at 8 gave ship 6' further from Sun than the D.R. position, the S.T.B. being S. 50 E. From 8 A.M. to 11.40 A.M. the ship ran S. 60 W. 34'. At 11.40 A.M. an observation of the Sun gave ship l'"2 nearer the Sun than the D.R. position, the S.T.B. being S. 7 E. Find the position of the ship at 11.40. [36 25^' N., 74 54' W.] 2. At 8.30 A.M. a ship's D.R. position was 52 12' N., 39 36' W., and the observed Z.D. 2 '4' greater than the calculated Z.D. S.T.B., S. 49 E. At 11.30 A.M. the observed Z.D. was 6 '7' less than the calculated Z.D. S.T.B., S. 1\ E. Run between sights, N. 38 W. 31'. Find the position at 11.30. [52 3C%' N., 40 23' W.] 3. At 1 P.M., when a ship's D.R. position was 30 45' N. 67 20' W., the calculated Z.D. of Sun's centre was 21 39' 30", and the observed Z.D. 21 36' 0". S.T.B., S. 43 W. Ship is steaming S. 70 W. at 12 knots. At 4 P.M. the calculated Z.D. was 57 12', and the observed Z.D., 57 10'. S.T.B., N. 87 W. Find position of the ship at 4 P.M. [30 31f ' N., 68 4|' W.] 219 4. At 9 A.M. a ship's D.R. position was 38 43' N., 27 41' W. The true Z.D. of the Sun's centre found by observation was 73 20' 20", and the calculated Z.D., 73 24'. S.T.B., S. 47 E. At 11*40 A.M. the true Z.D. of the sun's centre found by observation was 58 4', and the calculated Z.D., 58 2'. S.T.B., S. 3i E. Run between sights, N. 67 W., 20' '5. [38 50^ N -> 27 59' W.] 5. At 6 A.M., in D.R. 56 12' N.. 30 28' W., the true altitude of the Sun's centre obtained by observation was 4 ' 2' less than the calculated altitude. S.T.B., N. 70 E. The run from 6 A.M. to noon was S. 62 E. 47', and the latitude at noon obtained from a meridian altitude of the sun was 55 40' N. Find the longitude at noon. [29 16' W.] [Also obtain the ship's position without plotting by the use of traverse tables or otherwise.] To FIX A SHIP'S POSITION from one observation of the Sun and a bearing of a shore object. 1. A ship was in D.R. 47 56' N., 5 43' W. An observation taken at that instant showed that the ship was 5' nearer the sub-solar point than the D.R. position, and the S.T.B. was S. 65 W. After running 30' N. 30 E. (true) Ushant Light (48 28' N., 5 3' W.) was observed bearing N. 40 E. (true). What was the ship's position ? [48 12f ' N., 5 22' W.] 2. The ship was by D.R. in 49 37' N., 4 33' 30" W. when an observation of the Sun (true bearing due E.) gave true Z.D. 3 ' 2' greater than calculated Z.D. After running N. 40 E. 26' the Eddystone, 50 11' N., 4 16' W, bore N. 53 W. (true). Find the ship's position and the distance from the Eddystone. [50 9' N., 4 13' W. ; dist. 3'.] 3. The ship was by D.R. in 53 30' N., 4 18' 30" W., when an observation of the Sun (S.T.B. , S. 55 W.) gave true alt. T greater than calculated altitude. After running N. 28 W. 24', Chicken Rock Light, 54 3' N., 4 49' 30" W., bore N. 11 E. true. Find the ship's position and distance from the Light. [53 52J' N., 4 53' W. ; dist. 11'.] 4. A ship steering S. 2 E. by compass at 10 knots was in D.R. 37 46' N., 9 27' W., at 4 P.M., when the true Z.D. of the Sun was 74 53' and the calculated Z.D. was 74 58' (S.T.B., S. 61 W.). At 7 P.M. Cape St. Vincent Light, 37 2' N., 9 0' W., was sighted, bearing S. 47 E. by compass. Variation, 16 30' W. Deviation, 2 30' E. Find ship's position at 7 P.M., and distance from Cape St. Vincent. [37 10' N., 9 18' W. ; dist. 18'.] 220 5. A ship steering S.W. (true) at 12 knots was in D.R. 28 42' N., 14 50' W., at 2.30 P.M., when the true altitude of Sun's centre was 47 19' 20" and the calculated altitude 47 21' 15" (S.T.B., S. 63 W.). At 5.30 P.M. Highest Point of Isleta, Grand Canary I. (28 10' N., 15 25' W.), bore S. 50 W. (true). Find ship's position at 5.30. [28 16' N., 15 16^ W.] 6. A ship was in D.R., 46 22' S., 171 23' E., when an observation of the Sun gave observed Z.D. 3' greater than calculated Z.D. (S.T.B., N. 88 E.). After running 30', N. 52 W. by compass C. Saunders, 45 54' S., 170 45' E., bore N. 85 W. by compass. Variation, 17 E. Deviation, 1 W. Find ship's position and distance of C. Saunders. [45 56' S., 170 53' E. ; dist. 6'.] 7. The ship was in D.R. 24 57' N., 36 9' E., when an observation of the Sun gave ship 1' nearer the Sun than D.R. position (S.T.B., N. 75 E.). After running 15', S. 35 W. (true) the Dredalus Lighthouse (24 55' N., 35 51' E.^was observed, bearing S. 61 W. (true). What was then the position of the ship ? [24 58' N., 35 57' E.] 8. In D.R. 32 40' S, 71 10' W., at about 7.30 A.M. the obs. alt. 0^ was 30 12' 40", when chronometer showed 10 h 17 m 25 8 (slow on G.M.T. l h 49 m 35"). Declination, 22 50' S. Equation of time, 7 m 31 8 - to A.T. Semidiam., 16' 16". Index error, 1' 20" + . Height of eye, 22 feet. S.T.B., S. 81 E. Ship then ran 17', N. 66 W. (true) when a peak in 32 41' S., 72 0' W. was sighted, bearing S. 61 W. (true). Find ship's position. [32 25' S., 71 26' W.] 9. At 9 h A.M. on October 28th, 1912, in lat. D.R. 36 39' N., long. D.R. 2 26' W., from a ship whose course was S.E. (truej, speed 14 '5 knots, Cape de Gata, lat, 36 43' K, long 2 11' 30" W., was observed to be abeam. At about ll h 45 m A.M. the obs. alt. of the Sun's L.L. was 40 33' 0". Index error, 1' 50". Height of eye 28 feet. Chronometer time, 10 h 26 m 32". Error, l h 9 m 33" slow on G.M T. Find the ship's position at ll h 45 m A.M. [36 6' N., 1 47 V W.] GEOGRAPHICAL POSITIONS OF HEAVENLY BODIES. Find to the nearest 15" of latitude and longitude the geographical positions of the following bodies at the given times : 1. The Sun, February 14th, 1912, at 4 h 20 m G.M.T. [13 19' S., 61 24' W.] 221 2. The Sun, July 27th, 1912, at 14 h 40 m G.M.T. [19 7' N., 141 34' 45" E.] 3. The Sun, September 22nd, 1912, at 21 h 5 m G.M.T. [0 1' N., 41 52' E.] 4. The Sun, December 2nd, 1912, at 16 h 47 m G.M.T. [22 3' 30" S., 105 40' 45" E.] 5. Canopus, March 6th, 1912, at 14 h 10 m G.M.T. [52 39' S., 101 25' 45" W.] 6. Betelguese, June 10th, 1912, at 2 h 17 m G.M.T. [7 23' 30" N., 25 12' 45" W.] 7. Pollux, October 18th, 1912, at 17 h 35 m G.M.T. [28 14' 15" N., 3 55' E.] 8. Mars, April 14th, 1912, at 6 h 24 m G.M.T. [25 8' 45" N., 22 57' 15" W.] 9. Venus, January 17th, 1912, at 7 h 20 m G.M.T. [20 36' 8., 151 26' 30" W.] 10. Jupiter, May 25th, 1912, at 15 h 44 m G.M.T. [21 21' 45" S., 49 27' 15" W.] 11. The Moon, September 25th, 1912, at 23 h 34 m G.M.T. [0 24' N., 175 18' 45" W.] 12. The Moon, March 24th, 1912, at 18 h 20 m G.M.T. [27 52' 30" N., 164 42' 15" E.] Section 12. ERROR AND RATE OF CHRONOMETER. 1. One of your chronometers has stopped. How do you know whether it has broken down, or has been allowed to run down ? If the latter, how will you start it again ? 2. Why, when observing for errors of chronometer, do you use the artificial horizon and not the sea horizon? 3. On June 2nd you obtain the errors of your chronometer by A.M. and P.M. absolute altitudes as follows : 8 A.M. - 2 h 03 m 47 ' 5 8 slow on G.M.T. 4p.M. - - 2 h 03 m 41'O 222 On June 9th, using the same sextant, &c., you are only able to obtain forenoon sights about 8 A.M., which give you 2 h 03 m 06 s slow on G.M.T. (a) What is the chronometer's rate ? (6) What will be the chronometer's error at midnight June 10th ? [5-93 sees., gaining ; 2 h 2 m 53 -8 s slow.] 4. It being necessary to take a chronometer on shore for observations, how would you carry it, and what special precautions would you take ? 5. What is a " Mean Comparison " ? Find the Mean Comparison from the following com- parisons : Before landing. After returning. A. - 11 22 00 A. 1 15 00 D: W. - 8 19 13-5 D.W. - 10 12 14' 7 Deck watch time of middle sight 9 10 00 6. How should a chronometer be carried ? What precautions are necessary ? 7. Describe, with rough sketch, the construction of a chronometer box. 8. How would you take accurate time with a Deck Watch ? 9. What objection is there to taking sights -for error of chronometer on only one side of the meridian ? 10. How often should 8-day chronometers be wound, and why? 11. You have three chronometers. A.'s rate is 1' 20 sees, gaining, B.'s rate 2 '02 sees, gaming, C.'s rate 5 '50 sees, losing. The daily differences between A. and B. on five consecutive days were 4, 4, 4, 4'5, 4 ; and between A. and C., 4, 3'5, 3'5, 4'0, 3 '5. Which of the chronometers would you suspect is going wrong, and to what extent approximately ? 12. What is a " Mean Comparison," and why is it necessary ? What comparisons would you take for " Sun Equal Altitudes " ? 13. Describe how you would compare a Deck Watch with a chronometer. 14. Explain the method of winding and comparing chronometers. 15. What comparisons are necessary when taking equal altitudes, one day P.M., and A.M. the next ? 223 16. You have taken equal altitudes one day, and 7 days after take the forenoon set for another set of equal altitudes.; but in the afternoon the Sun is clouded over. How will you get your rate ? 17. What rules govern you as to when to consider a chronometer as unfit ? 18. Having drawn your chronometers, state in detail how you will stow them away. Which one will you call " A." ? 19. What comparisons are necessary when taking equal alts, for rating chronometers ? 20. In order to get reliable sights for rating chronometers, what conditions must the heavenly body used fulfil ? 21. Your " B." and " C." chronometers, having run down and stopped, how will you start them both at G.M.T. ? 22. What precautions are necessary when winding a chronometer ? State reasons. 23. Describe in detail how you would pack a chronometer. 24. What comparisons are necessary for (a) Sun Equal Altitudes, ? (b) Absolutes of E. and W. Stars ? 25. How are instrumental errors avoided when taking sights for error of chronometer ? 26. On April 3rd, errors of chronometers were obtained by equal altitudes, and on the 13th, A.M. sights were obtained, but the Sun was obscured for the P.M. sights. How should the rates of the chronometers be obtained ? Explain your answer. 27. Why should chronometers be wound daily at the same time ? 28. State the various methods of finding errors of chro- nometers, in the order of their accuracy. 29. On August 25th, 1912, at G.M. noon, a chronometer was slow on G.M.T. O h 14 m 12 s . September 7th landed at Port Adelaide, and in lat. 34 50' 30" S., long. 138 28' 15" E., at about 8 h 30 m A.M. local mean time, took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. m 26 8 50 58' 30" 2 m 58" 51 10' 50" 3 m 23 8 22' 10" Index error, 2' 30"-. Obtain the error and rate of the chronometer. [O h ll m 55 s slow ; 11 sees., gaining.] 224 30. February 25th, 1912, at G.M. Noon, the chronometer was slow on G.M.T. 2 h 47 m 15". March 18th, 1912, landed in lat. 34 50' 30" S., long. 138 29' 45" E., and about 8 A.M. mean time took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 7 h 5 6 m 4 2 8 46 12' 10" 57 m 15 8 19' 10" 57 m 46 8 25' 20" index error, 3' 20" + . Obtain the error and rate of the chronometer. [2 h 49 m 16 s slow ; 5-6 sees., losing.] 31. January 13th, 1912, at G.M. Noon, the chronometer was slow on G.M.T. l h 12 m 4 s . On January 27th landed at Batavia, and about 4 P.M., in lat. 6 8' 15" S., long. 106 48' 30" E., took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 7 h 39 ra 14 s 39 m 39 8 4Qm gs Index error, 2' 30"+. Obtain the error and rate of the chronometer. [l h 12 m 52 s slow ; 3-5 sees., losing.] 32. January 18th, 1912, at G.M. Noon, the chronometer was fast on G.M.T. O h l m 1 s . February 10th landed in lat. 35 2' 15" S., long. 117 54' 15" E., and about 8 A.M. mean time took the following observations to rate chronometer : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. O h 7 m 33 8 59 10' 40" 8 m 3 8 22' 30" 8 m 37 8 36' 10" Index error, 3' 10" +. From similar observations, taken at about 4 h 30 m P.M. on the same day, the chronometer was found to be 24 seconds slow on G.M.T. Find the error of the chronometer at Local Noon on February 10th, also the rate of the chronometer. [O h O m 25 s slow ; 3 8 sees., losing.] 225 33. March 1st, 1912, at G.M. Noon, the chronometer was slow on G.M.T. 2 h 43 m 5 8 . March 12th, landed at Palermo at about 8.30 A.M. mean time, in lat. 38 6' 30" N., long. 13 21' 15" E., and took the following observations to rate chronometer : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 4 h 52 m 26 s 48 0' 0" 52 m 56 s 10' 0" 53 m 28" 20' 0" Index error, 4' 10" . From similar observations, obtained about 3 h 50 m P.M. on the same day, the error of the chronometer was found to be 2 h 43 m O 8 slow. Find the error and rate of the chronometer on March 12th. [2 b 42 m 56 -5 s slow ; -77 sees., gaining.] 34. January 7th, 1912, at G.M. Noon, the error of the chrono- meter was l h 42 m 5 s fast on G.M.T. On January 18th, about 9 A.M. mean time, landed at Valetta, lat. 35 53' 15" N., long. 14 30' 45" E., and took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 9 h 45 m 8 8 34 54' 30" 45 m 40 s 35 5' 10" 46 m 14 9 16' 20" Index error, 2' 40"+. From similar observations about 3.30 P.M. the error of the chronometer was found to be l h 42 m 45 s fast on G.M.T. Find the error and rate of the chronometer at Noon, January 18th. [l h 42 m 44 '5 s fast ; 3-6 sees., gaining.] 35. May 24th, 1912, at G.M. Noon, the chronometer was slow on G.M.T. O h 14 m 2 s . June 7th, landed in lat. 51 43' 30" N., long. 5 1' 15" W., and about 8.30 A.M. G.M.T. took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 8 h 16 m 38 9 17 m 12 8 17 m 49 8 Index error, 3' 20" -. 226 From similar observations, taken about 4 P.M. on the same day, the error of chronometer was found to be O h 12 m 55 8 S!OAV on G.M.T. Find the error of the chronometer at Local Noon, June 7th, also the rate of the chronometer. [O b 12 m 52 -5 s slow ; 4-9 sees., gaining.] 36. April 10th, 1912, about G.M. Noon, the chronometer was fast on G.M.T. O h 56 m 8 8 . May 16th, landed in lat. 33 19' 15" S., long. 115 39' 30" E., at about 8.40 A.M. mean time, and took the following observations : Obs. alts. Sun's L.L. in Times by Deck Watch. Artificial Horizon. 12 h 44 m 18 s 39 44' 40" 44 m 40 8 55' 0" 45 m 17 8 40 4' 20" Index error, 4' 20"+. About Noon the Deck Watch showed 4 h l m 57 8 when the chronometer showed 5 h 14 m 18 s , and, from observations taken about 3.10 P.M. mean time on the same day, the error of the Deck Watch was found to be 14 m 55 8 slow on G.M.T. Find error chronometer, Noon, May 16th, and rate. [O h 57 m 22 -5 8 fast ; 2- 1 sees., gaining.] 37. On June 2nd, 1912, landed in lat. 34 19' 30" S., long. 115 40' 45" E., at about 3 P.M. mean time, and took the following observations : Obs. alts. Sun's L.L. in Times by Deck Watch. Artificial Horizon. 4 h 41 m 36 a 370 33 / 30" 42 m 6 8 25' 0" 42 m 37 8 16' 10" Index error, 3' 10" . At about 2 P.M. the Deck Watch showed 3 h 42 m 35 s when chronometer showed 3 h 28 m 16 8 . On May 5th, 1912, at G.M. Noon, the chronometer was 2 h 49 m 45 s slow on G.M.T. Find the error and rate of chronometer. [2 h 49 m s slow ; 1-6 sees., gaining.] 38. January 31st, 1912, at 4 P.M. M.T., landed at Portsmouth, Dominica (15 34' 30" N., 61 27' 45" W.), and took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 8 m 7 8 47 32' 20" 8 m 37 8 47 19' 30" 9 m 25 8 46 59' 10" The index error was 3' 10" . 227 From similar observation s> taken at about 8 h 20 m A.M. on the same day, the error of the chronometer was found to be 2 h 57 m 7 s fast on G.M.T. Calculate the error of the chronometer on G-.M.T. at Noon. [2 U 57 m 3-5 s fast] 39. May 16th, 1912, landed at Lamlash (55 32' 15" N., 5 7' 30" W.) and at about 8 h 30 m A.M., G.M.T., took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 7 h 6 1U 55 8 68 2' 30" 7 m 26 s 10' 50" 8 m 15 s 24' 10" The index error was 3' 30" + . From similar observations, taken at about 4 h P.M. on the same day, the error of the chronometer was calculated to be ]> 23'* 30 9 slow on G.M.T. Calculate the error of the chronometer on G.M.T. at Noon. [1 U 23 U1 24-5 S slow.] 40. October 4th, 1912, landed at Villajuan (42 35' 30" N., 8 46' 45" W.) and at about 9 h A.M., M.T.P., took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 6 h ll m 32 s 59 4' 50" 12 m 7 9 15' 0" 12 m 54 8 28' 40" The index error was 2' 30" . From similar observations, taken at about 2 h 35 m P.M. on the same day, the error of the chronometer was calculated to be 3 h 25 m 3 s slow on G.M.T. Calculate the error of the chronometer on G.M.T. at Noon. [3 U 25 m s slow.] 41. February 10th, 1912, landed at Whitehaven, and at about 6.15 P.M. G.M.T, in lat. 54 33' 15" N., long. 3 35' 45" W., took the following observations : Obs. alts. Procyon in Deck Watch Times. Artificial Horizon. 6 h 54 m 26 8 39 19' 40" 54 m 58 s 30' 20" 55 nl 33 s 41' 50" Index error, 2' 40"-. About 6.30 P.M. comparison : Deck watch, 7 h 10 m s . Chronometer, 5 h 18 m 3 s . Find the error of the chronometer on G.M.T. [l h 15'" 3 -5 s slow.] u 5777. Q 228 42. May 18th, 1912, about 9.40 P.M. (M.T.P.), lauded at Batavia (6 8' 30" S., 106 48' 30" E.) and took observations in artificial horizon : Eegulus (W. of meridian) 74 51' 40". D. W. 2 h 38 m 4 8 . Antares (E. of meridian) 87 23' 40". 2 1 ' 45 m 16 s . Index error, 3' 10" + . Find the error of Deck Watch on G.M.T. [O h 7 m 17 s fast.] 43. March 25th, 1912, about 10 P.M. (M.T.P.), landed at Coquiinbo (29 55' 45" S., 71 21' 15" W.) and took observations in artificial horizon : Spica (E.) 82 14' 0". D.W. 2 h 19 m 37 8 . Sirius (W.) 80 46' 40". 2 h 24 m 21 8 . Index error, 1' 40" . Find the error of Deck Watch on G.M.T: [O u 19 U1 47 -5 s slow.] 44. October 16th, 1912, about 10 P.M. (M.T.P.), landed at Acapulco (16 51' N., 99 55' 30" W.') and took observations in artificial horizon : Altair (W.) 65 36' 50". D.W. 4 h 40 m 37 8 . Saturn (E.) 56 9' 0". 4 h 46 m 14 s . Index error, 50" . Find the error of Deck Watch on G.M.T. [O u 2 m 41 s fast.] 45. January 17th, 1912, about 9.50 P.M. (G.M.T.), landed at Gibraltar (36 7' 30" N., 5 21' 30" W.) and took observations in artificial horizon : Saturn 101 21' 40". D.W. 10 h 4'" 41 s . Procyon 91 11' 0". 10 h 8 m 56 8 . Index error, 3' + . Find the error of Deck Watch on G.M.T. [O h 16 ln 41 -5 s fast.] 46. December 22nd, 1912, about 9.35 P.M. (M.T.P.), landed at Port Adelaide (34 47' S., 138 30' 45" E.) and took observations in artificial horizon : ft Ceti (W.) 95 33' 20". D.W. 12 h 12 m 28*. Sirius (E.) 91 35' 50". 12 h 16 m 49 8 . Index error, 2' 10" . Find the error of Deck Watch on G.M.T. [0" 6'" 25 s slow.] 47. July 20th, 1912, about 9.25 P.M. (G.M.T.), landed at Portland (50 34' N., 2 26" W.) and took observations in artificial horizon : Arcturus 88 25' 30". D.W. 9 h 24 m 19 8 . Altair 74 4' 0". 9 h 29 ra 47 8 . Index error, 30" . Find the error of Deck Watch on G.M.T. [O u 3 m 55 -5 s fast.] 229 48. February 15th, 1912, about 9.40 P.M. (Standard Time), landed at Malta (35 53' N., 14 30' 45" E.) and took observations in artificial horizon : Regulus (E.) 90 15' 30". D.W. 8 h 24 m 11 s . Aldebaran (W.) 95 31' 30". 8 h 30 m 32 s . Index error, 2' 20" +. Find the error of Deck Watch on G.M.T. [O h 13 m 18 s slow.] 49. November 21st, 1912, about 11 P.M. (M.T.P.), landed at Port Elizabeth (33 58' S., 25 37' 15" E.) and took observations in artificial horizon : Fomalhaut (W.) 76 33' 10". D.W. 9 h 9 m 47 s Siritts tE.} 77 10' 30". 9 h 13 m 51*. Index error, 1' + . Find the error of Deck Watch on G.M.T. [O h 5 m 48 -5 s slow.] 50. August 24th, 1912, about 9.20 P.M. (Standard Time), landed at Wellington, New Zealand (41 17' S., 174 46' 30" E.) and took observations in artificial horizon : Fomalhaut (E.) 99 13' 40". D.W. 9 h 46 m 46 s . Antares (W.) 92 46' 20". 9 h 52 m 17 8 . Index error, 1' 30" +. Find the error of Deck Watch on G.M.T. [O h 3 m 51 s slow.] 51. April 20th, 1912, about 9 P.M. (Standard Time), landed at Syracuse (37 3' 30" N., 15 17' 30" E.) and took observations in artificial horizon : Arcturus (E.) 87 40' 40". D.W. 6 h 41 m 19 s . Procyon (W.) 68 5' 40". 6 h 44 m 42 s . Index error, 1' . Find the error of Deck Watch on G.M.T. [l h 17 m 48-5 slow.] 52. April 18th, 1912, lat. 39 33' 45" N., long. 2 38' 15"E., the Sun had equal altitudes at the following times by Deck Watch : (A.M.) 6 h 17 m 48 s . (P.M.) O h 19 m 32 s . Find the error of the Deck Watch on G.M.T. at noon. [2 U 30 m 20 -5 s slow.] 53. March 12th, 1912, lat. 8 30' N., long. 13 14' 15" W., the Sun had equal altitudes at the following times by Deck Watch :- (A.M.) 9 h 51 m 53 8 . (P.M.) 3 h 55* 15 8 . Find the error of the Deck Watch on G.M.T. at noon. [O h 9 m 22-5*8low.] Q 2 230 54. July 18th, 1912, lat. 17 32' S., long. 149 34' E, the Sun had equal altitudes at the following times by Deck Watch : (A.M.) l h 5 m 39 B . (P.M.) 7 h 13 m 39 8 . Find the error of the Deck Watch on G.M.T. at local noon. [2 h l m 53 s fast.] 55. April 4th, 1912, lat. 50 48' N., long. 1 6' W., the Sun had equal altitudes when a Deck Watch showed (A.M.) ll h 23 m O. (P.M.) 5 h 23 m O 8 . Find the error of the Deck Watch on G.M.T. at noon. [2 h 15 m 11 s fast,] 56. In lat. 13 45' S., long. 172 17' E, the Sun had equal altitudes as follows : P.M. March 10th, 1912. D.W. time 5 h 4 m 13 s . A.M. March llth, 1912. 9 h 12 m 18 8 . Find the error of the Deck Watch on G.M.T. at local midnight. [O h 27 m 3 s fast.] 57. Find the rates of A., B., C. from the following : Time Ball, Devonport, at 1 P.M. G.M.T., 28th June, Deck Watch showed l h 05 m 03 s . Comparisons A. l h 09 m OO 8 B. l h 13 m 00 s C. l h 21 m 30 9 . D.W. l h 10 m 15 "2 s D.W. I 11 10 m 52 '8 8 D.W. l h ll m 37 '6 8 . On arrival Bermuda (64 W.), Time Ball, July 8th at 1 P.M., Mean time place, Deck Watch showed 5 h 22 m 10'4 8 . Comparisons A. 5 h 59 m 30 s B. 6 h 03 m 30 8 C. 6 h 13 m 30 8 . D.W. 6 h 01 m 46'2 8 D.W. 6 h 02 m 30'4 8 D. W. 6 h 03 m 09 2 s . [63 seconds gaining; '02 seconds losing; 9 '41 seconds, gaining.] 58. On 1st July, error of chronometers found by stars East and West, at Portsmouth, at 9 P.M. Chronometers slow on G.M.T. : A. l h 10"" 25 s . B. 2 h 05 m 20 s . C. l h 15 m 50 s . On 8th July again found by Sun Equal Altitudes at Portsmouth : A. l h 10 m 11 s . B. 2 h 05 m 40 8 . C. l h 15 m 24 8 . What will be the error of the Deck Watch at 9 A.M. llth July when the following comparisons were taken : A. 7 h 50 m 00 s . A. 7 h 50> 15 s . A. 7 h 50 m 30 s . B. 6 h 54 m 18 8 C. 7 h 45 m 05 8 D.W. 9 h 20 m 22 8 . ? [O h 19 m -17 -2 s fast.] 231 59. On 1st April Deck Watch showed O h 32 m 42" when Time Ball dropped at Portsmouth at 1 P.M. G.M.T. Following comparisons were then taken : A. 10 h 50 m 00 s . B. ll h 46 m 00". C. 9 h ll m SO 9 . D.W. O h 35 m 50 s . D.W. O h 37 m 20 9 . D.W, O h 39 ra 25 8 . On 7th April at Portland, at 10 P.M. G.M.T., Deck Watch was found to be O h 27 m 15 s slow on G.M.T. by Stars E. and W. Following comparisons were then taken : A. 7 h 56 m 30 9 . A. 7 h 56 m 45 9 . A. 7 h 57 m OO 9 . B. 8 h 51 m 30 s . C. 6 h 14 m 32 s . D.W. 9 h 43 m 03 9 . Find eiTors and rates of A., B., and C. [A., 2 h 13 m 18 s slow, losing 1 -57 seconds.] [B., l h 18 m 18 s slow, gaining 3-14 seconds.] [C., 3 h 55 m 31 s slow, losing 2 -82 seconds.] 60. When Time Ball was dropped at Portsmouth at 1 P.M. G.M.T., on 12th December, Deck Watch showed 12 h 50 m 0'32 8 . Following comparisons were then taken : A. 12 h 10 m 00 O 9 . B. ll h 48 m 30 s . C. 12 h 59 m 00 s . D.W. 12 h 56 m 04 ' 2 9 . D.W. 12 h 57 m 15 B . D.W. 12 h 58 m 30 6 s . What will be the rates of A., B., and C., having found by stars E. and W. at Bermuda (long. 64 W.), at 9 P.M. on 25th December, that A. O h 56 m 16 '6 s slow on G.M.T. B l h 18 m 17 'I 8 J. _L A .M. 48 44' N. 5 541' W. 2 .50 A .M. 49 58' N. 3 58' W. D, R . 4 A .M. 50 3' N. 3 27V W. ] PRACTICAL NAVIGATION. EXAMPLE XIV. CHART A. courses and bearings are by compass, unless otherwise stated. Index error for all observations, 2' 20" , and height of eye, 30 feet!] January 1st, 1912, at G.M. Noon, the chronometer was l h 47 m 20" slow on G.M.T., losing daily 1*6 seconds.. January 10th at 2 P.M., ship was in D.R. lat. 51 15' N., long. 6 10' W. Course, S. 32 W. Deviation, H E. Speed, 9 knots. About 4.15 P.M. Comparison Chronometer, 3 h l m s . Deck Watch, 4 h 45 m 20 9 . 4.35 P.M. Obs. alt. Mars, 39 56'. D. W., 5 h 5 m 32 s . Obs. alt. Polaris, 52 5'. D.W., 5 h 7 ra 10 s . True bearing of Mars, S. 74 E. 10.20 P.M. Bishop Rock Light 4 points on the bow. 11 P.M. Bishop Rock Light abeam. Altered course S. 43 W. Deviation, li E. January 11 8.30 A.M. Comparison- Chronometer, 7 h 15 m O 8 . Deck Watch, 8 h 59 m 19 8 . 9 A.M. Obs. alt. Sun's U.L. 8 39' 30". D.W., 9 h 29 m 23 9 . Obs. alt. Moon's L.L. 19 48' 30". D.W., Qh 3^m 2 s Moon's True Bearing, S. 50 W. R 2 246 Lay off the various courses, and obtain the observed positions at 4.35 P.M. and 11 P.M. on January 10th, and at 9 A.M. on January llth. [4.35 P.M. 50 52' N. 6 14' W. 11 P.M. 49 54' N. 6 36' W. -9A.M. 4838'N. 737'W.] PRACTICAL NAVIGATION. EXAMPLE XV. CHART B. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 2' 10" +, and height of eye, 30 feet.] March 30th, 1912, at G.M. Noon, the chronometer was slow on G.M.T. 2 h 42 m 5 8 , and losing daily 2 3 seconds. April 15th. Comparison about 8 A.M. Chronometer, 9 h 43 m 30 s . Deck Watch, 10 h 25 m 47 8 . Deck Watch gains 3 ' 6 seconds daily. 8.30 A.M. Ship was in D.R., lat. 33 8' S., long. 113 32' E. Obs. alt. Sun's L.L. 24 21'. D.W., 10 h 56 m II 8 . Course, S. 40 E. Deviation, 1 30' E. Speed, 14 knots. Noon. Obs. mer. alt. Sun's L.L. 46 25'. Altered course S. 29 E. Deviation, 20' E. Speed, 11 knots. .5 P.M. Altered course S. 50 E. Deviation, 3 E. Speed, 11 knots. 7 P.M. Obs. alt. Achernar, 21 52'. D.W., 9 h 18 m 49". Obs. alt. ft Centauri, 34 8'. D.W., 9 h 20 m 13 8 . Lay down the various courses and distances, and obtain 'the positions at Noon and at 7 P.M. April 18th landed in lat. 35 2' 15" S., long. 117 54' 15" E., and at about 3.30 P.M., mean time, took the following observa- tions : , Obs. alts. Sun's Times by Deck Watch. L.L. in Artificial Horizon. '5 h 39 m 6 9 44 5' 30" 39 m 42 s 43 53' 40" 40 m 17 s 41' 30" At about 3 P.M. on the same day the Deck Watch showed 5 h 4 m 23 8 , when the chronometer showed 4 h 21 m 41 B . Obtain the error and rate of the chronometer. [Noon 33 45-J,' S. 114 23^' E. 7r.M. 34 47' S. 115 15' E. Chron. 2 h 42 m 38 s slow, losing 1 75 seconds daily.] 247 PRACTICAL NAVIGATION. EXAMPLE XVI. CHART B. {All courses and bearings are by compass, unless otherwise stated. Index error for all observations was 4' 20" + ; height of eye, 25 feet.] April 10th, 1912, at G.M. Noon, the chronometer was fast on G.M.T. O h 56 m 8 s , and gaining daily 2 ' 5 seconds. May 14th 8 A.M. (about) Comparison Chronometer, l h 14 m s . Deck Watch, 12 h l m 45 8 . Daily rate of Deck Watch, 3 ' 2 seconds, losing. Noon. Ship was in D.R. 35 24' S., 113 33' E. Shaped course N. 35 E. Deviation, 4 40' E. Speed, 8 knots. 4 P.M. Obs. alt. Sun's L.L. 11 34' 30". Deck Watch, 8 h 5 m 34 s . 5.40 P.M. Obs. alt. Regulus 40 46'. Deck Watch, 9 h 45 m 42 s . True Bearing of Regulus, N. 20 E. 6 P.M! Altered course N. 21 E. Deviation, 2 E. Speed, 6 knots. May 15th 8 A.M. (about) Comparison Chronometer, l h 10 m 38 8 . Deck Watch, ll h 58 m 17 s . SA.M. Obs. alt. Sun's L.L., 12 22'. Deck Watch, 12 h ]. m 51 s , and Sugarloaf Rock bore N. 77 E. Deviation, 2 E. Increased speed to 9 knots. 8.45A.M. Altered course N. 78 E. Deviation, 1 20' E. Speed, 12 knots. 11.24 A.M. Casuarina Point 4 points on the starboard bow. 11.30 A.M. Casuarina Point abeam. Lay down the various courses and distances, and obtain the positions at 5.40 P.M. on May 14th, and at 8 A.M. and 11.30 A.M. on May 15th. [5.40P.M. 34 50' S. 1H16'E. SA.M. 33 35' S. 1H56'E. 11.30 A.M. 33 17V S. 115 38' E.] 248 PRACTICAL NAVIGATION. EXAMPLE XVII. CHART B. [All courses and bearings are by compass, unless otherwise stated. The Index error for all observations is 1' 20" and height of eye, 29 feet.] May 2nd, 1912, at local mean Noon, at Sydney, long. 151 12' 30" E., the chronometer was l h 45 m 20 s slow on G.M.T., gaining 1 '8 seconds daily. May 13th 4 P.M. Ship was due South (true) of D'Entrecasteaux Point, steaming N. 55 W. Deviation, 50' W. Speed, 14 ' 8 knots. 4.15 P.M. D'Entrecasteaux Point abeam. 7.15 P.M. Cape Leeuwin Light 4 points on bow. 7.45 P.M. Cape Leeuwin Light abeam. Altered course to N. 85 W. Deviation, nil. Speed, 15 '6 knots. 8 P.M. Comparison Deck Watch, ll h 50 m s . Chronometer, 10 h 24 m 57 s . 10 P.M. Obs. alt. of Antares, 49 46' 30". Deck Watch, l h 58 m 9 s . True bearing, S. 89 E. May 14th 1 A.M. Obs. alt. of Arcturus, 28 1' 40". Deck Watch, 5 h l m 6 s . True Bearing, N. 34 W. Altered course to N. 57 E. Deviation, 1 30' E. Speed, 15 knots. 5.10 A.M. Altered course to N. 4 30' E. Deviation, 30' E. 8 A.M. Altered course to S. 17 30' W. Deviation, 115' W. Speed, 13 '2 knots. 10 A.M. Altered course to N. 78 30' E. Deviation, 1E. Comparison Deck Watch, 3 h 32 m 10 s . Chronometer, 2 h 7 m 9". 11.45 A.M. Obs. alt. Sun's L.L., 37 44'. Deck Watch, 3 h 42 m 10 s . Bearing of Sugarloaf Rock to the south of Cape Naturaliste, S.E. (true). Lay down the various courses and distances, and find the positions at 7.45 P.M. on May 13th, and at 1 A.M. and 11.45 A.M. on May 14th. [7.45P.M. 3428'S. 115 4' E. 1 A.M. 3437'S. 11323'E. 11.45 A.M. 33 30 r S. 114 56' E.] 249 PRACTICAL NAVIGATION EXAMPLE XVIII. CHART B. [All courses and bearings are by compass, unless otherwise stated. The Index error for all observations is + 2' 30" and height of eye, 25 feetj] January 13th, 1912, at G.M. Noon, the chronometer was slow on G.M.T. l h 12 m 4 8 , and losing daily 3 ' 4 seconds. January 21st, 1912, at 1 A.M., the ship was in D.R. 35 4' S M 115 57' E., course N. 45 W. Deviation, 3 40' W. Speed, 12 knots. 1.30 A.M. Obs. alt. Spica 33 46'. Chronometer, 4 h 45 m 29 8 . True Bearing of Spica, N. 79 E. 2 A.M. Obs. mer. alt. of Eegulus 42 43'. 6 A.M. Altered course N. 24 W. Deviation, 1 30' W. Speed, 12 knots. 8.30 A.M. Obs. alt. Sun's L.L. 41 24' 30". Chronometer, ll h 49 m 13 s . Sun's True Bearing, N. 87 E. 11.30A.M. Obs. alt. Sun's L.L. 75 5' 0". Chronometer, 2 h 52 m 3 s . Lay down the various courses and distances, and obtain the D.R. and observed positions at 2 A.M and 11.30 A.M. On January 27th landed at Batavia, and about 4 P.M., in lat. 6 8' 15" S., long. 106 48' 30" E., took the following observations : Obs. alts. Sun's L.L. in Chronometer Times. Artificial Horizon. 7 h 39 m 14 s 66 25' 40" 39 m 39 s 13' 10" 40 m 6 s 65 59' 50" Obtain the error and rate of the chronometer. [D.R. 2 A.M. 34 56i' S. 115 45' E- 11.30A.M. 33 31' S. 114 11' E- Obs. 2 A.M. 34 56|' S. 115 38' E. 11.30A.M. 33 19^' S. 114 12^'E. Chron. l h 12 m 56^ s slow, losing 3 '75 sees, daily.] PRACTICAL NAVIGATIONEXAMPLE XIX. CHART B. {All courses and bearings are by compass unless otherwise, stated. Index error for all observations, + 1' 30"; height of eye, 40 feet.~\ March 27th, 1912, at G.M. Noon, the chronometer was 2 h 17 m 35 s fast, losing daily 3 ' 6 seconds. April 9th at 3 P.M. the ship was in the position by D.R. 35 5' S., 116 5' E., course N. 53 W. Deviation, 1 E. Speed, 9 knots. 6 P.M. Comparison Chronometer, 12 h 35 m O 8 Deck Watch, 10 h 21 m s 250 6.15 P.M. Obs. alt. Rigel 49 C 14' 20". Deck Watch, 10 h 37 m 11 s . Obs. alt. Achemar 29 47' 10". Deck Watch, 10 h 38 m 47". True Bearing of Rigel, N. 58 W. ; of Achemar, S. 38 W. 9.40 P.M. Cape Leeuwin Lt. was 4 points on the bow. 10.20 P.M. Cape Leeuwin Lt. was abeam. 11 P.M. Altered course N. 34 W. Deviation, IF E. April 10th 8.15 A.M. Comparison : Chronometer, 2 h 51 m 30 8 Deck Watch, 12 h 37 m 30" 8.30A.M. Obs. alt. Sun's L.L. 24 11' 20" Deck Watch, 12 h 52 m 40 8 . Obs. alt. Moon's U.L. 65 12' 10". Deck Watch, 12 h 54 m 19 s . Moon's True Bearing, N. 82 W. Lay down the various courses and distances, and obtain the observed positions at 6.15 P.M. and 10.20 P.M. on April 9th, and at 8.30 A.M. on April 10th. Compute the D.R. position at Noon on April 10th. [6.15P.M. 34 48' S. 115 38' E. 10.20P.M. 34 27' S. 115 5' E. 8.30A.M. 33 12^' S. 113 48' E. D.R. Noon. 32 47j' S. 113 25' E.] PRACTICAL NAVIGATION. EXAMPLE XX. CHART B. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 1' ; height of eye, 30 feetJ] July 7th, 1912, at G.M. Noon, the chronometer was 2 h 16 m 41 s fast, gaining daily 2 ' 4 seconds. July 20th, at 4 P.M., the D.R. position was 33 5' S., 113 25' E., course, S. 33 E. Deviation, 2 W. Speed, 10 knots. Obs. alt. Sun's L.L. 11 36' 40". D.W., 8 h 27 m 14". Obs. alt. Moon's U.L. 59 31' 0". D.W., 8 h 29 m 3 s . True Bearing of Sun, N. 56 W. ; of Moon, N. 33 E. 4.15 P.M. Comparison Chronometer, ll h 13 m s . Deck Watch, 8 h 52 m 32 s . July 21st 1.30 A.M. Cape Leeuwin Lt. was 4 points on the bow. 2.30 A.M. ,, ,, abeam. Altered course S. 51 E. Deviation, 1 W. Reduced speed to 9 knots. 251 6.15 A.M. Comparison Chronometer, l h 2 m s . Deck Watch, 10 h 41 m 28 s . 6.30. A.M. Obs. alt. Saturn 33 18' 30". D.W. 10 h 56 m 49". Obs. alt. Canopus 47 9' 20". D.W. 10 h 58 m 33 s . True Bearing of Saturn, N. 24 E. ; of Canopus, S. 48 E. Lay down the various courses and distances, and obtain the observed positions at 4 P.M. on July 20th, and at 2.30 A.M. and 6.30 A.M. on July 21st. Set a course at 6.30 A.M. to clear S.W. Reefs by 5 miles. Deviation, 1 W. [4 P.M. 33 7' S. 11332i'E. 2.30A.M. 3428i'S. 115 (/E. 6.30A.M. 34 49' S. 115 35' E. Comp. co., S. 60 E.] PRACTICAL NAVIGATION. EXAMPLE XXI. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, + 1' 20" ; height of eye, 30 feet.] On July 29th, 1912, at G.M. Noon, the chronometer was Qh ^jm 353 s i OWj losing 3 seconds daily. August llth, 1912, at 7 A.M. (S.A.T.) from the position Sheep Head bore N. 53 E. (true). Mizen Head bore S. 53 E. (true). Shaped course S. 13 W. Deviation, 1 W. Speed, 12 knots. 11 A.M. Comparison Chronometer, 10 h 57 m s . Deck Watch, ll h 47 m 31 s . 11.30 A.M. Obs. alt. 0. 54 0' 10". Deck Watch, 12 h 17 m 9 s . Sun's True bearing, S. 13 E. 3.30 P.M. Obs. alt. 35 17' 30". Deck Watch, 4 h 15 m 11 s . Sun's True Bearing, S. 70 W. 4 P.M. Altered course S. 20 W. Deviation, 1 W. 5 P.M. Sighted " B " Squadron. Courses and speeds as requisite for Battle Exercise. 6.30 P.M. Finished Battle Exercise. " B " squadron took station. 7 P.M. Shaped course S. 78 E. Deviation, 2 E. Speed, 10 knots. 8 P.M. Comparison Chronometer, 7 h 56 m s . Deck Watch, 8 h 46 m 33 s . 8.15 P.M. Obs. alt. Arcturus 38 55' 10". Deck Watch, Qh Qm 29^ Obs. alt, Vega 75 46' 20". Deck Watch, gh 9m 12 s True Bearing of Arcturus S. 74 W. 9 P.M. Altered course S. 70 E. Deviation, H E. 252 August 12th 6.30 A.M. Bishop Rock 4 points on the bow. 7 A.M. Bishop Rock abeam. Lay down the various bearings, courses, and distances, aiid find the observed positions at 3.30 P.M. and 8.15 P.M. on August llth, and at 7 A.M. on August 12th. [3.30 P.M. 49 51^' N. 9 30' W. 8.15 P.M. 49 47^' N. 9 7' W. 7 A.M. 49 48' N. 6 27' W.J PRACTICAL NAVIGATION.- EXAMPLE XXII. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error for all obsei'vations, 50" ; height of eye, Wfeet.] September 12th, 1912, at G.M. Noon, the Chronometer was 2 h 3 m 49 s fast, gaining 1'7 seconds daily. September 25th, 1912, at 4 P.M., position by account was lat. 49 25' N., long. 8 30' W. Course, N. 54 E. Deviation, 2 E. Speed, 10 knots. 6 P.M. Comparison Chronometer, 8 h 32 m O 8 . Deck Watch, 6 h 24 m 22 s . 6.22 P.M. Obs. mer. alt. Vega 79 4' 10". 6.24P.M. Obs. alt. Arcturus 30 21' 20". Deck Watch, 6 h 48 m 27". September 26th 5 A.M. Comparison Chronometer, 7 h 33 m O 8 . Deck Watch, 5 h 25 m 23 s . 5.15A.M. Obs. alt. Polaris 51 51' 40". Deck Watch, 5 h 39 m 15 8 . Obs. alt. Regulus 24 24' 40". Deck Watch, 5 h 40 m 57 s . True Bearing of Regulus S. 80 E. Altered course N. 41 E. Deviation, 2 E. Speed, 12 knots. 8 A.M. Smalls Lighthouse 4 points on the bow. 8.35 A.M. Smalls Lighthouse abeam. Lay%l^wn the various courses and distances, and find the observed "^positions at 6.22 P.M. on September 25th, and at 5.15 A.M. and 9 A.M. on September 26th. " [6.22 P.M. 49 45' N. 8 11' W. 5.15A.M. 5111VN. 624'W. 9A.M. 51 51' N. 547'W.] 253 PRACTICAL NAVIGATION. EXAMPLE XXIII. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error 'for all observations, - V 20" ; height of eye, 33 feet.] March 7th, 1912, at G.M. Noon, the chronometer was 3 h 6 ra 40 s fast, losing 3 ' 6 seconds daily. March 21st, at 5 P.M., the position by account was lat. 50 N., long. 10 W. Course, N. 73 E. Deviation, 2F E. Speed, 12 knots. 6.30 P.M. Comparison Chronometer, 10 h 26 s . Deck Watch, 7 k 23 m 31 s . 6.45P.M. Obs. alt. Procijon 44 31' 30". Deck Watch, 7 h 35 m 44 s . Obs. alt. Moon's L.L. 33 3' 0". Deck Watch, 7 h 37 in 21s. True Bearing of Procyon S. 16 E. 8 P.M. Increased speed to 13 knots. March 22nd 4Jt5 A.M. Comparison- Chronometer, 8 h 40 m s . Deck Watch, 5 h 37 m 32 s . 5 A.M. Obs. alt. a Cygni 56 48' 10 /; . Deck Watch, 5 L 50 m 29 s . Obs. alt. Polaris 50 54' 0". Deck Watch, 5 h 52 m 178> True Bearing of a Cygni N. 80 E. 6A.M. Altered course N. 33 E. Deviation, 2| E. Speed, 12 knots. 9 A.M. Tuskar Rock 4 points 011 the bow. 9.30 A.M. Tuskar Rock abeam. Lay down the various courses and distances, and find the observed positions at 6.45 P.M. on March 21st and at 5 A.M. and 9.30 A.M. on March 22nd. [6.45 I>.M. 50 5'N. .9 29' W. 5 A.M. 51 23V N. 6 37V W - 9.30A.M. 52 -ll'N. 6 4' W.] 254 PRACTICAL NAVIGATION. EXAMPLE XXIV. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 2' ; height of eye, 20 feet.] July 10th, 1912, at G.M. Noon, the chronometer was l h 47 m 9 8 fast, losing 3 ' 2 seconds daily. July 17th, at 11.30 P.M., shaped course S. 67 E. Deviation, 1 E. Speed, 13 knots : from the position Roche Point Light bore N. 8 W. Daunt Rock Light bore N. 85 W. Estimated current till 3 A.M., knot E.N.E. (true). July 18th 3.15 A.M. Comparison Chronometer, 5 h 36 m s . Deck Watch, 3 h 53 m 2 s . 3.30A.M. Obs. alt. Saturn 27 13' 40". Deck Watch, 4 h 8 m 1C 8 . Obs. alt. Markdb 52 58' 40". Deck Watch, 4 h 9 m 49 s . True bearing of Saturn, S. 86 E. 7.15 A.M. St. Ann's Head Lighthouse 4 points on the bow. 7.45 A.M. ,, abeam. Stopped engines. Used engines as necessary to maintain this position till 4 P.M., when convoy took station. 4 P.M. Shaped course S. 52 W. Deviation, 1| W. Speed, 10 knots. Estimated current till 10 P.M., f knot N.E. (true). 8 P.M. Comparison Chronometer, 10 h 20 m s . Deck Watch, 8 h 37 m 6 s . 8.30 P.M. Obs. alt. Moon's L.L. 14 14' 40". Deck Watch, 9 h 7 m 47 8 ' Moon's true bearing, S. 76 W. 9 P.M. Obs. alt. Polaris 50 19' 50". Deck Watch, 9 h 38 m 148 Lay down the various courses and distances, and obtain the D.R. and observed positions at 3.30 A.M. and 9 P.M. [D.E. 3.30 A.M. 51 40 N. 6 47' W. 9 P.M. 50 53' N. 5 52' W. Obs. 3.30 A.M. 51 38V N. 6 46' W. 9 P.M. 50 56 f N. 5 53' W.] 255 PRACTICAL NAVIGATION. EXAMPLE XXV. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 2' 20"; height of eye, 26 feet-.] October 3rd, 1912, at G.M. Noon, the chronometer was 3 h 14 m 47 s slow, gaining 2 ' 4 seconds daily. October 21st, at 4 A.M., position by account was lat. 49 20' N., long. 9 50" W. Course, N. 81 E. Deviation, 2 E. Speed, 13 knots. 8 A.M. Comparison Chronometer, 5 h 9 m O 9 . Deck Watch, 8 h 27 m 18 s . 8.30 A.M. Obs. alt. Sun's L.L. 14 28' 30". 8 h 56 m 19 s . Sun's true bearing S. 53 E. 11.40 A.M. Obs. alt. Sun's L.L. 29 6' 20". 12 h 5 m 57 8 . Noon. Altered course N. 82 E. Deviation, 2 E. 5.45 P.M. Comparison Chronometer, 2 h 48 m s . Deck Watch, 6 h 6 m 15 s . 6 P.M. Obs. alt. Moon's U.L. 17 19' 30". Deck Watch, 6 h 2im 17s. Moon's True bearing, S. 45 E. Obs. alt. Altair 48 V 20". Deck Watch, gh 22 m 57 s True bearing of Altair, S. 2 W. 10 P.M. Lundy Island Light bore N. 43 E. Hartland Point Light bore S. 65 E. Lay down the various courses and bearings, and obtain the observed positions at Noon, 6 P.M., and 10 P.M. [Noon. 50 3i' N. 7 41' W. 6 P.M. 50 3S f N. 5 58' W. 10 P.M. 51 3' N. 4 46' W.] 256 PRACTICAL NAVIGATION. EXAMPLE XXVI. CHART C. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 1' 30" ; height of eye^ 40 feet] January 14th, 1912, at G.M. Noon, the chronometer was l h 4 m 37 s fast, gaining 2 ' 5 seconds daily. January 27th, at 5.50 A.M. Eddystone Light bore N. (true), distant 3 miles. Course, S. 87 W. Deviation, 1 W. Speed, 12 knots. 8.35 A.M. Lizard Light 4 points on the bow. 8.50 A.M. Lizard Light abeam. Altered course N. 61 W. Deviation, W. 11.20 A.M. Altered course N. 33 E. Deviation, 1 E. Noon. Seven Stones Light bore S. 65 W. Longships Light bore S. 31 E. 4 P.M. Comparison Chronometer, 5 h 4 m O 8 . Deck Watch, 3 h 56 m 3 s . 4.10 P.M. Obs. alt. Sun's U.L. 6 35' 40". 4 h 6 m 24 s . Obs. alt. Moon's U.L. 45 21' 10", 4 h 8 m 10 s . True Bearing of Sun, S. 50 W., of Moon, S. 51 E. 8.15 P.M. St. Ann's Head Light bore N.W. (true), distant 2 miles. Used engines as necessary to maintain this position. 9 P.M. Shaped course N. 77 W. Deviation, 1 W. Speed, 10 knots. January 28th 7.30 A.M. Comparison Chronometer, 8 h 34 m s . Deck Watch, 7 k 26 m 1 s . 7.40 A.M. Obs. alt. ft Leonis 32 5' 50". 7 h 3701 26 8 . Obs. alt. Jupiter 16 29' 50". 7 h 39 m 15". True bearing of ft Leonis, S. 73 W. ; of Jupiter, S. 14 E. Put clocks back 25 m to Dublin Mean Time. Lay down the various courses, bearings, and distances, and obtain the observed positions at Noon and 4.10 P.M. 011 January 27th, and at 7.40 A.M. on January 28th. Estimate the time by ship's clock when the Fastnet will be abeam. [Noon. 50 10' N. 5 53| f w - 4.10 P.M. 50 54V N. 5 29V W. 7.40A.M. 51 27^ N. 7 55' W. Fastnet abeam, 1.40 P.M.] 257 PRACTICAL NAVIGATION. EXAMPLE XXVII. CHART D. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 2'; height of eye, 20 feet.] November 9th, 1912, at G.M. Noon, the chronometer was l h 2 m 46 s slow, gaining 4' 8 seconds daily. November loth, at 6 P.M., the position by account was lat. 40 S., long. 173 45' E. Course, S. 45 E. Deviation, nil. Speed, 15 knots. 8.30 P.M. Comparison Chronometer, 7 h 30 ra s . Deck Watch, 8 h 36 m 58 s . 8.40P.M. Obs. alt. y Pegasi 34 50' 0". Deck Watch, 8 h 49 m I 8 . Star's true bearing, N. 3F E. Stephen's I. Light bore S. 28 W. 11 P.M. Altered course S. 9 E. Deviation, 1 W. November 16th 2 A.M. Cape Campbell Light abeam. Altered course S. 49 E. Deviation, nil. 4.15 A.M. Obs. mer. alt. Procyon 42 29' 30". Obs. alt. Spica 10 23' 40". Deck Watch, 4k 24 m 45 s . True bearing of Spica, S. 85 E. . 7.45 A.M. Comparison Chronometer, 6 h 46 m O 8 . Deck Watch, 7 h 53 m 3 s . 8 A.M. Obs. alt. Sim's L.L. 34 22' 40". Deck Watch, 8 h 8 m 20 s . Sun's true bearing, N. 84 E. 11.30 A.M. Obs. alt. Sun's L.L. 64 6' 40". Deck Watch, lib. 37 m 29 s Lay down the various courses and distances, and obtain the observed positions at 8.40 P.M. on November 15th, and at 4.15 A.M. and Noon on November 16th. [8.40 P.M. 10 31' S. 174 13' E. 4.15A.M. 42 11' S. 174 52' E. Noon. 43 47' S. 176 21' E.] 258 PRACTICAL NAVIGATION. EXAMPLE XXVHI. CHART D. [All courses aud bearings are by compass, unless other-wise stated. Index error for all observations, V 30" ; height of eye, 32 feet.] November 18th, 1912, at G.M. Noon, the chronometer was 2 h 7 m 49 s slow, losing 3 ' 2 seconds daily. December 3rd, at 5 P.M., the position by account was lat. 44 S., long. 176 15' E. Course, N. 47 W. Deviation, 2 E. Speed, 11 knots. 6.30 P.M. Comparison Chronometer, 4 h 27 m s . Deck Watch, 6 h 37 m 53 s . 6.45 P.M. Obs. alt. Sun's L.L. 7 20' 10". Deck Watch, 6 h 53 m 9 8 . Sun's true bearing, S. 66 W. 8.15 P.M. Obs. alt. a Andromedae 17 13' 20". Deck Watch, 8 h 22 m 47 9 . Star's true bearing, N. 11 W. December 4th 3.30 A.M. Comparison Chronometer, l h 24 m O 8 . Deck Watch, 3 h 34 m 56 s . 3.45A.M. Obs. alt. Canopus 68 3' 20". Deck Watch, 3 h 53 m 10 s . Obs. alt. Sirius 56 47' 30". Deck Watch, 3 h 54 m 47 s . True bearing of Sirius, N. 47 W. 7 A.M. Cape Campbell Light 4 points on the bow. 7.45 A.M. Cape Campbell Light abeam. Altered course N. 9 W. Speed, 12 knots. Deviation, nil. Lay down the various courses and distances, and obtain the observed positions at 8.15 P.M. on December 3rd, and at 3.45 A.M. and 7.45 A.M. on December 4th. Estimate the time when the Brothers Light will be 4 points on the bow. [8.15 P.M. 43 31' S. 175 51V E - 3.45 A.M. 42 20' S. 174 55 r E. 7.45 A.M. 41 39' S. 174 27^' E. Brothers Light cm the bow abt. 10.10 A.M.] 259 PRACTICAL NAVIGATION. EXAMPLE XXIX. CHART D. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, 4- V 20" ; height of eye, 33 feetJ] August 19th, 1912, at G.M. Noon, the chronometer was 3 h 16 m 24 s slow, losing 3'2 seconds daily. August 24th, at 6 P.M., the position by account was lat. 44 S., long. 175 E. Course, N. 22 W. Deviation, nil. Speed, 9 knots. Comparison Chronometer, 3 h 6 m s . Deck Watch, 6 h 25 m 10 s . 6.6 P.M. Obs. mer. alt. Jupiter 67 1' 50". Obs. alt. Spica 39 18' 30". Deck Watch, 6 h 33 m 24*. True bearing of Spica, N. 63 W. August 25th 5.45 A.M. Comparison Chronometer, 2 h 51 m s . Deck Watch, 6 L 10 ra 9 s . 6 A.M. Obs. alt. Aldebaran 31 27' 0". Deck Watch, 6 h 25m i 7 8 True bearing of Aldebaran N. 5 E. 7.30 A.M. Obs. alt. Sun's L.L. 8 24' 20". Deck Watch, 7 h 54 m 49 s . Altered course to N. 20 W. Deviation, nil. Increased speed to 11 knots. Noon. Brothers Lighthouse bore N. 88 W. 12.30 P.M. bore S. 35 W. Lay down the various courses and distances, and obtain the observed positions at 6.6 P.M. on August 24th, and at 7.30 A.M. and 12.30 P.M. on August 25th. Set a course at 12.30 P.M. to clear Stephen's Island by 5 miles, allowing deviation 1 W. [6.6P.M. 43 57' S. 174f>8'E. 7.30 A.M. 41 58V S. 171 37' E. 12.30P.M. 41 2V S. 174 34' E. Course, N. 52i 3 W.] u 5777. 260 PRACTICAL NAVIGATION. EXAMPLE XXX. CHART L). [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, + 30"; height of eye, 26 feet.] March 10th, 1912, at G.M. Noon, the chronometer was 4 h 7 m 24 s fast, gaining 3 seconds daily. March 23rd, at Noon, 'the position by observation was lat. 41 14' S., long. 176 17' E, Course, S. 35 W. Deviation, 1| W. Speed, 11 knots. 3.30 P.M. Comparison Chronometer, 8 h O m s . Deck Watch, 3 h 48 m 42 s . 4 P.M. Obs. alt. Sun's L.L. 21 41' 50". Deck Watch, 4 h 18 ra 11 s . Cape Palliser Lighthouse bore N. 69 W. Altered course S. 23 W. Deviation, U W. 6 P.M. Reduced speed to 10 knots. 6.45 P.M. Obs. alt. Moon's L.L. 10 14' 40". Deck Watch, Jh 9m 948 Obs. "alt. Achernar 40 13' 0". Deck Watch, 7 h 3 m 50 s . True bearing of Moon, N. 46 W. ; of Achernar, S. 43 W. March 24th. 5 A.M. Comparison Chronometer, 9 h 30 m O 9 . Deck Watch, 5 h 18 m 38 8 . 5.10 A.M. Obs. alt. a Centauri 61 22' 0". 5 h 27 m 47 s . Obs. alt. a Gruis 40 34' 0". 5 h 29 m 3 s . True bearing of a Centauri, S. 40 W. ; of a Gruis, S. 58 E. Increased speed to 12 knots. Lay down the various courses and distances, and obtain the observed positions at 4 P.M. and 6.45 P.M. on March 23rd, and at 5.10 A.M. on March 24th, and estimate the time of arrival at the entrance to Akaroa harbour. [4p.M. 41 43^ S. 17533'E. 6.45p.M. 42 7' S. 175 7' E. 5.10 A.M. 43 27' S. 173 39' E. Arrive about 8.35 A.M.] 261 PRACTICAL NAVIGATION. EXAMPLE XXXI. CHART D. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, + 1' 20" ; height of eye, 41 feet.'] September 22nd, 1912, at G.M. Noon, the chronometer was 2 h 17 m 46 s fast, losing 2 '4 seconds daily. October 3rd, at 4 A.M., the D.R. position was lat. 40 5' S., 173 10' E. Course, S. 69 E. Deviation, H E. Speed, 10 knots. 4.45 A.M. Comparison : Chronometer, 7 h 20 m O 9 . Deck Watch, 5 h 5 m 49 8 . 5 A.M. Obs. mer. alt. Moon's U.I,. 20 59' 0". Obs. alt. Fomalhaut 11 34' 30". Deck Watch, 5 h 22 m 10 s . True Bearing of Fomalhaut, S. 60 W. 9.30 A.M. Altered course S. 48 E. Deviation, E. 1 P.M. Altered course S. 15 E. Deviation, 1 W. 3.30 P.M. Altered course S. 49 E. Deviation, i E. Increased speed to 12 knots. 6.30 P.M. Comparison Chronometer, 9 h 3 m s . Deck Watch, 6 h 48 m 56 s . 6.50P.M. Obs. alt. Venus 14 32' 10". Deck Watch, 7h Qm 27s. Obs. alt. Altair 39 20' 30". Deck Watch, 7 h 10 m 56 s . True Bearing of Venus, S. 86 W. Lay down the various courses and distances, and obtain the observed positions at 5 A.M. and 6.50 P.M., and the Compass bearing of Stephen's I. Light at 9.30 A.M. [5A.M. 40 9' S. 17320'E. 6.50P.M. 42 2^' S. 175 6' E. Bearing, 8. 28 W.] PRACTICAL NAVIGATION. EXAMPLE XXXII. CHART D. [All courses and bearings are by compass, unless otherwise stated. Index error for all observations, - 1' ; height oj eye, 45 feet.] February 1st, 1912, at G.M. Noon, the chronometer was 2 h 17 m 47 s fast, losing 4*2 seconds daily. Ship's clock is keeping New Zealand Standard Time. February 10th, at 7 P.M., the position by account was lat. 40 S., long. 173 E. Course, S. 76 E. Deviation, l-- E. Speed, 18 knots. 7.30 P.M. Comparison- Chronometer, 10 h 17 m s . Deck Watch, 7 h 56 m 23 s .7.45P.M. Obs. alt. Mars 25 28' 50". Deck Watch, 8 h ll m 2 s . Obs. alt. Procyon 33 5' 30". Deck Watch, 8 h 12 m 59 s . True bearing of Mars, N. 15 W. ; of Proetjoti, N. 47 E. 10.20 P.M. Stephen's Island Light bore S. 19 E. 10.40 P.M. S. 34 W. Midnight. Altered course to S. 10 E. Deviation, 1 W. February llth 2.30 A.M. Cape Campbell Light 4 points on the bow. 3 A.M. ,, ,, ,, abeam. Altered course to S. 7 W. Deviation, H W. 6.30 A.M. Comparison Chronometer, 9 h 16 m s . Deck Watch, 6 h 55 m 28 s . 6.45 A.M. Obs. alt. Moon's U.L. 69 33' 20". Deck Watch, 7 h 12 m 37 8 Obs. alt, Sun's L.L. 16 50' 0". Deck Watch, 7 h 14 m 25 8 Moon's true bearing, N. 16 W. 10 A.M. Obs. alt. Sun's L.L. 49 45' 50". Deck Watch, 10 h 28 in 12 8 . East Head bore S. 54 W. Laj 7 down the various courses and bearings, and obtain the observed positions at 7.45 P.M. and 10.40 P.M. on February 10th, and at 3 A.M., 6.45A.M., and 10 A.M. on February llth. Compute the D.R. position at Noon on February llth. [7.45P.M. 40 9' S. 17311'E. 10.40 P.M. 40 36^' S. 174 8' E. SA.M. 41 44' S. 17430'E. 6.45A.M. 42 48' S. 17353'E. 10 A.M. 43 42' S. 173 22' E. D.R. Noon. 44 15V S. 173 4' E.] 263 STATION POINTER. Obtain positions from the following fixes, explaining fully why those marked with a * are faulty : CHART A. I. Lundy Island Light. 37 Bull Point. 32 Hartland Point. [513VN. ; 4 47V W.] 2. Bull Point. 59 Hartland Point. 38 Lundy Island Light. [Indeterminate.] 3. Tintagel Head. 62 Pentire Head. 26i Trevose Head. [50 42' N. ; 4 58' W.] 4. Wolf Light. 43 Longships Light. 14^ Runnelstone Buoy. [49 53V N. ; 5 39' W.] *5. Wolf Light. 84 Longships Light. 33 Runnelstone Buoy. [Indeterminate.] 6. North Manacle Rock. 48 St. Anthony Point. 93^ Dodman Point. [oO 7' X. ; 4 52^ W -l 7. Eddystone. 94| Rame Head. 43 Mewstone. [50 14' Js. ; 4 9' W.] *8. Eddystone. 59 Rame Head. 25 Mewstone. [Indeterminate.] 9. Bolt Tail. 20 Bolt Head. 26 Start Point. [50 4' N. ; 3 49' W.] 10. Prawle Point. 11 Start Point. 38i Mewstone. [50 8' N. ; 3 24' W.] CHART C. II. Sheep Head. 15 Mizen Head. 4<) J Fastnet. [5 1 3 18V N. ; 9 57^ w -] 12. Galley Head. 45 Seven Heads. 44 Kinsale Head. [51 27' X. ; 8 38V VV 13. Daunt Rock Lightship. 43 Roche Point. 88 Bally- cottin Island Light. [51 43' N. ; 8 6V VV.] 14. Daunt Rock Lightship. 22 Roche Point. 44 Bally- cottin Island Light. [Indeterminate.] 15. North extreme, Capel Island. 35 Ram Head. 41 Mine Head Light. [51 51V ^. ; 7 C 34V W.] 264 16. North Bishop. 15 South Bishop Light. 110 Smalls Light. [51 49V N - ; 5 39 V W 17. St. Govan's Head. 6?i Caldy Island Light, 70i Worms Head Lightship. [51 30' N. ; 4 44^' W.] 18. Brown Willey. 51 Pentire Head. 31 Trevose Head Light. [50 41' N. ; 4 58' W.] 19. Longships Light. 62^ Wolf Light. 95 Seven Stones Lightship. [50 5V N. ; 5 52' W.] 20. Wolf Light. 46 Longships Light. 107 Lizard Light. [49 55' N. ; 5 36V W. 265 PART IV. -HEAT AND STEAM. HEAT AND STEAM. 1. What is the generally accepted explanation of the nature of heat and temperature ? Define the " Specific Heat " of a substance. If the specific heat of air be '2375, how many cubic feet of air can be raised one degree in temperature by one British Thermal Unit, the specific volume of air being 12 ' 38 cubic feet at 32 F. ? [52-1 c. ft.] 2. Describe the graduation of a thermometer and name the three scales used. Explain how readings on the Centigrade scale can be converted into those on the Fahrenheit scale. Convert the following : 300 F., F. into Centigrade readings. 15 C., 180 C. into Fahrenheit readings. At what temperature is the reading on a Fahrenheit ther- mometer a number twice as large as that simultaneously taken on a Centigrade thermometer ? [149 C. ; 17'7 C. ; 59 F. ; 356 F. ; 320 F. = 160 C.] 3. Describe a method of determining the linear coefficient of expansion of a metal rod, and show what is the connection between the linear coefficient and the cubical coefficient of expansion. The coefficient of linear expansion of steel is 0' 000012 per 1 C. Calculate, approximately, the increase in volume of a cylindrical shaft when it is plunged into boiling water, if at C. it is 10 feet in length and 3 inches in diameter. [3 - 05 c. ins.] 4. State Joule's Law with reference to the internal energy of a gas. Imagine a vertical cylinder of area 3 square feet, closed at the bottom and open at the top, fitted with an airtight piston, and that 1 pound of air is confined in the space between the piston and the bottom of the cylinder. The weight of the piston is 900 pounds, and the specific heat of air under constant pressure is ' 24. If the air in the cylinder is originally at 60 F., find the gain in internal energy if it be heated to 120 F., given that 1 pound of air occupies 12 ' 38 cubic feet at atmospheric pressure (14'7 lbs./in. 2 ) and at 32 F. [10-4 B.T.U.] 266 5. An air-cliainber of capacity 10 cubic feet is filled witli air at a pressure 2,000 lbs./in. 2 (gauge) and at temperature 80 F. Given that 1 Ib. of air occupies 12 ' 38 cubic feet at atmo- spheric pressure and at 32 F., estimate the weight of air in the chamber. [101 pounds.] 6. State Charles's Law on the behaviour of a gas when heat is applied to it. A quantity of gas which occupies 100 cubic feet at 25 lbs./in. 2 pressure (gauge) and temperature fiO F. is found to occupy 39*1 cubic feet at 105 lbs./in. 2 (gauge) and 150 F. Calculate the absolute zero of temperature as measured on the Fahrenheit scale. [- 461 F.] 7. State the laws of Boyle and Charles for the expansion of gases, and express them by an equation. A cubic foot of hydrogen at atmospheric pressure and 32 F. weighs 0'0056 pound. Calculate the constant in the above equation. [767.] 8. A gas-engine, with cylinder 8 inches diameter and 16 inches stroke,- compresses to 60 pounds per square inch by gauge. Find the clearance volume if the gas is compressed according to the law pu 1 ' 4 = constant. Take atmospheric pres- sure as 15 pounds per square inch. Find also the temperature at the end of compression if the initial temperature is 60 F. [373 cub. ius. : 364 F.] 9. State the Law connecting the pressure, volume and temperature of a given mass of gas. A torpedo has an air-chamber of 10 cubic feet capacity ; recently the air-pressure of the charge has been increased from 2,000 to 2,500 lbs./in. 2 How many more pounds of air can be carried under the latter condition at temperature 60 F. ? [26 Ibs.] 10. The ratio of compression in the Diesel engine is about 15 ; assuming the compression to be adiabatic, estimate the temperature produced from air originally at 60 F. Adiabatic law is pt? 1 ' 408 = constant. 11. State Charles's Law for a permanent gas. Given that 1 pound of air occupies 12 ' 38 cubic feet at 32 F. and 15 pounds pressure per square inch absolute, calculate the weight of air contained in an air-reservoir of 25 cubic feet capacity at 80 F. and 2,000 pounds pressure per square inch (by gauge). [247'1 Ibs.] 267 12. A certain Hornsby-Akroyd engine has the following dimensions: stroke, 17 inches; diameter, 14 ' 5 inches ; clear- ance volume, 1,170 cubic inches. Find the pressure of the charge at the end of compression, assuming atmospheric pressure at the beginning and the operation carried out so that PF 1 ' 3 is a constant. [72-151bs,/in. 2 ] 13. A piston, 12 inches diameter, compresses air in a cylinder from atmospheric pressure to 100 pounds per square inch absolute pressure. If the length of the cylinder barrel be initially 12 inches, through what distance must the piston move, if the air be compressed in such a manner that PV may be taken as constant ; and what work will be done by the piston during the compression, if it is open to the atmosphere on the other side. [10-23 ins. : 1,770 foot-lbs.] 14. A ball of copper (specific heat ' 095) of weight 5 pounds is heated in a furnace and then dropped into a gallon of water at temperature 50 F. ; the temperature of the water rises to 120 F. Find the temperature of the furnace. [1594F.] 15. Define the " specific heat " of a body and state Regnault's Law for the specific heat of a gas. If 1 pound of coal gives out 14,500 B.T.U.s in burning, and it requires 24 pounds of air for complete combustion, estimate the mean rise in temperature of products of combustion if there were no losses of any kind. Specific heat of air (fe p ) = ' 24. [2416F.] 16. Define the terms Specific Heat and Latent Heat. Boiler steam at a temperature of 372 F. is led into an iron tank containing 5,160 pounds of water. If the amount of steam condensed is 115 pounds and the initial and final temperatures of water and tank 49 '6 and 74 '5 F. respectively, find the dryness fraction of the steam, the weight of tank being 876 pounds and the specific heat of iron j. [984.] 17. What is meant by specific heat ? If at high temperatures the specific heat of iron may be taken as '1053 + '000071 t (Cent.), what is the temperature of a red-hot iron ball weighing a kilogramme which, when plunged into 10 kilogrammes of water, raises its temperature from 12 Cent, to 24 Cent. ? [894 F.] 18. State Regnault's Law on the Specific Heat of a gas. If the relative weights of air and oil in the explosive charge of a Diesel Engine are as 80 : 1, estimate the rise in temperature 268 on ignition, assuming the specific heat at constant pressure for air to be '2375, and calorific value of oil = 20,000 B.T.U. per Ib. [1040 F.] 19. Define the Specific Heat of a substance. Distinguish between the specific heat of a gas at constant pressure and at constant volume, and illustrate your answer by reference to a steam superheater, a Diesel engine, and petrol motor engine cycles. 20. A shaft of 6 inches in diameter rests in bearings at its ends, and is acted on by forces which cause a pressure of ] ,000 pounds on each bearing. If the shaft makes 100 revolu- tions per minute and the coefficient of friction is ' 01, calculate the H.P. lost in friction and the amount of heat generated per hour. [0952 H.P. ; 212-3 B.T.U.] 21. Define a British Thermal Unit, and state what is meant by its mechanical equivalent. An engine uses 20 pounds of water per H.P. per hour, the feed temperature being maintained at 100 F. and the boiler temperature at 360 F. What proportion of the heat used by the boiler is converted into useful work, the total heat of evaporation of steam at 360 F. being 1,190 B.T.U. ? [11 '33 per cent.] 22. A turbine uses 12 ' 5 pounds of water per S.H.P. per hour, the feed temperature being maintained at 100 F. and the boiler temperature at 360 F. : what percentage of the heat taken up in the boiler is converted into work, the Total Heat of Steam at 360 F. being 1,190? [18 '14 per cent.] 23. State the First Law of Thermodynamics. The collars of a thrust shaft run in an oil bath, of which the temperature remains constant. The horse-shoe collars are cooled internally by water circulation. If 10 tons of water are circu- lated per hour and the mean rise of temperature be 20 F., estimate the power absorbed by the bearing in friction. [176 H.P.] 24. State the First Law of Thermodynamics and define the " Mechanical equivalent of heat." An engine is supplied with 200 pounds weight of steam per hour, the " Total Heat " of the steam at admission being 1,200 B.T.U.s per pound, and at exhaust 1,000 B.T.U.s per pound. If 75 per cent, of the heat abstracted from the steam is turned into useful work, what is the horse-power of the engine ? [11-79H.P.] 269 25. Give the value of the latent heat of steam at atmospheric pressure and show, by the aid of a formula, how the latent heat varies with the pressure at which the steam is formed. Dry steam at 200 lbs./in. 2 pressure (gauge), (temperature 388 F.), is passed into a tank containing 100 gallons of water at 60 F. When the temperature of the water has risen to 80 F. it is found by careful measurement that the tank contains 101 ' 74 gallons of water. Calculate from the above data the latent heat of steam at 200 lbs./in. 2 pressure. [842 B.T.U.s per pound.] 26. What is meant by " Latent heat of fusion " and " Latent heat of vaporisation " ? If 20 pounds of ice are dropped into 10 gallons of boiling water the temperature falls to 158 F. when the ice is just melted. Determine the latent heat of fusion of ice, neglecting any possible losses of heat. [144 B.T.U.s per pound.] 27. Compare the evaporative efficiencies of two kinds of coal, one evaporating 9 pounds of steam containing 10 per cent, of moisture at a temperature of 324 F. per pound of coal from water at 60 F., and the other 8 pounds of steam per pound of coal from feed-water at 104 F., the steam having a temperature of 350 F. and a dryness fraction of '95, the efficiency of the furnace being the same in each case. Give the pounds of water evaporated per pound of each coal from and at 212 F. [9-90: 9'431bs.] 28. If a pound of coal on complete combustion gives out 14,200 British Thermal Units of heats, and 70 per cent, of this heat passes into the water in a boiler, how many pounds of water will be evaporated per pound of coal burnt ? The feed- water temperature is 100 F. and the temperature of the steam 300 F. [9 Ibs.] 29. Define Sensible and Total Heat, and give an expression for determining the latter in terms of the temperature of the steam. In a trial the temperature of the Condenser steam, which was found to be dry, was 125 F. and the initial and final tempera- tures of the circulating water, 45 ' 5 F. and 97 ' 3 F. respectively. The air pump discharge was 20 " 45 pounds per minute and temperature, 100 F. Find the weight of circulating water per minute. [415-4 Ibs.] 30. Diy steam of 150 pounds absolute pressure (t = 358 F.) (v = 2 ' 96) expands adiabatically (n = 1 ' 135) to 1 pound absolute (t= 102F.)(u =330-4). 270 Find the dryness fraction at the lower pressure, and compare the area of the theoretical diagram in the expansion from 10 pounds (t = 193 '3) (v = 37 '83) to 1 pound with the diagram for the whole expansion. [739: -57.] 31. Explain the differences between the expansion of steam (i) when it expands adiabatically, (ii) when it remains dry and saturated, (iii) when it expands hyperbolically. If steam at 120 pounds per square inch (gauge), and of dryness fraction ' 95 expands adiabatically to a pressure of 25 pounds (gauge), estimate the dryness fraction at the lower pressure, assuming a suitable law of expansion. The volume of one pound of dry steam at 120 pounds pressure - 3 ' 28 cubic feet and at 25 pounds pressure = 10 ' 35 cubic feet. [883.] 32. What is meant by the " dryness fraction" of steam ? The absolute pressure of steam before wire-drawing is 235 pounds per square inch (T = 395 F.), and after wire-drawing is 80 pounds per square inch (T = 311 F.). If the temperature after wire-drawing be 320 F. and the specific heat of steam at constant pressure be ' 6, calculate what the dryness fraction of the steam was before wire-drawing, all losses being neglected. [976.] 33. One pound of water at 32 F. is placed under a frictionless loaded piston in a vertical cylindrical vessel. Describe fully what takes place as heat is gradually applied until the whole of the water is just converted into steam. State the amount of heat that must be applied at each stage of the transformation, supposing the piston loaded so as to produce a pressure of 235 pounds per square inch, in addition to that due to the atmosphere (T = 400 F.). [S = 368 : L = 834.] 34. Quote Regnault's formula for the total heat of saturated steam. Steam is generated in a boiler at an absolute pressure of 160 pounds per square inch, from feed water at 110 F., and the temperature of the steam is 363 F. Calculate the heat of formation per pound from the temperature of the feed water. [1,113 B.T.U.] 35. Distinguish between the conditions of steam described as (a) Dry and saturated, (6) Wet, (c) Superheated. How much heat is required to produce 1 pound weight of super- heated steam at 180 lbs./in. 2 pressure and temperature 522 F. from feed water at 120 F., the boiling point at 180 lbs./in." being 372 F., and the specific heat of steam at constant pressure = '5. [1,181 B.T.U.] 271 36. Define the tenn " Total Heat of Formation " of steam at one temperature from water at another temperature. Obtain numerical results for the total heat of formation of steam from water at 100 F., at (1) 380 F. (2) 160 F. |~(1) 1,128 B.T.U.l |_(2) 1,063 B.T.U.J 37. Compare the quantity of heat necessary to produce a pound weight of steam in the following conditions : (a) Dry and saturated, from feed at 100 F., steam formed at 406 F. (6) Superheated, from feed at 120 F., steam formed at 380 F. and superheated to 520 F., the specific heat of steam being ' 5. [(a) 1,136; (i) 1,178.] 38. During the consumption trials of an engine the dryness fraction of the steam Avas obtained by condensing some steam from the main steam pipe in a tank of cold water. Deduce from the following data the dryness fraction of the steam : Weight of water in tank, 1,000 pounds. Temperature of water in tank, initially, 60 F. Weight of water in tank, finally, 1,010 pounds. Temperature of water in tank, finally, 70 F. Steam pressure, 100 Ibs. sq. in. (temperature, 327 ' 6 F.). Latent heat corresponding to 100 Ibs. sq. in., 883 '8 B.T.U.s [838.] 39. Give an account of the addition of heat to water and formation of steam under constant pressure. Define the terms Volume per pound of steam, Relative volume of steam, and quote the results for atmospheric pressure. [26-36.1 L 1,648. J 40. If a triple expansion engine have the following particulars : Diameter of H.P. cylinder, 20 inches ; clearance, 25 per cent. of stroke volume, and cut-off at '6 of the stroke, Diameter of L.P. cylinder, 50 inches ; clearance, 15 per cent. of stroke volume, calculate the total ratio of expansion of the steam, and the mean pressure for the theoretical PV diagram, i.e., expansion hyperbolic. What proportion of this P m is realised practically ? [371 i>\ : about '5.] 41. Compare the efficiencies of two boilers, one of which supplies 9 ' 5 pounds of dry steam from 100 F. and at 400 F., and 272 the other 10 pounds of wet steam, dryness fraction ' 95, under the same conditions, both per pound of coal burned. [/", at 400 F. - 835 B.T.U.s.] [" 10,782 "I LlO,932j 42. Explain what is meant by " wire-drawing," and describe its effect on (a) wet steam, (6) dry and saturated steam. Steam of dryness fraction '8 and pressure 200 lbs./in. 2 (gauge) is passed through a reducing valve which reduces the pressure to 150 lbs./in. 2 (gauge). Estimate the dryness fraction of the steam on the low-pressure side of the valve given : p in Ibs./in. 2 (gauge). i where p b is the back pressure. Calculate the work performed per minute in the cylinder of a single-acthig steam engine, 14 inches in diameter and 12 inches stroke, when the initial steam-pressure is 70 pounds absolute, the back pressure 3 pounds absolute, the revolutions per minute 200, and the ratio of expansion 3 '5. [Log e 3'5 = 1'253.] [42-06 lbs./in. 2 : 1,294,500 ft.-lbs.] 56. A single cylinder double-acting engine drives a centri- fugal pump which lifts 18,000 gallons of fresh water per minute to a total height of 20 feet. Assuming that the ratio of B.H.P. to I.H.P. is '9 and that the efficiency of the pump is '6, find the 275 mean effective pressure, the cylinder diameter being 14 inches, stroke 1 foot, and revolutions 300 per minute. [72-21bs./in. 2 .] 57. The cylinders of a tri-compound expansion engine are 10 inches, 16 inches, and 22% inches in diameter and of 18 inch stroke. Steam is cut off in the H.P. cylinder at one-half stroke, the pressure on admission being 150 pounds per square inch absolute. Neglecting losses and assuming hyperbolic expansion, what is the mean pressure referred to the low-pressure cylinder ; and what horse-power would be developed by each cylinder when the engine was running at 210 revolutions per minute and the vacuum was 24 inches ? [46-1 lbs./in. 2 : H.P. 78, I.P. 106, L.P. 166. Total 350 H.P.] 58. Define Work and Power, and investigate the formula for the I.H.P. of an engine TTTP - LKP '- 3000' A steam cutter has a 2-cylinder simple engine. Diameter of cylinder, 6 inches. Stroke, 6 inches. If the mean effective pressure be 50 pounds per square inch and the revolutions be 400 per minute, estimate the I.H.P. developed. [34-27.] 59. Calculate the H.P. developed by a six-cylinder engine, the dimensions being as follows : - Diameter of cylinders, 4 inches, Length of stroke, 6 inches, Revolutions per minute, 1,000. Mean effective pressure, 50 pounds per square inch. The cylinders are single-acting. [57-13.] 60. Define Energy, Work and Power, and deduce an expression for the indicated horse-power of an engine. A modern locomotive has two cylinders, 20 inches diameter, 26 inches stroke, and at full speed runs at 280 revolutions per minute. If the mean effective pressure under these conditions be 40 pounds per square inch, estimate the I.H.P. of the engine. [924.] 61. How does the method of conversion of the heat energy in the steam into mechanical work differ in the case of the turbine from that in the reciprocating engine ? Dry steam at 160 pounds gauge pressure is expanded through aDe Laval conical nozzle adiabatically to 2 pounds per square inch u 5777. T 276 absolute. Find its issuing velocity and kinetic energy per pound, having given t l = 370 degrees, 2 = 126 degrees, and dryness fraction at end of expansion = ' 75. [4050 ft./sec. : 254,050 ft./lbs. = 113-4 ft./tons.] 62. Give an account of the broad principles on which the action of the Parsons steam turbine depends ; and show that for the same efficiency, initial and final conditions of the steam, the periphery speed is proportional to /=, when n is the number v n of fixed and moving vanes, and hence deduce the necessity for cruising turbines in warships. 63. The velocity of steam at exit from the guide-blades of a turbine is 500 feet per second in a direction making an angle of 30 with the plane of the blades. If it impinge on blades rotating at 200 ft./sec., draw the velocity diagram and state the relative velocity of steam and rotating blades. [342 feet per sec.J 64. Given that the loss of heat during the passage of steam through the guide-blades of a turbine is 4 B.T.U.s per pound, estimate the increase in velocity of the steam. [ = 223-/B.T.C. = 446 ft./sec.] 65. State a formula for the maximum weight of steam discharged through an orifice in a given time. Safety valves on a boiler are designed to be able to get rid of all the steam produced when the boiler is under full firing, with the stop valve shut. The boiler in question is capable of supplying a 800-H.P. engine with 18 pounds of steam per H.P. per hour. The safety valves blow off at a pressure of 275 lbs./in. 2 (gauge) and the maximum lift of the valves is -^ih of their diameters. [Note. The contracted area of flow may be taken as ' 75 of the total area of the orifice.] Calculate (a) the total clear area necessary for the discharge of steam, (6) the diameter of the valves, if the area is divided between two valves. [1-287 sq. ins. ; 2 '86 inches.] 66. Given that a slide-valve is to have 2 inches maximum opening and \ inch lead and that cut-off occurs at ' 7 stroke, find the necessary travel of the valve and the angle between the eccentric arm and the corresponding crank arm, if the valve takes steam on the outside edges. [7-ft inches; 127.] 67. Given that the outside lap of a slide-valve is 1 inch, the dead, f inch, and the mean cut-off at "75 stroke, draw Zeuner's diagram and find the travel and angle of advance. 277 Also, if there be $ inch, inside lap, estimate the percentage of the stroke at which release occurs. [4| inches ; 35 degrees ; 94 per cent.] 68. By means of Zeuner's Valve diagram determine the inside lap necessary to give release at ' 9 of the stroke of the piston, given : Travel of valve - - 6 inches, Cut-off - . - ' 7 of the piston stroke, Maximum opening - - 1^ inches. When will compression occur? [None ; '9 stroke.] 69. Draw the complete Zeuner's Valve diagrams for a slide- valve when given Valve travel - - 8 inches, Angular position of crank at cut-off 120 after dead point, release 150 admission 10 before stating the inside and outside lap and lead obtained. 7 " - 70. Given the following details of the working of a slide- valve : Travel - - 6 inches, Lead - - - \ Mean cut-off - - ' 75 stroke, draw Zeuner's diagram for the valve in question and find from it the outside lap, maximum opening, angle of advance, and angle of admission. [1-23 inches ; 1 '11 inches ; ; 36 degrees ; 1 1 degrees before dead centre.] 71. State the Law connecting the total resistance to the passage of a ship through the water with speed, and hence deduce the relation between Coal Consumption and Speed. The coal consumption of a ship steaming at 10 knots is 70 tons per day : estimate the consumption at 12 knots. [121 tons per day.] 72. If the total coal capacity of a ship is 2,000 tons and the coal consumption for all purposes at 10 knots is 72 tons per day, calculate the radius of action at 8 knots, if the consumption for auxiliary purposes in both cases is 12 tons per day. [8,989 miles.] 73. What is the shortest time in which a ship can reach a port 1,500 sea-miles distant, if she has 750 tons of coal available and her consumption at 12 knots is 96 tons per day ? [102-1 hrs.] T 2 278 74. What is meant by the terms " effective horse-power " and " indicated thrust " ? In a good example of a twin-screw ship, what percentage of the Indicated Horse-Power should be realised as Effective Horse-Power? The I.H.P. at 23 knots of a twin-screw cruiser is 22,000. The projected blade area of each propeller is 64*5 sq. ft. Find the thrust per square inch of projected blade area. [About -6: 10-07 lbs./in. 2 .] 75. Explain the meaning of the term "moderate speed"; and state the approximate relations existing between the I.H.P. , the speed of the vessel, and the coal consumption, at moderate speeds. A war-vessel burns 32 tons of coal per day for main engines when steaming at 12 knots ; her bunkers stow 600 tons. What is the quickest time in which a port to port passage of 2,880 sea-miles can be made if only 500 tons of coal are available for main engines ? State also about how much of the coal burned in each case would be used for auxiliary purposes. [180 hours : 40 tons : 4.5 tons.} 76. What is the apparent slip of a screw propeller ? If a propeller have a pitch of 24 feet and the revolutions be 100 per minute, calculate the 'apparent slip when the ship is travelling at 20 knots. [15-6 %.] 77. It is generally assumed that the coal consumption for a ship is proportional to the I.H.P. developed by the engines ; show that then the coal consumption for a certain voyage will vary as the square of the speed at which it is made. 78. Two similar vessels, A and B, make the same voyage out and home. A makes a uniform speed of 10 knots. B makes a speed of 12 knots on the outward journey and 8 knots on the return journey. Which vessel will have the greater total coal consumption for the main engines, and which will make the round voyage in the quicker time ? 7? 7? [Consumption, -j = 1 '04 : time, -7 = 1 '02.1 jT\. A 79. Give an account of the resistances to motion of a ship through the water, and state what assumptions are made in the statement that the I.H.P. required for a given speed varies as. the cube of that speed. 279 80. What is meant by the Economical Speed of a ship ? If given a curve of coal consumption, and speed for a ship, show that at the speed for which the tangent to the curve passes through the origin the speed is the most economical. 81. Explain the meaning of (a) " Indicated Thrust " and (6) " Effective Thrust." Mention the causes of the difference between (a) and (&). At full power the total I.H.P. developed by the engines of a twin-screw ship, whose speed is 18 knots, is 16,000. It is estimated that only ' 6 of this is usefully employed in driving the ship. If the total effective ahead bearing surface of each thrust block is 2,000 square inches, calculate the pressure in lbs./in. 2 on this surface. [43-321bs./in. 2 .] 82. State the approximate law connecting the Total Resistance to a ship's motion through the water with the Speed. From this deduce the relation between the Coal Consumption per hour and the Speed of the ship. A certain ship has a coal capacity of 1,500 tons and can steam 4,800 sea-miles at a speed of 8 knots. How far can she steam at a speed of 12 knots, the consumption of coal for auxiliary purposes being 12 tons per day in each case? [2,483 miles.] 83. Describe briefly the elements of the total resistance to a ship's motion through the water, and state the approximate law connecting the Total Resistance with the Speed. A certain battleship is detached to take a sister ship in tow and must only develop 4,100 I.H.P. , with which power she could steam alone at 11 knots. Find the speed at which she will tow the other vessel. [8-73 knots.] 84. Define the terms " pitch," " projected area " and "expanded area " as applied to a screw propeller. Answer either one of the two following parts (a) A ship is fitted with four propellers, equal power being obtained from each shaft. The total effective horse- power required to drive the ship at 21 knots is 14,700, and the projected area of each propeller is 4,500 in. 2 Estimate the pressure in lbs./in. 2 on the projected area. ' (b) The effective horse-power referred to in case (a) above is ' 6 of the horse-power transmitted by the shafts when making 320 revolutions per minute. If each, shaft is 10-inch diameter, estimate the maximum stress in the material of the shaft due to torsion. [12-66 lbs./in. 2 : 6,144 lbs./iu. s .] 280 85. In relation to a screw propeller, define the following quantities : Pitch, diameter, length, angle, blade area and projected area. With a certain pattern of screw propeller a slip of 20 per cent, had been found to give good results : what should be the pitch of such a screw to drive a ship at 30 knots, the revolutions being then 350 per minute ? [10ft. 10 ins.] 86. What is meant by the slip of the screw propeller ? The propellers of a Torpedo Boat Destroyer have a pitch of 9 feet, and at full power run at 400 revolutions per minute, the speed of boat being then 30 knots. Calculate the slip per cent. [15-56%.] 87. Define the terms "slip," "pitch," and "speed of screw " as applied to screw propellers. The full speed of a ship is 19 knots when the engines are running at 120 revolutions per minute. If the pitch of the propellers is 18 feet 6 inches, estimate the percentage slip. 88. Explain how the total heat of combustion of a fuel is theoretically estimated, and how it may be experimentally determined, pointing out in the latter case possible errors and how you would allow for them. The composition of a sample of coal is Carbon - 91 per cent. Hydrogen 4 Oxygen - 2 '5 Ash - - 2'5 Calculate its evaporative power from and at 212 F. [16-03 Ibs.] 89. The boilers of two similar cruisers of 22,000 I.H.P. are fitted to burn oil fuel in conjunction with coal. Each vessel has 2,000 tons of coal and 400 tons of oil fuel available for fast steaming. They are fitted with different types of boilers, but the working pressure is 275 pounds per square inch in each case. In Cruiser A the equivalent evaporation from and at 212 degrees F. per pound of .coal is 10 '75 pounds, and in Cruiser B 11 '25 pounds. At 70 per cent, full power the water consumption in A is 19 '25 pounds per I.H.P. per hour, for all purposes, and in B 18 ' pounds. At 15,400 I.H.P. the speed of each vessel is 21 '5 knots. 281 Find the radius of action of each vessel at this speed whilst burning coal and oil in the ratio of 7 : 3, and also their total radii as long as the above quantity of coal lasts. [Take 3 Ibs. of oil = 4 Ibs. of coal.] [(1) 2,562 : 2,866 miles.] [(2) 4,425 : 4,950 miles.] 90. Taking the composition of Welsh coal and oil fuel respectively as 90 per cent Carbon, 6 per cent. Hydrogen, and 4 per cent. Oxygen, 86 per cent. Carbon, 14 per cent. Hydrogen, calculate the amount of air theoretically required for the complete combustion of 1 pound of fuel in each case. [Note. Com- position of air, by weight, is 77 per cent Nitrogen and 23 per cent Oxygen. The atomic weights of Carbon, Oxygen and Hydrogen respectively are 12, 16 and 1.] [12*36 pounds ; 14'84 pounds.] 91. State the conditions under which the maximum efficiency of a heat-engine may be obtained and hence show that under no circumstances could the efficiency be unity. 92. What is meant by the "Thermal Efficiency" and the "Mechanical Efficiency" of an Engine? Give approximate numerical values for each, for a modern triple expansion steam engine. If 1,100 Thermal Units of heat be expended in producing each pound of steam and 18 pounds of steam are required by an engine per brake horse-power per hour, find the efficiency of the steam used by the engine. [15/ c ; 90/ D ; 12-85 c / c .] 93. In making a trial of a certain oil-engine the following data were obtained : Indicated horse-power 120 Oil used, pounds per hour - 40 Calorific value of oil, B.Th.U.s. per pound - 19,500 Jacket cooling water, pounds per minute 43 Temperature at inlet, Fah. 48 outlet, Fah. 144 Pounds of air used per pound of fuel - 50 Temperature of exhaust gases, Fah. - 460 ,, ,, engine room - 54 Specific heat of exhaust gases- 0'25 282 Estimate the percentages of the total heat of combustion of the oil rejected in cooling water and in exhaust gases, and find the Thermal efficiency of the engine. [31-75 / ; 26-54 / ; 39-15 / .] 94. Define "Thermal Efficiency" and "Mechanical Effici- ency." The following particulars refer to a steam-engine plant used for ship propulsion : Coal burnt per day at full power 300 tons Calorific value of coal per pound - 14,500 B.T.U. Feed-water used per I.H.P. per hour 18 Ibs. Total I.H.P. at full power - 15,000 Full speed of ship 18 knots. Resistance of ship at full speed - 60 tons. Steam pressure at boilers 250 Ibs./in. 2 Temperature of steam = 406 F. Dryness fraction of steam ' 95 Temperature of feed water 100 F. circulation water at inlet 60 F. ., outlet 90 F. (a) Find the thermal efficiency of the boilers. (6) If each pound of steam loses 30 B.T.U.s by radiation during its passage from the boilers to the engines, estimate the thermal efficiency of the engine. (c) If the work done is only equivalent to 85 per cent, of the heat abstracted from the steam during its passage through the engines, the remainder being lost by radiation, &c., estimate the weight of circulating water supplied per hour. (d) If the mechanical efficiency of the engine is ' 85, find the efficiency of the propeller. (e) What is the overall efficiency of the propelling plant ? [(a) -728; (6) -133; (c) 3,607 tons ; (rf) -582; (e) -0479.] 95. Define the terms Thermal Efficiency amd Mechanical Efficiency. An electric generating engine uses 20 pounds of steam per kilowatt per hour, and each pound of steam required 1,080 B.Th.U.s to produce it. Find the Thermal Efficiency of the plant ; also, if the mechanical efficiency of the engine is ' 85 and the electrical efficiency of the dynamo is ' 9, estimate the steam consumption in pounds per I.H.P. per hour. [15-8 / ; 11-41 Ibs.] 96. From the trials of a marine engine the following data were obtained : I.H.P. - - 500 Pounds of feed-water used per I.H.P. per hour - 16 Steam pressure at engines - 2001bs./in. 2 (gauge). Temperature of feed water ' - - 100 F. circulating water at inlet - 60 F. ,, water at outlet - 85 F. Assuming that the steam is supplied dry and that the work done is 80 per cent of the heat abstracted from the steam during its passage through the engines, estimate (a) The weight, in tons, of circulating water supplied per hour. (6) The thermal efficiency of the engine. p in Ibs./in 2 .(gauge.) *Fh. Sfrom32Fh. L. Specific Volume. 200 388 356 840 2-1 [() 132-8 tons; (6) 14-1 / .] 97. A steam-engine is supplied with steam of temperature 360 F., and expands the steam down to a temperature of 100 F., using 13 pounds of steam per H.P. per hour. Feed temperature, 80 F. Compare this performance with a perfect heat-engine working between the same limits of temperature, and point out as far .as possible where the losses take place in the practical case. Are any of these losses avoidable ? [17-14%; 31-67%.] 98. Distinguish between the thermal and mechanical effici- ency of an engine. Compare the efficiency of the following engines : (a) A reciprocating engine using 1 ' 75 pounds of coal per I.H.P. per hour ; mechanical efficiency, ' 95. Calorific value of coal, 14,000 B.T.U.s per pound. (6) A steam-turbine using 1 pound of oil fuel per brake H.P. per hour. Calorific value of oil, .20,000 B.T.U.s per pound. (c) A gas-engine using 45 cubic feet of producer gas per brake H.P. per hour. Calorific value of gas, 200 B.T.U.s per cubic feet. [() 9-87%; (ft) 12-73%; (c) 28-28 %.] 99. Compare the performances of the three following engines as regards heat efficiency and cost per H.P. per hour : (a) A steam-engine using 1 ' 5 pounds per I.H.P. per hour of Welsh coal of calorific value 14,500 B.T.U.s per pound, at 15s. per ton. 284 (6) An oil-engine using '75 pounds per I.H.P. per hour of oil fuel, calorific value 20,000 B.T.U.s per pound, at 60s. per ton. (c) A gas-engine using 20 cubic feet per I.H.P. of gas per hour, calorific value 600 B.T.U.s per cubic foot, at 2s. per 1,000 cubic feet. [Efficiency: (a) -117; (6) -171; (c) '213. Cost: (a) -'12 ; (6) -24 ; (c) -48 of a penny per H.P. hour.] 100. Determine the " Total Heat of Formation " of 1 pound of superheated steam of temperature 500 Fah. under following conditions : Temperature of feed water entering boiler - 130 Fah. ,, evaporation in boiler - 400 Fah. Specific heat of steam - * 48 If 11 pounds of this steam are used by a Turbine per B.H.P. per hour, what is the efficiency of the Turbine ? [1,152B.T.U. ; 20 / .] 101. An engine uses 14 pounds of steam per hour per I.H.P. The steam used is at an absolute pressure of 150 pounds and is superheated 150 F., the temperature of the feed water being 95 F. Determine the amount of heat given to the steam per horse- power hour by the boiler and superheater, the amount turned into work, and the thermal efficiency of the engine. The saturation temperature at 150 pounds absolute pressure = 358 F., and the specific heat of steam = '48. [17,386 B.T.U. ; 2,545 B.T.U. ; 14-64/ .] 102. A modern boiler under trial conditions was found to evaporate 12 pounds of water from and at 212 F. per pound of coal. Explain the meaning of the statement and calculate the efficiency. [Calorific value of fuel, 14,800 B.T.U.s per pound.] [78-3/ .] 103. State and explain the Second Law of Thermodynamics. What form does the indicator diagram of a perfect heat- engine take, and under these circumstances what would be the efficiency of the engine working between the temperatures 3 90 and 100 b F. ? [34-1%.] 104. The consumption of steam for all purposes in a Ship is 21 '0 pounds per I.H.P. per hour and the corresponding coal consumption is 2 ' 05 pounds per I.H.P. per hour. Given that the temperature of the feed water is 72 F. and that the boilers deliver steam having 3 ' 4 per cent, of moisture at a pressure of 285 196 pounds per square inch absolute [T = 386 F.], calculate the efficiency of the boilers, assuming the calorific value of the fuel used to be 15,000. [77-1%.] 105. A triple expansion engine, using boiler steam, at 200 pounds pressure by gauge, uses 16 pounds of steam per I.H.P. per hour ; and a compound engine, working with 75 pounds boiler pressure, uses 20 pounds of steam per I.H.P. per hour. Compare the efficiencies of the two engines, supposing the temperature of the feed-water to be 110 F. in both cases. The temperature corresponding to 200 pounds pressure by gauge is 387 F., and to 75 pounds, 319 F. [14-2 / ; H'6/ .] 106. Investigate a formula for the maximum efficiency of a perfect heat-engine. What is the maximum efficiency of an engine working between the temperatures of 400 F. and' 100 F. ? [34-8 / .] 107. Explain the terms " dry saturated steam," " superheated steam." An engine uses 20 pounds of dry saturated steam per I.H.P. per hour, the temperature of steam in boiler is 400 F., and the temperature of the feed- water is 110 F. What percentage of the total heat in the steam is utilised in the engine ? [ll-3/ .] 108. A marine steam-engine developing 11,500 I.H.P. uses 195,500 pounds of steam per hour. Assuming a boiler pressure 220 lbs./in. 2 (by gauge), dryness fraction of steam 98 per cent., and temperature of feed 100 Fah., determine the efficiency of the engine. Temperature corresponding to 220 lbs./in. 2 pressure may be taken as 396 Fah. [13-4/ .] 109. State and explain the first and second laws of thermodynamics. The quantity of dry steam at a pressure of 215 pounds per square inch (gauge), temperature 393 F., supplied to a steam- engine is 15 pounds per H.P. per hour, and the temperature of the circulating water available is 60 F. Compare the performance of this with that of an ideally perfect engine. [U-5/ ; 39/ c .] 110. An evaporator is required to produce 30 tons of " make-up feed " per day and the working density is not to be greater than 25 degrees as measured by a Service hydrometer. What quantity of water must be blown down per hour ? [1,866 pounds.] 286 111. Referring to the previews example, if the temperature of the water in the evaporator is 220 F. and the temperature of the sea- water is 60 F., what proportion of the heat supplied to the evaporator is wasted by blowing down ? [8 7 per cent.] 112. Steam at 20 lbs./in. 2 and having a diyness fraction of 0*8 is admitted to the coils of an evaporator from the closed exhaust system ; the coil drains are adjusted so that the drainage water is at 200 Fah. Calculate the number of pounds of secondary steam produced per pound of primary steam, if the feed-water temperature is 80 Fah., and the secondary steam at 1 lb./in. 2 and dry. Given for dry steam that : p lb./in 2 . t Fah. S, from 32 Fah. L. 1 217 185 962 20' 260 228 932 [732 Ibs.] 113. Describe briefly the construction and mode of action of an evaporator. What is generally considered to be a suitable working density for such, and why ? An evaporator produces 30 tons of "make-up feed " per clay. How much brine must be pumped overboard every hour in order that a density of 40 may be maintained in the evaporator ? [866 Ibs.] 114. Give an account of the objections practically to the use of sea-water as feed for the boilers in a ship. What is the approximate amount and composition of the residue after evaporating sea-water to dryness ? If a boiler contain 4 tons of salt water and is supplying steam for 1,000 H.P. with a salt feed, how much impurity will be left in the boiler at the end of a 12 hours' run if no brining were allowed ? Will all this impurity be solid matter ? [6,000 + 280 Ibs., if engine uses 16 Ibs. of steam per H.P. hour.] 287 PART V.-ELECTEICITY. 1. An incandescent lamp takes ' 5 ampere at 100 volts. How many lamps of this description would a 5-H.P. engine keep burning ? [74.] 2. Four cells are joined up to an external resistance of 4 ohms. The resistance of each cell is ' 2 ohm and the E.M.F. of each cell is 1 ' 5 volts. What will be the fall of voltage in the external resistance (a) When all the cells are placed in series ? (&) When all the cells are placed in parallel ? (c) When the cells are placed two in series and two in parallel ? [5 volts ; 1 '48 volts ; 2-8 volts.] 3. Three wires, whose resistances are 1, 4 and 12, are joined in parallel. A fourth wire (whose resistance may be neglected), containing 10 cells in series, is now joined in parallel with the other three. If each cell has an E.M.F. of 1 ' 2 volts and internal resistance '2 ohm, find the current passing through the wire whose resistance is 4 ohms. [ T 9 T ampere.] 4. Three wires whose resistances are 2, 6, 12 ohms are joined in parallel. A fourth wire, whose resistance may be neglected, containing 10 cells in series, is now joined in parallel with the other three. If each cell has an E.M.F. of 1*2 volts and an internal resistance of ' 2 ohm, find the current passing through the wire whose resistance is 6 ohms. [ ampere.] 5. A galvanometer of 1,000 ohms resistance is required to be shunted so that the total resistance of the circuit is lessened by 400 ohms. What is the resistance of the shunt required ? [1,500 ohms.] 6. State Ohm's Law. A small 4-volt accumulator battery is to be charged fromlOO-volt mains by using 52-watt 16-candle power 100-volt lamps as resistances. 288 Sketch the accumulator battery and the precise arrangement you would adopt if the charging current is to be 1 ' 5 amperes. When so arranged, what are the watts per candle consumed per lamp if the candle-power of a 16-c.p. lamp varies 1 candle per volt when near 100 volts ? [3 in parallel.] 7. How do you proceed, in the laboratory, to measure a low resistance, such as an armature ? The winding of a two-pole armature with the brushes lifted is disconnected at some point and the resistance between the disconnected ends is measured and found to be ' 2 ohm. If the winding is connected up again and the brushes lowered, and the machine set running, what will now be the resistance between the brushes ? [05 ohm.] 8. A battery has the D.P. at its terminals lowered from 10 to 6 volts when the terminals are connected by a wire of 10 ohms resistance. What is the resistance of the battery ? What assumption must you make ? [6f ohrne.] 9. A cell of E.M.F. 1 ' 5 volts, and having some internal resistance, is connected to a wire of resistance 2 ohms and the current flowing is measured. On doubling the resistance of the wire the current is found t6 be two-thirds of its former value. What is the resistance of the battery ? What assump- tion do you make as to the electromotive force of the battery in this calculation ? [2 ohms.] 10. Sketch a circuit, showing 2 cells each of E.M.F. 1 ' 5 volts and internal resistance ' 2 ohm, coupled in series, supplying current to a cylindrical coil of resistance 2 ohms. Find the amount of current flowing and its direction. Indicate the parts of the circuit where heat is produced and where magnetic effect is produced. Find also the D.P. between the ends of the coil. [1-25 amperes ; 2'5 volts.] 11. A battery of E.M.F. 4 volts and resistance 1 ohm is connected by copper wires of resistance ' 2 ohm to a piece of platinum wire of resistance 2 ohms. Find (a) The current in the copper wire ; (6) The current in the platinum wire ; (e) The difference of potential between the battery terminals. In which portion of the circuit is the greatest amount of heat developed ? [(a) 1' 25 amperes; (6) 1*25 amperes; (c) 2*75 volts.] 289 12. A galvanometer forms part of a circuit and has a resistance of 800 ohms. With a certain battery in circuit and galvanometer unshunted the reading is 101 scale-divisions. The instrument is now shunted with a resistance equal to its own, and the new reading is 100 scale-divisions. What is the resis- tance of the rest of the circuit ? Explain the apparent pecu- liarity of the results given. [8-08 ohms.] 13. A certain ammeter, reading a maximum of 20 amperes, has 40 turns of wire on its coil. Length of mean turn is 6 inches and diameter of wire is ' 2 inch and its specific resistance per inch cube is ' 7 microhm. What is the waste of energy at full reading? If electricity costs 2d. per B.O.T. unit, what would be the cost of running the ammeter per thousand hours ? [2-15 watts ; 4'3rf.] 14. How does the capacity of a condenser depend upon its size and upon the material between the coatings ? What must be considered when choosing this substance, and how would its requisite properties be measured ? If a condenser of 1 microfarad capacity is connected to a cell of E.M.F. 2 volts by wires having a resistance of O'l ohm, the cell resistance being negligible, what will be the initial charging current, and what will be the energy finally stored in the condenser in ergs ? [20 amperes ; 20 ergs.] 15. If a condenser of capacity 0'25 microfarad is charged up to a certain voltage and contains 25 microcoulombs, find the voltage to which it is charged, and the joules stored up in the condenser. [10 volts 12-5 x 10-5 joules.] 16. How would you combine 4 condensers, each of capacity 1 microfarad, to form a capacity of ' 75 microfarad ? 17. A battery of constant E.M.F. 1 ' 5 volts and internal resistance 0'2 ohm is attached, suddenly, to the terminals of a condenser. State what happens, and sketch a curve showing how the current varies with the time, indicating the numerical values of the initial and final currents. If the condenser has a capacity of one microfarad, how many ergs of work will it perform when discharged ? [7-5; 0; 11-25.] 18. What are the Laws of Electrolysis ? How much would it cost, per ton, to electrically deposit copper if electrical energy may be purchased at Id. per unit, if the depositing bath needs a pressure of 2 volts and has a resis- tance of ' 01 ohm ? Sketch the arrangement. 290 [Ampere electrochemical equiv. of copper = O'OdOSS gm. and 1 pound = 454 gms.] [3 11*. 4d., nearly.] 19. An E.H.F. of 3 volts is required to force a current of 1 ampere through a voltameter containing acidulated water. If the work required to separate one gramme of hydrogen is 142,000 watt seconds and the electro-chemical equivalent of hydrogen is '. 00001035, find the resistance of the voltameter. [1-53 ohms.} 20. What must be considered when choosing the size of a cable for supplying electric light at a distance ? Apply your reasons to finding the sections in the following cases : (a) 200 electrical horse-power at a pressure of 200 volts is to be delivered at a point 300 yards away from a dynamo which is only capable of producing a terminal D.P. of 205 volts. (6) The same power is to be sent the same distance at the same delivered pressure, but, the dynamo not being provided, one may be purchased producing any pressure desired. [Sp. res. of copper = ' 66 microhm per inch cube.] [(a) 2-127 square inches ; (6) '746 square inch.] 21. An arc lamp, needing 10 amperes and 60 volts, is 200 yards from a dynamo which produces a terminal D.P. of 100 volts. The cable supplied is 0'12 inch diameter. Find the value of the regulating resistance to be placed in series with the lamp to fulfil the conditions. How is the candle-power of such an arc lamp measured ? [N.B. p = '66 microhm per inch cube.]- [3 -16 ohms.] 22. If the distance between feeding centres be 400 yards and one lamp using ' 3 ampere is connected per foot run of the cable, find the resistance of the cable per foot if the maximum difference of the D.P. across the terminals of any two lamps is not to exceed 2 volts. [0000185 ohm.] 23. What two important considerations guide one in select- ing a suitable cable, for electric lighting or power purposes, from the dynamo to a distant building? What size of cable would in the following circumstances give the requisite voltage ? A dynamo, producing a terminal D.P. of 120 volts, supplies a building 300 yards away in which are fixed 200 100-volt 291 60-watt incandescent lamps and two 4-horse-power electric motors, also supplied at 100 volts. [Resistivity of copper = ' 66 microhm per inch cube.] Find also the current density in this cable. [ 18 square inch.] 24. An arc lamp taking 10 amperes at 50 volts is connected across supply mains at 110 volts with a resistance in series with the lamp. What must be the value of this resistance ? How many Board of Trade units are consumed if the lamp burns for three hours ? If the price of energy is Qd. per unit, how much will be paid for the energy wasted in the resistance at the same time ? [6 ohms; 3.3; 10'Srf.] 25. Explain, with a sketch, the construction and working of a Nernst lamp. State the reason for any appliances you include, and calcu- late the approximate cost of running 10 such lamps, each of oO c.p., when energy is sold at 6d. per unit. [efficiency 1*2 watts per c.p., 3 - 6r/. per hour.] 26. Explain the construction and working of an accumu- lator battery fitted for supplying one hundred 100-volt 16 c.p. lamps. If the efficiency of the lamps is 3 watts per c.p., what would be the maximum discharge current taken ? [56 amperes.] 27. Sketch the circuits of a shunt dynamo supplying incandescent lamps. If there are forty 100-volt lamps, each having a resistance of 166 ohms, how many amperes must the dynamo supply ? [24* I amperes.] 28. A lighting circuit is run at a pressure of 100 volts, the lamps available are marked for 50 volts. Show how they should be arranged, and calculate the total current required to be supplied to run ten lamps, if the resistance of each *be 84 ohms when incandescent. [3 amperes.] 29. A searchlight 300 yards from a dynamo is supplied with a current of 150 amperes ; allowing a current density of 1,000 amperes per square inch, what diameter of cable would you employ? If the D.P. at the lamp terminals is 80 volts, what will be the D.P. at the dynamo terminals? What are the watts consumed in the arc and what is the electric horse-power given out by the dynamo ? [Specific resistance of copper is ' 7 microhm per inch cube.] [94-3 volts, 12,000 watts ; 18-i) H.P.] u 5777. U 292 30. What are the laws of the magnetic circuit and what is the practical use of the conception ? An iron ring, having a mean diameter of 40 cms. and of cross-section 10 sq. cms., partially wound with wire, is mag- netised by a current of 2 amperes so that the total flux round the ring is 10 5 lines. If the ring is now cut through in one place and a slice of iron, 1 cm. thick, removed, what must be the value of the current now sent through the coil if the flux is to remain as before ? [> - 2,300.] [38 amperes nearly.] 31. What do you understand by the law of the magnetic circuit ? What is the least number of turns of wire required to produce a magnetic flux of 6 X 10 5 lines in an iron ring if made of round bar of 40 sq. cm. cross-section and of mean diameter 100 cm. ? A battery capable of giving 10 amperes is available and p = 525. When is hysteresis of importance in connection with the magnetisation of iron ? [714.] 32. What do you understand by flux, flux density, and magnetic saturation ? Write down the law of the magnetic circuit which is analogous to Ohm's law. Find the ampere turns necessary to produce a flux density of 10,000 in an air gap ' 5 cm. long. [3,981.] 33. The case of a drum armature for a two-pole machine is 18 inches in diameter, with a 3-inch hole through it, and is 20 inches long parallel to the shaft. It runs at 600 revolutions per minute, and is worked at such a density that the hysteresis loss is 5,000 ergs per c.c. per cycle. Find the hysteresis loss in watts. [405-3.] 34. A closed soft iron ring, of 30 cms. mean diameter and 6 sq. cms. cross-section, is wound uniformly with 300 turns of insulated wire. Supposing the following relations to exist in iron of this quality : ft - - 10,200 12,000 13,700, u - 2,000 1,500 1,000, calculate the current in amperes at which the total flux is 78,000 lines. 293 If this ring is cut in two across a diameter, how will the force with which one half clings to the other half vary with the flux ? [2 '72 amperes.] 35. Define the terms magnetomotive force and total magnetic flux, and state the law connecting them in the magnetic circuit. Calculate the flux in an iron ring of 5 sq. cms. in cross-section on which are wound 200 turns of wire conveying 10 amperes, the mean diameter of the ring being 20 cms. and the permeability of the iron 1,000. [200,000 approximately.] 36. What do you understand by the magnetic circuit ? Calculate the ampere turns necessary to cause a flux of 4,000 lines per sq. cm. in an anchor ring of iron of 10 sq. cm. section and mean radius of 20 cms. [ft = 2600.] [154.] 37. Sketch the circuits of a compound-wound dynamo suitable for supplying either incandescent lamps or for charging accumulators. A generator is supplying energy at the rate of 4 Board of Trade units per hour, and the D.P. at the terminals of the generator is 100 volts. State a probable value of the efficiency of a carbon lamp, and, assuming this value, find how many 8 c.-p. 50- volt carbon filament lamps are supplied by the above generator. [125 lamps.] Sketch the arrangement of the lamps. State a probable value of the efficiency of a metallic filament lamp, and, assuming this value, find the available candle-power in this case. [2,857 c.p.] 38. What is the fundamental formula for dynamo design, giving the E.M.F. in terms of three variable quantities ? A two-pole shunt dynamo is to give 30 amperes at a terminal D.P. of 110 volts. The resistance of the armature is ' 15 ohm and that of the shunt coil is 55 ohms. Find the E.M.F. generated in the machine. If there are 500 conductor bars on the armature, the sectional area of the coil being 100 sq. c.m. and the speed 20 revolutions per second, find the total armature flux and the average flux density in the air gap. [114-8 volts ; 11,480 per D c.m.] 39. Sketch the circuits of a compound dynamo in which the series coil can be rendered useless by the insertion of a plug, 294 Sketch the characteristics you would expect (a) With plug in, (b) With plug out. Such a machine, generating 20 kilowatts, with the plug in, has a terminal D.P. of 120 volts and an electrical efficiency of 90 per cent. If the armature and field coil losses are equal, determine (a) the resistance of the field winding, and (b) the current in the armature. [() 14 *4 ohms ; (b) 158-3 amperes.] 40. A dynamo has to charge 50 accumulators in series. What type of dynamo would you employ, and what E.M.F. ought it to supply ? State your reasons. When the battery has been discharged, what will be the D.P. at its terminals? If this battery delivers 400 ampere- hours at an E.M.F. of 100 volts, what is the energy of discharge in ft. Ibs. ? [125 volts : 90 volts ; 1 -06 X 10.] 41. A shunt dynamo producing a terminal D.P. of 150 volts is used to charge 60 storage cells, each having an E.M.F. of 2 ' 1 volts and a resistance of ' 001 ohm. If the leads have a resistance of ' 2 ohm, what will be the current generated ? [92 '3 amperes.] 42. Given a ring armature 10 inches long, 10 inches diameter, 1 inch radial thickness of iron, what number of turns would you use to give 110 volts? The speed is 1000 revolu- tions per minute, and the flux density equals 8,000 lines per sq. cm. [About 640.] 43. If a dynamo gives 100 volts at 1,000 revolutions per minute, at what speed must it run to charge 54 cells at 50 amperes, each cell having a back E.M.F. of 2 ' 2 volts and ' 005 ohm internal resistance, the leads having resistance of * 01 ohm ? [1,328 revolutions per minute.] 44. What is the fundamental formula of the dynamo ? An armature running at 18 revolutions per second lias 288 external conductors and a resistance of ' 025 ohm. Find the total flux so as to give 70 amperes at 100 volts at the terminals. [1,962,770.] 45. An 80-volt dynamo is driving a motor taking 200 amperes through leads whose resistance is ' 05 ohm. What will be the voltage at the motor terminals ? [70 volts,] 46. A shunt motor, with an armature resistance of ' 01 ohm and supplied with a constant pressure of 100 volts, creates a back 295 E.M.F. of 10 volts at 100 revolutions a minute, and is running at 970 revolutions a minute. If the load increases until the current through, the armature is 50 amperes greater than before, what will be the new speed ? [965 revolutions per minute.] 47. The resistance of a series motor is 1 ' 5 ohms and the D.P. at the terminals is 105 volts. What will be the current and electrical efficiency when the back E.M.F. is 75 volts ? [20 amperes ; 71*4 per cent.] 48. A slrunt motor with an armature resistance of ' 02 ohm creates a back E.M.F. of 10 volts per 100 revolutions, and with a voltage of supply of 80 volts requires 100 amperes to pull a certain load. The load is doubled, what is the relation between the new and the old speeds ? [New speed = 20 revolutions per minute less than old speed.] 49. A capstan motor is supplied at 80 volts, and takes a current of 400 amperes when working at 40 H.P. What is the efficiency of the motor ? [93*25 per cent.] 50. Describe the apparatus you would employ to measure the B.H.P. of a motor in the laboratory. If, in a shunt motor, the armature resistance is ' 08 ohm, the field resistance is 40 ohms, and the current supplied is 20 amperes at a terminal D.P. of 100 volts, calculate the electrical efficiency. [86-3 per cent.] State a probable value of the mechanical efficiency of the above motor. Taking this probable value of the mechanical efficiency, find roiighly what will be the B.H.P. of the motor in question. [1-8 H.P.] 51. A series motor under a given load takes 50 amperes at 400 volts and has a speed of 800 revolutions per minute, the resistance of the motor being 0'8 ohm. It is required to have a starting and regulating resistance such that the starting current cannot exceed 30 amperes, and the speed may range from 800 to 200 revolutions per minute. Find the values of the resistances required. [At speed 200 rev./min., resistance = 10^ ohms. At speed 400 rev./min., resistance = 7^ ohms. At speed GOO rev./min., resistance = 4 ohms.] 52. A shunt motor which has been wound for 110 volts is used at 220 volts. What changes would you expect to find in the 296 speed, armature current for a given load, the field current, and the losses in the armature and field ? 53. An electrically driven train, weighing 200 tons, travels 20 miles along a line which rises 500 feet in the distance, the frictional resistance being 15 pounds per ton. Find the kilowatt hours required to do the work, the effi- ciency of the motor and gearing being taken to be 75 per cent. [271-6 kilowatt, hours.] 54. How would you test a shunt motor that gives by brake 10 H.P. to determine its condition as regards insulation and its electrical and commercial efficiencies at full load ? Assuming a probable value of the latter efficiency, how many amperes at 200 volts would such a motor probably need ? [46 '6 amperes, nearly (assume efficiency of 80 per cent.).] 55. A motor, wound for 100 volts, actuates a lift. If a mass of 900 pounds is being raised uniformly at the rate of 5 ft./sec., what is the current of the motor ? If on starting the accelera- tion is uniform and the motor acquires its final speed in two seconds, what is the extra current needed during acceleration ? The combined efficiency of motor and gearing may be taken as 70 per cent. [87 '2 and 6' 7 amperes.] 56. Explain the function of a motor starting resistance. A series motor produces a back E.M.F. of 94 volts and is supplied with 100 amperes at a D.P. of 100 volts. If ^-horse-power is lost in various frictions, what will be the brake horse-power and the electrical and mechanical efficiencies ? [12-1 H.P. ; 94 per cent. : 96-02 per cent,] 57. Aii electrically-driven train, weighing 100 tons, travels 10 miles along a line which rises 600 feet in the distance, the tractive force on the level being 12 pounds per ton. Find the number of kilowatt hours required to do the work, efficiency of motor and gearing being taken at 82 per cent. [91 nearly.] 58. Current is supplied to a shunt-motor at 100 volts. Resistance of the armature is '5 ohm. What is the back E.M.F. when the rate of total external work done by the motor is equivalent to 4,800 watts ? [60 or 40 volts.] 59. Given a 100-volt motor, with a separately excited and constant field, and an armature resistance of ' 02 ohm, which takes 80 amperes with a certain load. If the load be doubled, what will be the relation between the new and the old speeds ? [Vj : V 3 = 98-4 :96-8.J 297 60. A circuit has a resistance of 100 ohms and a self- inductance of 2 henrys, and a condenser in series. The frequency being 100, the phase difference is found to be zero. What must be the capacity in microfarads, and what is the value of the impedance ? [l-26m.f. ; 100 ohms.] 61. A coil of resistance 120 ohms and self-induction ' 2 henry is placed in series with a condenser of capacity 20 micro- farads. If an ammeter in series with the arrangement reads ' 2 ampere when the frequency is 50, what would a voltmeter read if placed (a) across the whole ? (fe) across the terminals of the coil ? (c) across the terminals of the condenser ? [30-8 volts; 12-56 volt; 31 -8 volts.] 62. Define frequency, lag, and power factor, as applied to alternating current supply. The power supplied to an alter- nating current circuit is measured by an ammeter, a voltmeter, and a wattmeter simultaneously. The readings are respectively 50, 100, and 3,500. Sketch the pressure and current curves (assuming sine waves) in correct proportion and relative position. What is the value of the wattless current in this case ? [35 amperes.] 63. If you have an inductance of ' 001 of a practical unit and a capacity of ' 01 X 10 of a practical unit, show fully, by sketches, how you could arrange them in series to form the sending circuit of a tuned wireless station, and what will be the wave-length thus produced. [5,960 metres.] 64. What is meant by the wave-length of an electric wave as used in wireless telegraphy ? What will be the length of the wave radiated by a system having a capacity of 1 microfarad and an inductance of ' 5 henry ? Sketch a typical tuned receiving circuit, explaining the details. [1-34 x 10 metres.] DATE DUE PRINTED IN U.S.A. vw ."JiViwi