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EL 
 
 SU] 
 
SYLLABUS OF 
 
 ELEMENTARY MECHANICS, 
 
 BY 
 
 JAMES LOUDON, M.A., 
 
 Professor of Physics, University of Toronto; President of the University, 
 
 WITH 
 
 SUITABLE EXERCISES AND EXAMPLES 
 
 PREPARED BY 
 
 C. A. CHANT, B.A., 
 
 Lecturer in Physics, University of Toronto. 
 
 TORONTO: 
 ROWSELL & HUTCHISON, 
 
 PRINTERS AND PUBLISHERS. 
 
 1892. 
 
 two ^L \ : >u f. r " " TORONTO. 
 
T^ 
 
 Entered according to the Act of Parliament of Canada, in tlie year of our Lord 
 one thousand eight hundred and ninety-two, by Rowsell & Hi xciiisox, in the 
 Office of the Mitiister of Agriculture. 
 
PREFACE. 
 
 The acriompanyiug Syllabus is an outline of the work dealt 
 with for some years by Professor Loudon in his lectures on 
 Elementary Mechanics, and it is hoped that the present publi- 
 cation will prove a convenience. The usual division into Statics 
 and Dynamics has not been made, but these two parts are 
 considered together. 
 
 In the Examples, the object aimed at was adaptability rather 
 than originality, and thus some familiar exercises may be found 
 in the collection. It is believed that this part of the work 
 will supply the student with an adequate means of testing his 
 knowledge. 
 
 It will be noticed that the difference between a pound mass 
 and a pound force has been emphasized V)y using a distinct 
 abbreviation in each case. This was suggested by an instructor 
 of wide experience, and will be found a valuable expedient. 
 
 Though primarily intended for the use of students in Physics 
 at the University, it is probable that teachers in other institu- 
 tions will find this little work useful. 
 
 Professor Loudon's numerous other duties prevented him 
 from revising any part, either before or during the time of 
 printing. 
 
 C. A. C. 
 
 Toronto, October 1st, 1892. 
 
USEFUL RESULTS. 
 
 1 gram = 15-43 grains. 
 1 kilo. = 2-20 lbs. avoir. 
 
 1 ounce (avoir.) = 28 J grams. 
 1 lb. " = 45 1 
 
 (( 
 
 1 inch = 2 5 cms. 
 
 1 mile =16 kilometres. 
 
 1 cm. = -3937 inches. 
 
 v/^= 1-414214. 
 ^T= 1-732051. 
 
mum OF ELEMBTACT MECHAMMS. 
 
 THE MOTfON OF A PARTfCLP np . 
 
 t wi ., '^''IILLR, OH A POINT. 
 
 to the otl,e,.. '""'- ''■"'' P^"'" - --1 t„ ,„„vo .elativ;!^ 
 
 2. ^«'<"^ity of a particle «<„„,;„,.,„,„^ , 
 
 3. Such velocity ,„ay be ,■ "" """■"""• ''"«""'■ 
 
 ■notion. '""" '^".y l»'>it m the ,Ii,.ectio„ of 
 
 Examples. 
 
 (1) Horizontal nibtiou in line AP- , •• 
 
 ;:^^^^'-up%,.ep,.o.ent:r^;;.:i;7r-'^^ 
 
 tuTie by vertical line Ot, '^"^'^^ ^"'^«v ''^"^I 
 
 °' XV 
 
 Fi..'. I. 
 
 (2) Vertical motion in line ^/> !i ■.• 
 
 time ren,-P........ k . ^^' ^''^^^''^'^^•^ increasin.. 
 
 ^•epresentcd by horizontal line. 
 
 o .> 
 
SYLLAItUS OF 
 
 ' nr 
 
 Fi(!. 2. 
 
 (?,) Motion in oiirve J /' ; vcloeilios chiinging in both 
 ii)!i«>uitudo and difcetion ; time not rt'pivsontod. 
 
 Fig. 
 
 4. Strictly speaking a change of velocity requires time, and 
 does not take i)lace instantaneously. The so-called Instantaneous 
 change from IW to BC /Fig. 1) should therefore be represented 
 as in the anne.xed figure, the time being re[.resented by BB\ 
 
 B 
 
 . . »•*• 
 
 ^ 
 
 Fio. 4. 
 
 r 
 
< nr 
 
 EL EM EXT A li Y MECIIA XICS. 
 
 d 
 
 j.'lie velocities in collisions, etc., are represented thus. 
 
 5. Velocity, nvIkmi constunt, is calculated tVoui the formula 
 
 .V 
 V — J 
 
 where t may have any vmIuc, liiri^e or small. When the velocity 
 varies, the same lormula gives iippi'oximatiuns which heconie 
 closei' iis f is diminished. 
 
 The use of the Chronograph in measuring small intervals 
 of time. 
 
 6. Component velocity defined. 
 
 7. When the velocity alters in magnitude, or direction, or 
 both, there is said to be a chaixje of velocity. Hate of" change of 
 velocity is called accelei'ation or i-etardation. 
 
 8. When a velocity AH becomes AC in time t, BC is called 
 the chang(! of velocity in time t. 
 
 9. By the accelei'ation at a point, or indiud, is meant the 
 change that would take place in unit time, the acceleration 
 remaining constant for that time. 
 
 10. If the acceleration (/') is constant, and the motion recti- 
 tineal, and if velocity v become v in time t 
 
 f= 
 
 V — V 
 
 11. If / is constant, and the motion curvilinear, then the 
 value of / depends on the magnitude and directions of tiie 
 velocities at the beginning and end of any time. 
 
 Thus if in t seconds the velocity AJl at P (Fig. 5) becomes 
 AC at Q, 
 
 PC 
 /= — , (acceleration constant) 
 c 
 
 which can be calculated when AB, AC and the angle A are 
 known. 
 
 r 
 
SYLLAfiUS OF 
 
 « ««• 
 
 Fid. r>. 
 
 12. The component acceleration in a specified direction is the 
 acceleration of the corresponding' component velocity. 
 
 13. When the acceleration varies in magnitude, or direction, 
 or both, and a velocity A/i at P (Fig. G) becomes AC at Q in 
 
 time f, then ^ approximates more and more closely to the 
 
 acceleration at P as BC and f are taken smaller and smaller. 
 
 In uniform motion in a circle the acceleration at any point 
 
 is towards the centre and equal to — . 
 
 Fig. 6. 
 14. If the unit of time be changed from 1 second to t seconds, 
 the numerical value of the acceleration will be changed from/ 
 toft''. 
 
 r- •'^ ■ 
 
 t- 
 
ELEMEXTAKY MECHANICS. 
 
 9t%- m 
 
 t- 
 
 CALCULATION OF SPACE DESCRIBED IN RECTI- 
 LINEAL MOTION. 
 
 15. If V is constant, s = vt 
 
 16. H / is constant, and initial velocity = 0,8= J/ii^ , 
 being equal to the space described in t seconds with the average 
 velocity ^/t. If time and velocities be represented geometrically 
 (Art. 3) s will be equal to the area of the triangle Of v. 
 
 
 (Area=^yi52) 
 
 Fig. 7. 
 
 Is 
 
 In such cases of motion, since / = -y, it follows that f 
 
 may be measured by twice the space from rest in 1 second. Also 
 since the spaces described in 1, 2, 3, ... .seconds are respectively 
 
 2 ' 2 ' 2 ' 
 
 it follows that the distances traversed in the 1st, 2nd, 3rd, .... 
 seconds are as the odd numbers 1, 3, 5 .... 
 
 17. Let / be constant, and initial velocity be u. Here the 
 final velocity is w + ft, and s = id + ^ ffi , being equal to the 
 space described, with average velocity u -f J/^ in time t seconds. 
 In this case the space described will be represented geometrically 
 by the area Otvu. 
 
SYLLABUS OF 
 
 Ot -s^ t ^ Ou = 11 
 
 tv = u -f f/\ 
 (Area = nt -f h /V) 
 
 Fig. 8. 
 
 18. If a particle F moves horizontally tVoin a point with 
 an initial velocity u, and its velocity is subject to a constant 
 vertical acceleration <j, its iiorizontal velocity subsequently n^- 
 mains u, whilst its vertical velocity in time t becomes gt. Hence 
 the horizontal distance described by the particle is ut, whilst the 
 vertical distance is I gt'^ . 
 
 -_ ^ M A 
 
 Fi«. 9. Fig. 10. 
 
 Thus if P be the position of :he particle at the end of t 
 seconds, PN = ut, PM = hgt:^ . 
 
 Since ON also = hgfi , the time of flight from to P is 
 the same as the time of fall from to .V. 
 
 Experiment to illustrate. 
 
 Fig. 10 shews the component velocities (and also the 
 resultant velocity) of the body when at P. 
 
 19. If the particle is projected with velocity 7i at an angle a 
 with the horizon, and there is a constant vertical acceleration a, 
 then the horizontal velocity remains u ros a ihroughoi't ""he 
 
ELEMENT A R Y MECUA Xia^. 
 
 motion, whilst the vertical velocity at the eii.I of time t has 
 become ?< sin a — yf 
 
 Jill = /( sin '/ 
 6'// = II sin a. 
 
 f/f- 
 
 tf cos cl 
 
 H 
 
 Fi(t. 12. 
 
 Hence the horizontal distance P S\ in tinu; /, = u cos n.. f, 
 whilst the vertical distance PM = u sin a.t i iit- . 
 
 The vertical velocity, which = u sin a —yf^ becomes in 
 u sin a 
 
 time 
 
 y 
 
8 
 
 i^YLLABUS OF 
 
 MASS, DENSITY AND SPECIFIC GR VITY. 
 
 20. Tlie nmss of ca body is the qiuuitity of matter contained 
 tlieiein, and is measured l)y the number of unit masses it will 
 counterbalance in the process of weishino-. 
 
 The density, when uniform, is the mass of volume 1. 
 ThusJ/=i>r. 
 
 If D be the density, supposed uniform, of a substance, 
 and Z>„ the density of a standard substance (water), then the 
 specific gravity of the former is 
 
 Hence D = D^s, may be calculated from a table of 
 specific gravities when J)^ is known. 
 
 In the C. G. S. system D^ = 1, and therefore D = s . 
 
 MOMENTUM. 
 
 21. Momentum = mv, its direction being the direction of v 
 and its Rense the sense of v. Momentum may be represented 
 by directed finite straight Hues. 
 
 22. Component of momentum defined. 
 
ELEaMEXTA li Y MECHA MCS. 
 
 d 
 
 FORCE. 
 
 23. When a momentum mv, represented by AB (Fig. 5), 
 changes in time t to AC, BC is called the change of momentum 
 in the time t, and Force is said to act. 
 
 24. The rate of change of momentum at a point is the measure 
 of the/o»re there, mf, where m is the mass of the particle and/ 
 its acceleration. 
 
 25. Application of Newton's first and second Laws of Motion 
 to these cases. 
 
 26. Effects produced by a constant force acting for a specified 
 time. 
 
 UNITS OF FORCE. 
 
 27. If the units of length, mass and time be respectively a 
 foot, a pound (avoir.) and a second, the corresponding unit force 
 is called a j^oundal, or the British absolute unit of force. 
 
 Since the force of gravity on 1 lb. (mass) = 1 x 32, this 
 force (the pd.) is equal to 32 poundals. 
 
 28. If the unit mass is 32 lbs., the units of length and time 
 being a foot and a second as before, the corresponding unit force 
 is called a pound-force, or the British gravitation unit of force. 
 
 Thus, since the mass of 1 lb. would now be measured by 
 -3*2, the force of gravity on 1 lb. would be equal to ^^^ X 32 = 1, 
 the gravitation unit. 
 
 29. The French absolute unit of force is called a dyne, the 
 corresponding units of length, mass and time being a centimetre, 
 a gram and a second (the C. G. S. system). 
 
 2 
 
10 
 
 ^^YLLABUS OF 
 
 30. Tlio French gravitation unit of force is a kilogram-force, 
 tlio cor!'...sponding units of length, mass and time being a metre, 
 9*81 kilograms and a second. 
 
 1 kilo-force = 981,000 dynes. 
 
 31. The change of momentum in a speciHed direction is the 
 change in the corresi)oiidii)g component of momentum. Thus, 
 if the momentum AJi becomes AC, so that the change of momen- 
 tum is BCy the horizontal change of momentum is DC, and the 
 vertical change - BD. 
 
 Fig. 13. 
 
 For DC = AC — AB', the difference between the hori- 
 zontal components of AB, AC, whilst — BD =: CC — BE', the 
 difference between their veriical components. 
 
ELEMENTARY MECHANICS. 
 
 11 
 
 FORCES ACTING .SIMULTANEOUSLY. 
 
 32. When two finite forces act sinmltaueously on a particle, 
 the change of momentum which tliey would produce if continued 
 constant in any time, is such tliat its component in the direction 
 of either force is the same as if that force, continued constant, 
 acted singly for that time. 
 
 Thus two constant forces AB, BC, acting on a particle /' 
 at rest, would produce in unit time a momentum AC, the mo- 
 menta which they would separately produce in the same time 
 being AB, BC ; in half the unit time they would produce a 
 momentum AE — I AC, the components of which are AD, DK ; 
 and so on. 
 
 •K « 
 
 As the same effects would be produced by the single force 
 AC, it follows, as before, that the two finite forces AB, BC are 
 equivalent to A C, 
 
 33. The propositions of arts. 24, 31, 32 are involved in New- 
 ton's second Law of Motion which asserts that : 
 
 Change of momentum is proportional to the impressed 
 force, and takes place in the direction of the straight line in 
 which the force acts. 
 
 34. The Parallelogram of Forces. 
 
 35. Composition of any number of forces, (Graphical method). 
 
12 
 
 SYLLABUS OF 
 
 Example: The forces AB, BC, CD, DE, acting at a 
 {joint P, are equivalent to the single force AU. 
 
 Fig. 15. 
 
 36. Resolution of Forces. 
 
 37. The Triangle and Polygon of Forces. — Experiment. 
 
 38. Conditions of equilibrium, forces in one plane. 
 
 The algebraic sums of the components of the forces in 
 any two directions must separately varish. 
 
ELEMENT A R Y MECIJA NWS. 
 
 13 
 
 GRAVITATION. 
 
 39. According to Newton's law of gravitation, two particles 
 of matter attract each otlier with a force whose direction is the 
 line joining them, and whose intensity varies directly as (or may 
 be measured by) the product of their masses divided by the 
 square of the distance between them. 
 
 Thus F is j)roportional to 
 
 inin 
 
 0^2 
 
 m 
 
 m 
 
 Fig. 16. 
 The attraction of a body of sensible dimensions on a par 
 tide is accordingly the resultant of the attractions of its com- 
 ponent particles. 
 
 When the body is a spherical shell of uniform density its 
 attraction on an external pai tide is found by calculation to be 
 the same as if its mass were all concentrated at its centre. 
 
 Thus the attraction of shell of mass M on unit mass at P 
 
 is pp2^ which becomes — -- if the particle is on the surface. 
 
 Fig. 17. 
 If the particle be situated anywhere within the shell the 
 attraction is zero. 
 
 Hence it follows that the attraction of a sphere made up 
 of shells, each of uniform density, on a particle outside is the 
 
u 
 
 SYLLABUS OF 
 
 same as if the iiimss of tlie spliere were concentrated at its 
 centre ; whilst its attraction on a particle below its surface is the 
 attraction on it of the sj.lieie wliose radius istlic distance of the 
 particle from the centre. 
 
 Fig. 18. 
 
 40. The efl'cct of setting a sphere rotating round a diameter 
 is to introduce on a surface particle central forces tending towards 
 the axis (Art. 13). Hence, at the ecpiator if ?>i^ represent the 
 weight of a body, li the reaction of the earth, v the velocity of 
 the body and r the earth's radius. 
 
 V 
 
 m — ss= mg — M. 
 r 
 
 Therefore i? = when v"- =-. gr, in which case it will be found 
 that V would be about 17 times the velocity due to the actual 
 rotation of the earth. 
 
 41. Methods of determining r/. 
 
 At the equator g = 32-088 ; at the poles ^ = 32 2-52. 
 
ELEMEX TJ U Y ME( '//J MC:S. \ 5 
 
 WORK AND ENERGY. 
 
 42. If tlie point of application of a force moves, the force is 
 said to do w&rk. 
 
 The work clone hy a constant force F whose point of appli- 
 cation is displaced from A to />' is measured by /"cos OAB, where 
 is the angle between AB and the direction of /'. 
 
 Fi(4. 19. 
 If C be the foot of the perpeiidicidar from B on the line of action 
 of F at A, since AU cos = AC, the vyork is also measured by 
 F-AC, AC being the displacement estimated in the direction 
 of /'. 
 
 43. Since cos 0, or AC, may be -f , - , or 0, the work may 
 be + , — , or 0. 
 
 ^' 
 
 I't(j. 20. 
 
 44. When F is variable, an approximation to tlie work done 
 may be had by dividing the finite displacement into displace- 
 .nents so small that for each of them the force may be considered 
 constant, and estimating the work coiTesponding to each. 
 
 45. The unit of iris the work done bv unit force workin- 
 through unit length in its own direction and sense. " 
 
16 
 
 SYLf.A/irs or 
 
 In the British absoluto, system, the unit force ))eing a 
 poundal, the unit work is a foot-poundil. Tlie foot poiuulal is 
 therefore appr-oxiniately the work done by gravity wlien ^oz. 
 descends vertically 1 foot. 
 
 In the French absolute system (0. (i. S.), the unit foi'ce 
 being a dyne, the unit of work is an er(j. 
 
 In the British gravitation system, the unit force being a 
 pound-force, the unit of work is the foot-pound. 
 
 In the French gravitation system the unit force being the 
 weight of a kilogram, the unit of work is the kilograni-nietre. 
 
 46. The amount of work may be represented graphically by 
 setting off in one line lengths equal to the displacements esti- 
 mated in the direction of the forces, and drawing perpendiculars 
 thereto to represent the corresponding forces. The sum of the 
 ai-eas, of which these lines are adjacent sides will represent the 
 woi'k done. 
 
 Fig. 21. Fig. 22. 
 
 Thus if F, the force = Oy, and the displacement in the 
 direction oi F = Ox, W = rectangle yOx. If F diminishes from 
 OB to 0, and displacement = OA^ W ■= triangle BOA, 
 
 47. Work done by a blow. 
 
 48. The power^ or activity of an agent in its rate of doing 
 
 work, and is measured by the work that could be done iu unit 
 
 time. 
 
 The practical unit of i)Ower is 1 Horse-power, which is 
 
 equivalent to 33,000 foot-pounds per minute. 
 
 A Watt is equivalent to 10,000,000 ergs per second ; and 
 
 746 Watts = 1 Horse-power. 
 
ELEMENTARY MECIIAXICS. 
 
 17 
 
 • B 
 
 49. Tlie kinetic energy of a particle whose mass is m aiul 
 velocity is ?', is I mv- . 
 
 50. Wlien work is done on a particle, or system of uncon- 
 nected particles, the amount of work dotio or consumed is an 
 exact equivalent of the gain or loss of kinetic onerrjy. 
 
 Applicati(ms of this principle to establish .some of tii(> 
 laws of falling bodies. 
 
 51. Kx.'iniplcs : 
 
 (DA neavy particle m falls from rest thi-oiiirh a ]w\<A)t 
 /I ; the velocity attained being v, the gain of energy 
 IS -^ 7)11'- , and the work done by the wt'ight mg is 
 
 . • . J mv- = 7nf/Ji, or v- = 2f/h. 
 
 (2) How high will m rise, if its initial vertical velocitv 
 
 is 1 28 ft. per second ? 
 
 Here initial energy is Im (128)2 ^ f„^f^i eneigy is 
 ; word done arfninM gravity is ni>jh = 32 ,nh, if 
 h be the required height ; 
 . • . 32/«/^ = hm (128)2 , or h = 250 feet. 
 
 (3) If initial velocity downwards = n, and final = r, 
 
 then .J 771V- — h „m- = m(j's, where; s is the dis- 
 tance traversed. 
 
 Since in this ca.se i' = it + r^t , we get, on substi- 
 tuting for r, s= nt+ I gfi , as in Art. 17. 
 
 (4) If 7)i falls down a su)ooth inclined plane from 
 rest at A, and reaches B with velocity r, then 
 
 ^ mv- = in(j sin wAJi 
 
 = my AC. 
 
 Ym. 23. 
 
18 
 
 fiYLL Alius OF 
 
 Here the forccH are lutj and the rouction of the 
 plane. The effective part of the former in the 
 direction of motion is n\(f sin a, that of R is since 
 the surface is smooth. 
 (5) If m be projected witli velocity a at an anj^de a 
 with the horizon, the h)ss of eneigy wlien it 
 reaches tlie highest point of its path is 
 
 A rmi^ — A mil' cos^ a , 
 since u cos a is the vehicity there (Art. 19). If /t 
 is the lieight of this point 
 
 A i)w!^ cos'^ a = mylt ; 
 
 A init^ 
 
 and h = 
 
 ?^- sin- « 
 
 (6) If CJ represent a simple ])endnlum which starts 
 from the ))osition CA, its energy in the position 
 Cliy h mv^ , is equal to the work done by the 
 weight rng, i.e., nu/ BI). 
 
 Hence v'^ = 2rj BD =^ 2(/l (1 — cos C). 
 
 Observe that the tension 2'does no work, since the 
 displacement is |)erpendicularto its direction ; and 
 also that the string is supposed to be without 
 wTight. 
 
 T 
 
 • 4 
 
 •- » 
 
 Fig. 24. 
 
ELEMENTARY MECHANICS. 
 
 19 
 
 CENTRE OF MASS. 
 
 52. If the masses of a number of particles be denoted by 
 
 w, , m^ , . . _ and their distances from a plane by x , ,v 
 
 the distance .»■ of their centre of nvm, from the plane i's 
 
 -^ ''^h^x -f 'ni.y, -f ■ ■ . ■ 
 "'i 4- in^ -f . . . . 
 Two particles m, mi are said to have a centre of mass, or 
 centre of inertia, G, whose position is found from tlie formula 
 
 OG =s ''**^ "^ '"^ 
 
 Where x, x' are the distances of m, m', measured in the same 
 direction, from a point in the 
 
 8 
 
 line joinina- them. JHstaiiees measured to the left of must be 
 aflfected with a " — " siLni. 
 
 if coincides with A, the formula becomes 
 
 AG^'!'LA±. 
 
 53. If in the preceding case m and oi' move uniformly alon- 
 AB with given velocities u and u', the velocity of G will be 
 
 mu 4- m' w' 
 m -I- m' 
 
20 
 
 SYLLABUS OF 
 
 COLLISION. 
 
 54. If two particles, moving in the same line, collide, impul- 
 sive forces are brought into play, by the action of which the par- 
 ticles acquire at a certain instant a common velocity. If w be 
 this common velocity, then with the notation of Art. 53, the 
 change of momentum of m' during the first part of thf collision 
 is m!iv — m'u', and tliis represents the impulsive force or action 
 of m on m'. 
 
 Again the change of m^H momentum is mio — mu, and 
 this rej)resents the reaction of m' on m. Hence, by Newton's 
 Third Law, m'w — m'v' = — (mw — mu), and 
 
 mu + m'u' 
 
 tv = , 
 
 m -f 7n 
 
 which is the velocity of the centre of mass. After acquiring 
 this common velocity, the particles either move on together as 
 one mass, or, as generally happens, they separate with velocities 
 V, v' which can be determined by applying Newton's experimental 
 law, according to which 
 
 v' — V = 3{u — U'), 
 
 where e is a constant, depending on the nature of the particles. 
 This constant, which is called the coefficient of restitution, is al- 
 ways < 1. For glass it is 1^. 
 
 55. Let m, moving with velocity u, strike a similar particle 
 m at rest. 
 
 Then (I) m (w — u) = — m^v, or iv = 
 
 (2) m (o — vj) = — m (v' — w), 
 or v' -\- V = '2w = u. 
 
 2 
 
 Also r' — y = eu, 
 
 • ■ • ^' = o (1 — ^h and V = - (1 4- e}. 
 
miEMENTABY MECHANICS. 
 
 21 
 
 Hence iie = \ nearly, t- = and v' = „ nearly • or the 
 second Ul. would approximately move off with the li^oi 
 
 ExperiQient. 
 ^ ^. Graphical re|.re«entation of in,pact in the prece,li„g case, 
 
 ti^e, .■epres;:::r'r;ty:;:: '«:;: str °" *"^ "-^ -^ 
 
 /^^, the pe,,,endicuh,r tZn tl, ft/ I t, I"''"'' ""' 
 .second after impact. The veloeUv of , ' ™ "*^ °^ "'^ 
 
 equal to the perpendicular; :/L:, T*" '"""''■ ''^ 
 velocities heco»e e,nal to that o. the ^ i"lr' «' ^ ">« 
 
 / 
 
22 
 
 SYLLABUS OF 
 
 EQUILIBRIUM OF RIGID BODIES. 
 
 57. When the particles of a solid body are subjected to the 
 action of balanced forces, the relative positions of the particles 
 become more or less altered, and the body is said to be strained. 
 These strains bring internal forces or stresses into play which 
 disappear with the external forces. In order to avoid the con- 
 sideration of such internal forces we shall suppose solids to be 
 perfectly rigid, such, i.e., that the action of external forces pro- 
 duces no change in the relative positions of the component parti- 
 cles. 
 
 External forces acting on a body are of two kinds : — 
 
 { 1 ) Those which affect every part of the body ; e.g. the 
 attraction of gravity, magnetic attractions or re- 
 pulsions. 
 
 (2) Surface pressure, and attractions and repulsions 
 which are distributed over the whole or part of the 
 boundary ; e.g. the pressure of bodies in contact, 
 the pressure of the suri'ounding air ; attraction or 
 repulsion of electrified pith balls. 
 
 Forces of the former kind are specified as being so much 
 per unit mass ; forces of the latter kind as being so much per 
 unit area. 
 
 58. Moment of a force about a point defined : *•' -f " and 
 " — " moments. 
 
 59. When a number of forces act on a rigid body in the 
 same plane they will be in equilibrium if both of the following 
 conditions are satisfied : — 
 
 (1) The algebraic sum of the forces resolved in any 
 
 two directions must vanish. 
 
 (2) The algebraic sum of the moments of the forces 
 
 about any point in their plane must vanish. 
 
 60. Deduction of the preceding conditions from the Principle 
 of Energy. (Art. 60). 
 
 J 
 
 1 
 
ELEMENTARY MECHANICS. 
 
 23 
 
 PARALLEL FORCES. 
 
 61. To find the resultant R of two parallel forces P, Q, acting 
 at given points of a body. 
 
 Fig. 27. 
 
 Since — H, P,Q are in equilibrium, we have (Art. 59) 
 
 (1) Resolving in direction ^Z', 
 
 (2) Taking moments about C , 
 
 Q X AB 
 
 P X AC =^ Q X B'C . ' . AC =: 
 
 P+Q 
 
 CENTRE OF GRAVITY. 
 
 62. If the weights of the component particles of a body be 
 taken to be parallel, their resultant weight, which is their sum, 
 may be taken to act at a determinate point called the centre of 
 gravity (6'). 
 
 Thus if the particles be in one plane, the body on being 
 supported at G will be in equilibrium. 
 
 Hence, taking moments about a point 0, 
 
 Mg. X == m^g. x^ + m^g. x^-^ 
 
 where J/= m, + m^ + . . . . and a-, , :^-^ , . . . . a; , denote the dis- 
 tances of the vertical forces at m^ , m,^ , and G from . 
 
 « 
 
li 
 
 24 
 
 SYLLABdS OF 
 
 63. Centre of gravity of a straight wire, a triangular plate, etc. 
 
 64. Experimental determination of G by marking the vertical 
 through different points of suspension. 
 
 65. Stability and instability of equilibrium. 
 
 66. Balancing couples. 
 
 67. Examples of equilibrium, — the lever, inclined plane, etc. 
 
 68. The Balance — sensibility, stability. 
 Double-weighing. 
 
 FRICTION. 
 
 69. Hitherto the surfaces of bodies in contact have been sup- 
 posed perfectly smooth, so that tlie action at any point of the 
 surface could take place only along the normal there. In nature, 
 however, the surfaces of bodies are more or less rough, and the 
 direction of the mutual pressures between two bodies may 
 assume any position between the normal and a line inclined to 
 it at an angle whose magnitude depends on the nature of the 
 bodies in contact. The tangent of this angle £, which is some- 
 times called the angle of repose, is called the coefficient of 
 
 friction (//). 
 
 Thus, for earth on earth, damp clay, 
 ^ = 45°, /. = 1. * 
 
 For timber on metals, 
 
 £ varies from ll°-3 to 31^, /x from 0-2 to 0-6. 
 
 For timber on timber, 
 
 £ varies from ll'^'S to 26°-5, /x from 0-2 to 0-5. 
 
 When the action, R, between the bodies is resolved into 
 two forces, one, iV^, along the normal, and the other along the 
 surface of contact, the latter is called friction (F). 
 
ELEMEXTARY MECHANICS. 
 
 25 
 
 Friccion therefore assumes its greatest value, and tlie 
 body is on the point of moving, when the direction of li makes 
 the angle e with the normal. In this case 
 
 ^ = xV. tan e = /.. N. 
 70. Exam])les : 
 
 (1) Ten pounds resting on a level plane is on the point 
 of moving under the action of P applied at '60^ ; 
 
 P- = h 
 
 ^N 
 
 '10 
 
 Fig. 28. 
 
 Resuh ing vertically and horizontally, 
 ^ + - = 10, 
 
 /' = 
 
 /V8 
 
 
 .-. P== i-5,F== 3-9 (approx.) 
 
 (2) If 7^ = 3 in the i)receding example, resolving 
 
 horizontally, we iind /' = 3 x cos 30° = 2-G 
 nearly. 
 
 (3) W, resting on a rough inclined plane, is on the 
 point of moving down. Eesolving along and 
 perpendicular to the plane, we ha\e 
 
 F = W sin a 
 iV == ]r cos u . 
 • ■ • F = y tan a . 
 But F = y tan .- ; hence a in this case nntst equal 
 the angle of reposo, and its tangent is the Cf.effi- 
 cient of friction. 
 
26 SYLLA B US OF EL EMENTA R Y MECHANICS. 
 
 (4) Graphical representation of friction . 
 
 A given weight IF, placed on a rough level surface, 
 is oil the jjoint of moving, owing to the action of 
 a pull or a pu^sh. 
 
 Fig. 30. Fig. 31. 
 
 P is represented by BC, BC , BC" 
 
 The force of friction by CD, C D\ CD" 
 
 The least value of P is BC , making an angle s 
 with the horizon. 
 
 1 
 
ice, 
 
 lOf 
 
 ^^AAMPLES L\ JJECHANICS. 
 
 NOTE. 
 
 (1) As "pound" is ambieuous, and can reoresent .,>h., 
 Jor., the following clistinguishing^bbrev'tiorar:" ' '''^'' ^^ ^ 
 
 I /6. means 1 pound xAiass. 
 
 ».at«u:t t tT.: ^If """=■ '"^ ^••*'^ "^ «-^''^- '■■"^ otherwise 
 
 le £ 
 
 VELOCITY. 
 
 1. Give instances of the relative motion of two points : 
 
 (1) VVIiere tlie distance only varies • 
 
 (2) Where the direction of the line joining changes ; 
 
 (3) Where these two variations are sinn.itaneous 
 
 «,ir' " '' """-""'•^ '" '"^•' *"« -'-'*y «' « point, or 
 
 velochv *';%'^^«.r'''« '"' "" •"'"■"« " '™" iB moving with a 
 
 da th rfT' ''""'■' '' -'-'-•■y i-reases its°veIoc ty 
 '"Hi at the end of the minute is moving with a velocitv nf J-i' 
 ".■ les an hour. Find the velocity at th< end of L 30 i / 
 ^">d at the end of the 45th second. """''' 
 
 the 1, "^ ''""," " """"'^ ■'" ^ '*'■*'■«'" "»<'• A t the beginning of 
 the .m ,„,„„,y „f ,0 feet per second is suddenly Wvenl 
 
 tion of "o f T """ ' "'""' " "■'"•'■'^ "' '•'- "PP°-'^ di-'ee- 
 
 «t 
 
28 
 
 EXAMPLES IN MECH ASICS. 
 
 4. A body is thrown vertically upwards with a velocity of 600 
 feet per second ; the velocity regularly decreases to zero, and then 
 increases in the same manner. Draw a diagram representing the 
 velocities from the time the body leaves the ground until it 
 again reaches it. 
 
 5. A j)article is moving uniformly in a circle. From a point 
 draw lines to represent, in magnitude and direction, the velocity 
 at the different points of the path. 
 
 6. A rubber ball is dropped IVom a height h feet, falls for t 
 seconds, and strikes a horizontal surface with a velocity of v feet 
 per second ; it rebounds with half that velocity, and in t seconds 
 more again strikes the )»laue. Draw a diagram representing the 
 velocities. 
 
 7. A ball, moving with a velocity of 50 feet per second, is 
 struck witli a bat and leturned in the same straight line with a 
 velocity of 125 feet per second. Represent the velocities 
 geometrically. 
 
 8. A [)oint lias displacements 9 feet, 10 feet, 11 feet, 12 feet, 
 in four consecutive seconds. Find its average velocity for the four 
 seconds, for the first three second.s, and for the last three seconds. 
 
 9. A body is displaced 5 feet, 3 feet, 1 foot, — 1 foot, — 3 feet, 
 in five consecutive seconds. Shew that the average velocity for 
 the five seconds is 1 foot per second. 
 
 10. The spaces passed through in five consecutive seconds were 
 20 yards, 24 yards, ^^ yards, 32 yards, 36 yards. Shew that 
 the average velocities for the middle second, the three middle 
 seconds, and the five seconds are all equal. 
 
 11. A body starts with a velocity of 40 feet per second, and 
 during every second its velocity is increased by 9 feet per second. 
 Find the velocity at the end of 8 seconds. 
 
 12. A ball is moving along a smooth, horizontal surface with a 
 velocity of 10 feet per second. After moving for one second it 
 

 SXAMPLES IX MUCHANICS. jO 
 
 it 'tt th ''"^ T f '';"°"''' '*'°""^ ''^ '5 f««' P^-- -<=<""! i-' given 
 ■t , at the end of the second second it is ng.i„ struck and u 
 
 add,t.onal velocity of 25 feet pe. second is gFven it ; a 'tC e d 
 of he next second 20 feet ,,e.- second is added to he ve oci 
 and after this the ball moves fieelv PM ti,„ '"^^elocirx 
 in 10 seconds from the beginr.in./ ' ''"'"' "■'"■"•^'"' 
 
 ins!f„; il's 7ltr' " •"""V"""'-'^^" ""■f<"-™'->'' At a certain 
 
 14. iJofine componcuf, vdoc.it ij. 
 A iiian is vvalkiiu/ in n \r t? i- 
 
 4 ,n,les per hour ; Hnd the component velocities d-.e N „7l e 
 ■hi., respectively. ^ ^''. ^lau (iiie 
 
 15. A body is moving i„ a straight line with a velocity of 10 
 
 30 ' cr'of m':ir —^ »'■ -^^ --^^ ^" « --;: 
 
 J?d "Retf '"T" 'r '^.'*™'«'""- -i"' a velocity . feet per 
 second Resolve ,ts velocity along two lines at right angles one 
 of which makes an angle with the direction of motion 
 
 carrLfoSrtLtet:;f::r°\^ ''"^ -■"-• '« 
 velocity over the gLnd.'r ^ dtr ^0.10^" ''' ^^^' 
 
 velootvtf'sT'^'V' """""°" '''""^ *''" '"''S"''^' of "-"'b" with a 
 
 irrrtiqrrdr '"" "^ ^""■^""-^ --'''- 
 
 19. Find the resultants of the following pairs of velocities : 
 
 (a) 12 feet a second, and 16 feet a second, mutuallv at 
 right angles ; 
 
 («) « feet a second, and ,„ miles an hour, at right angles, 
 speed unchanged. What velocUy^ :a:S:n;^*:i.r:::::i ^^ 
 
30 EXAMPLES tN MECHANICS. 
 
 21. Find tlie resultants of the following pairs of velocities : 
 
 (n) 10 f(3et a .second, .'ind 20 feet a second, at an angle 
 
 of 60". 
 (h) 3 feet a second, and 3>/»2 feet a second at an angle 
 
 of 45"; 
 (c) 3 feet a second, and 3 feet a second at an angle 
 
 of 150'^ ; 
 
 22. A train is moving with the velocity of 20 miles per houj', 
 and the conductor throws out a parcel with a horizontal velocity 
 of 16.9 feet per second, in a direction at right angles to the line 
 of motion of the train. Find the resultant velocity of the parcel. 
 
 23. A spherical shot is rolling directly across the horizontal 
 deck of a ship, with a velocity of 10 feet j)er second ; find where 
 it would strike the side, supposing the ship, which is going at 10 
 miles per hour, to be suddenly stopped when the shot is 20 feet 
 from the side. (Neglect friction). 
 
 24. Three velocities are given simultaneously to a particle, 
 one 60 feet per second N., another 88 feet per second S. 60° W., 
 and the third 60 feet per second S. 60'' E. Find the magnitude 
 and direction of the resultant velocity. 
 
 25. A boat is rowed with a velocity of 6 miles per hour 
 straight across a river which Hows at the rate of 2 miles per 
 hour. If its breadth be 300 feet, find how far down the river 
 the boat will reach the opposite bank, below the point to which 
 it was originally directed. 
 
 26. A river one mile broa i is running at the rate of 4 miles 
 per hour, and a steamei-, moving at the rate of 8 n)iles per houi', 
 wishes to go straight across. How long will the steamer take to 
 perform the journey, and in what direction must it be steered 1 
 
 27. A balloon, rising vertically with a velocity of 10 miles 
 ))er houi", is carried by the wind over a horizontal distance of 100 
 yards in 20 seconds. Find the velocity of the balloon. 
 
 r 
 
 1 
 
r 
 
 EXAMPLES IN MEciIAXlt'S. 
 
 n 
 
 ACCELERATION. 
 
 1. Define accelemtion at a point. Kxplain ti.e necossity of 
 the phrase at a point. ^ 
 
 tion. At the beginning it has a velocity of 20 feet per second 
 an. at the end of 10 seconds its velocit/is 1.0 feet ^ J 
 
 -bind the acceleration. 
 
 beg,„„n,g of the 7 .sec-onas its velocity was 4. feet t' .1 ^^ 
 Fn.a the accelerat,on, unci the velo.i,,, after 3 «eeo,ul' n.o.e 
 lOo;, '^ P;';-"* "'»™» ^^'itl' "mfo,-,uly inereasi,,. velocity; at 
 
 llrutiorfr"''"^ '? "•■" P^"- »™™'"' -"- • -O"" 
 raoie u lb 10-1 feet per seeontl. Fiiul 
 
 (a) how far it goes i„ 1 second fron, 12 o'elock ; 
 
 (h) how far it goes in 2 spcoiuls ; 
 
 (o) how far it goes between 12.(.ll and 12.02 o'clock 
 
 at 2 45 1! " >'"'?■'■' 1'°'"' ''"' ■' ^''^'""'-^ "'■ ■ f-'^' !"■'• -^eco-Hl ; 
 3.30^ acceleration bemg constant ? What is the acceleration ! 
 
 6. A point, moving with nniform retarjation, has, at a eei- 
 urn instant, a veloacy of 30 feet per second ; after 5 se onds it 
 has a velocity of aO feet per seco„,l \V1 'it • '"°""^ '* 
 
 When will it ,„ ■ 1, "■'" *'" " ™™e '» '''•'St ? 
 
 Wiien will ,t again be moving at 30 feet per second ? 
 
 feet n ^ ''"'",*• ,"^"""8 «' •'' '^^''^in instant at the rate of 11 
 
 hou, ? ' '"""'"= "' "" '■'"'^ <"■ «0 '»il- per 
 
 seoot .^;^"7";'.P°"'' -.■""^''"S -i"' - velocity of 20 feet per 
 city lias It at 2. lo p.m., acceleration being constant I 
 
:\-2 
 
 KX AMPLER /X MKCnAXrCS. 
 
 V ' 
 
 9. A Uocly lias an initial velocity of 40 feet per second, and 
 in ') seconds it tmversns 300 teet. Find tlio accelenition, Hup- 
 poscd unifonn. 
 
 10. With 1 second iind 1 foot as units of time and space 
 respectively, tli(! accelcirati'jn is '<)'2. What is its measure wIkmi 
 a minute is .suljstituU'd lor a second ? 
 
 11. Which is the greater, an acceleration of 2,400 yards a 
 minute per minute, or 2 feet a second per second ? 
 
 12. A body starts from rest, and at the end of a second it is 
 moving with a velocity of 32 feet per second. If the increase 
 is uniform, wliat is the space traversed 1 
 
 13. llepresent geometrically the space passed over by a body 
 moving with constant acceleration : (a) starting from rest ; (6) 
 starting with a velocity u. 
 
 14. A point, in a certain interval of 3 seconds, passes over 300 
 4 feet ; in the next 4 i-econds its displacement is /64 feet. Shew 
 
 tliat the acceleration is 26 feet a second per second. 
 
 15. In a certain interval of 5 seconds a point goes 250 feet, 
 and in the next 8 seconds it goes 608 feet further. Find its 
 acceleration. 
 
 16. A point starts from rest under constant acceleration, and 
 alter 10 seconds it is moving at the rate of 25 feet per second. 
 How far does it go in the 20tli second of its motion 1 
 
 17. Prove that the spaces traversed in the 1st, 2nd, 3rd, etc., 
 seconds by a body moving freely under gravity, are proportional 
 to 1, 3, 5, etc., respectively. 
 
 18. A sphere of glass rolls down a smooth plane inclined at 
 30' to the horizontal. Its velocities at two points are 70 cm. 
 per second, and 140 cms. per second respectively. Find the 
 distance between the two points (^=980). 
 
 \ 19. A body is projected horizontally from the top of a tower 
 with a velocity of 100 feet per second. How far will it be from 
 the point of projection at the end of 2 seconds ? 
 
EXAMPLES ly MECHANICS. 
 
 nd 
 
 id. 
 
 lal 
 
 .at 
 
 IIQ. 
 
 Ihe 
 
 rev 
 )iii 
 
 20. A bullet is projected at an angle of 60' with the horizon- 
 tal, and with a velocity of 000 feet per secoiu'. Find its position 
 in 5 seconds, und its distance from the p(jint of projection. 
 
 21. A rifle is pointed horizontally, with its barrel 5 feet above 
 a lake. When discharged it is found that tiie ball strikes the 
 water 400 feet off. Find a[)proxiniately the muzzle velocity of 
 the ball. 
 
 22. Two bodies fall from heights of 49 feet and 81 feet, 
 and reach the ground simultaneously. What was the ititorval 
 between the instants of starting '? 
 
 23. A point starts from rest, and has an acceleration of 40 
 feet a second per second. Find the distance it passes over in the 
 fourth half-second of its motion. 
 
 24. A heavy body is projected in a horizontal direction from 
 the top of a tower. Prove that the vertical distance dropped 
 through varies as the square of the li'^rizontal space traversed. 
 Hence deduce the curve traced by the body. 
 
 25. Prove that the time of falling from the highest point of a 
 vertical circle down any chord is the same. 
 
 20. Find (by No. 25) the straight line of quickest descent from 
 a point F to a given straight line in the same vertical plane. 
 
 27- A stone, falling from rest for .5 seconds, passes through a 
 pane of glass, thereby losing \ of its velocity, and reaches the 
 ground 3 seconds afterwards. Find the height of the glass. 
 
 28. Two particles slide down two straight lines in a vertical 
 plane, starting simultaneously from their point of intersection. 
 Prove that the line joining them at any time is equal to the space 
 through which a particle would have moved in the same time, 
 along a line whose inclination to the horizon is the angle between 
 the given lines. 
 
 6 
 
u 
 
 EXAMPLES i:^ MECHANICS. 
 
 \ 
 
 PROJECTILES. 
 
 1. A heavy particle is projected in a horizontal direction 
 from any height. Prove that the path is a parabola. 
 
 2. The projection is in any direction other than the vertical. 
 Shew that the trajectory is a parabola. 
 
 3. Find the directrix, and shew that the velocity at any 
 point is equal to that which would be acquired by a particle 
 falling freely from the directrix to that point. 
 
 4. Find the focus, the vertex, and the velocity at tlie vertex. 
 
 5. A particle is projected with a velocity 60 feet per second 
 in a direction making 60° with the horizon. Determine the 
 position of the directrix, and the length of the latus rectum of its 
 path. 
 
 6. A particle is projected with a velocity of 2,000 feet per 
 second, at an inclination of 30° to the horizon. Find the 
 magnitude and direction of the velocity at the end of 10 seconds. 
 
 7. A bullet is projected with a velocity of 1,000 feet per 
 second, at an elevation of 15°. Find the range on the horizontal 
 plane, neglecting the resistance of the air. 
 
 8. What is the greatest height to which a particle will rise 
 if projected at an elevation of 30°, with a velocity equal to that 
 ^vhich it would gain hy falling freely through a vertical height 
 of 100 feet ? 
 
 9. Find the range in vacuo of a rifle-bullet projected with a 
 velocity of 1,200 feet per second, the direction of projection 
 making with the horizon an angle whose sine is y^. 
 
 10. Find the direction in which a stone must be thrown with 
 a velocity of 80 feet per second, in order to strike a small bird on 
 the top of a vertical pole, 20 feet higher than the point of pro- 
 jection, and 30 feet in front of it. 
 
 1 
 
^'XAMpLJfJs ly MECHANICS. 
 
 U 
 
 11. If particles be nroieofefl fiw^r,. fi,^ 
 
 13. If u and V are tlie velocities at the ends of n fno.l i i 
 - .. ...ojeotne. pat,.. ...a ...e i.nWa, vlX^fr i"' 
 
 4- 
 
 1 
 
 u' 
 
 V' 
 
 
 i^'jcii 11/ bi.iits with a velocifv d ^0f^. « t. 
 
 o^- '^0^ .bove the horizo,:7 ^n f^^ I^I^T' " ^^^ ^^^^^ 
 
 boito.n of the cliti. ^''*^ ^"^^""^^ ^'^^ ^''^^ 
 
 15. A shot leaves a ornn at the i-ifA r^f rna n . 
 Calculate the g.-eatestdi:tanee t Ih it 00^ I '" V"' 
 
 also the height to which it would rC ^""^ "' '""' 
 
% 
 
 EXAMPLES ry MECHANICS. 
 
 I 
 
 MASS, DENSITY AND SPECIFIC GRAVITY. 
 
 1. Define )nass and density. 
 
 Calculate the density of wute!-, the units being 1 foot and 
 1 lb., having given tbe fact that 1 c. in. of water contains 
 
 12S 
 
 3X6 
 
 oz. 
 
 2. If unit mass be 1 gin. and unit length be 1 cm., what is 
 the unit volume'? and what will be the measure of the density 
 of water 1 
 
 3. The .s. g. of mercury is IMO. Find the number express- 
 ing its density \\\ the English (foot-lb. -second) system, and 
 also in the French (centimetre-gram-second) system. 
 
 4. The s. g. of platinum is 21*5, when water is the standard 
 substance. What will be the s. g. when mercury is taken as 
 standard ? Will the measure of its density be cuanged ? 
 
 5. A cylindrical bar of silver is '1 in. in diameter. Find 
 jhe mass per unit of volume (density), and also the mass per unit 
 of length (the line density.) The s. g. of silver is 10*5. 
 
 6. A cubic foot of granite (s.g. 2'5) is moving with a velocity 
 of 50 feet a second. What is the momentum ? What is the 
 unit of momentum used 1 
 
 7. A cannon ball, weighing 18 lbs. is projected vertically 
 U})wards with a velocity of 250 feet ))er second. Find its?- 
 momentum at the end of 2^ seconds. 
 
 8. If it were projected at an angle of 60'' with the hor^ iontal, 
 after 3 seconds, what would be its vertical and its hoii antal 
 components of momentum ? Deduce the entire momentum. 
 
 9. Compare the momentum of a 15-lb. cannon ball, mov- 
 ing at the rate of 300 feet per second, with that of a 3-ounce 
 bullet moving with a velocity of 700 yards })er second. 
 
 10. Find the ratio between the momentum of an 81-lb. 
 ball moving at the rate of 100 feet per second, and that possessed 
 by a cubic foot of ice, »,g. "9, which has fallen freely for i\ seconds. 
 
 1 
 
 4 
 
 9 
 
 ^ 
 
EXAMPLES IN MECHANIC:, 
 
 37 
 
 '■J 
 
 i\ 
 
 FORCE. 
 
 1. Give Newton's First Law of Motion. 
 
 2. A mass of 4 lbs. is moving .ectilineally with a velocity of 
 
 wia is the wl T"T^ *''' -^ ^^'°"^"^- °f 20 feet per second 
 llr Lcond ; "'^ "^ '""""""»'" ' ^^'>»' - 'he change 
 
 3. What is the acceleration in Ex. 3 « Shew th»t tl,« .1 
 
 of momentum per second (or f„ce) = .ass f alS;: '^ 
 
 a — rftl""^"'""'^ '-"""/'-^•' «"" «'-v that it gives 
 
 7. If the force in Ex. 5 acted for 10 seconds on a mass of 5 
 ite. at rest, what would be the velocity generated ! 
 
 8. Two cubical bodies pmpI. r^f i ik 
 
 I'orizontal table Aft!- 9 ? V '""''' "■"'' °" * ''"»°"' 
 
 velocity of 3 f ei nt " r°"f',"'' "'•'* ''^ '""""=" ^'"' " 
 8 feet per second 7 ' ""'' *''' ■''"""'' "'"' '^ ^^'^^''^ "f 
 
 "f each' bodT 300, :;'' 1 ^'""""^ ,'"" "'""^^ '" ™°"-'- 
 velocities u^itirli!;"'""'''''"'" "•''''' ^^"' "^ '"^ 
 
 -con L;':r ""■ "™""- ^ ™"^'""' '■"'- -*» on it for 8 
 city ofi' /Af !" '' """"=' '" " ''^' ^- ''"«■'->■ -ith a velo- 
 
 omentun,, and also the rate of change of momentum. 
 
 1 
 
t 
 
 38 
 
 EXAMPLES IN MECHANICS. 
 
 10 Three separate experiments are made. A ball is let fall 
 freely for 1 second, thus obtaining a velocity of 32 feet per second. 
 Then it is taken and placed on a smooth horizontal table, and a 
 stretched elastic string is attached ; in ^ second it has obtained a 
 velocity of 32 feet per second. Again it is taken and struck with 
 a club, which gives in i^(f(,th of a second a velocity of 32 feet per 
 second. Find 
 
 (a) the whole change of momentum in each case ; 
 
 (6) the rate of change per second, and thus compare the 
 second and third forces with gravity. 
 
 UNITS OF FORCE. 
 
 1. Assuming 1 lb. (avoir.), 1 foot and 1 second, as our 
 fundamental units of mass, length and time, respectively, deduce 
 the units of velocity, acceleration, momentum ; and also of force. 
 
 2. What velocity will the force of gravity (or weight) give 
 to a lb. marfs in 1 second? What velocity will a poundal 
 give in 1 second ? Compare the two forces. 
 
 3. If we take as fundamental units 32 lbs. mass, 1 foot 
 and 1 second, what is unit of force ? 
 
 4. What are the three arbitrary units assumed in the French 
 
 (C.G.S.) system ? What is the unit force ? 
 
 5. Show that 1 kilo-force = 981,000 dynes. 
 
 6. If 1 kilo-force = 2i pds.. express 1 ounce force in dynes 
 
 {y = 981). 
 
 7. A force of 2 poundals acts on a mass of 2 lbs. for 5 
 seconds. Find the velocity generated. 
 
 8. Compare 10 poundals with 1 pd.-force, and find the velocity 
 generated if a force of 3 pds. acts on 5 lbs. mass for 8 seconds. 
 
I 
 
 ■'M 
 
 -: * 
 
 EXAMPLES IN MECHANICS. 39 
 
 9. A body of mass 6 lbs. is acted on by a force of 30 
 poundals ; find its velocity and momentum at the end of half a 
 minute from rest. 
 
 10. A forcfe equal to 1 pd. acts upon a ton mass ; what 
 acceleration is produced, and what will be the velocity at the 
 end of 10 minutes from rest. 
 
 11. A force of 10 dynes acts on 10 grams mass at rest for 10 
 
 seconds ; find the velocity generated. 
 
 12. A force F acts on 150 grams for 10 seconds, and produces 
 in it a velocity of 50 m. per second ; compare F with the weight, 
 of a gram. 
 
 13. If a body of 10 kilos, mass be acted upon for one minute 
 by a force which can just support 125 grams, what momentum 
 will it acquire ? 
 
 14. Compare the amounts of momentum in (1) a 06-lb 
 weight which has fallen for 2 seconds from rest, and (2) a cannon 
 ball of 12 lbs. moving with a velocity of 900 feet per second. 
 
 15. A body of mass 16 grams is acted upon by a force of 3 
 dynes for 5 seconds. Find the velocity and momentum acquired, 
 and the space passed through from rest. 
 
 16. A 7-lb. weight, hanging over the edge of a smooth 
 table drags a mass of 49 lbs. along it; find the acceleration 
 and the distance moved through in 5 seconds from rest. 
 
 17. A body rests on a smooth horizontal plane and a force of 
 30 dynes, actn.g along the plane, in 12 seconds imparts to it a velo- 
 city of 120 cms. per second ; what is the mass of the body ? 
 
 18. A mass of 15 lbs., lying on a smooth, flat table is 
 ac ed upon by a force along the table of 60 poundals. How far 
 will it move in G seconds ? 
 
 19. If the force in Ex. 18 were 5 pds., what would be the 
 distance ? 
 
f 
 
 40 
 
 EXAMPLES IN MECHANICS. 
 
 I 
 
 20. A force which can statidally support 50 pds. acts 
 uniformly for one minute on a mass of 200 lbs. Find the 
 velocity and momentum acquired Vjy the body. 
 
 21. A bucket weighing 25 lbs. is attached to each end 
 of a rope over a pulley, and in one bucket is poured a gallon, (10 
 lbs.) of water. Find the space passed through in 5 secoi.ds, 
 neglecting the mass of the pulley. 
 
 22. A ])article is on a smooth plane inclined to the vertical 
 at 60°. Find the space j)assed through in 2 seconds from rest. 
 
 23. A mass of 8 lbs. is on a smooth plane inclined to the 
 horizontal at an angle of 60°, and a force of 6 v/T pds. 
 pushes it up the plane. Find the velocity generated in 3 seconds, 
 and the space passed through from rest. 
 
 24. Two forces are a])])lietl, one to one mass and the other 
 to another mass three times as great, and produce velocities 
 30 and 20 respectively. Compare the forces. 
 
 25. A body of mass 3 is moving with velocity 2. What 
 impulse, acting perpendicularly to the original direction of 
 motion, will be necessary to turn the direction of motion 
 througl- an angle of 60° % 
 
 26. From a balloon at a given height, and rising vertically 
 with a given velocity, a stone is let fall. Find the velocity with 
 which the stone will strike tJie earth, neglecting the resistance 
 of the air. 
 
 27. A force A acts on a mass 30 for 5 seconds, and generates 
 a velocity 9 ; B acts on a mass 40 for 2 seconds, and generates 
 a velocity 10. Determine the ratio between the impulses of the 
 forces, and compare the magnitudes of the two forces. 
 
 28. If a force act on a body for 3 seconds from rest and 
 generate a velocity 60, determine the acceleration. What accel- 
 eration could this force produce in another body of double the 
 mass 1 
 
ds. acts 
 
 
 Find the 
 
 
 ach end 
 
 
 lion, (10 
 
 I 
 
 secoi-ds, 
 
 
 vertical 
 
 
 Q rest. 
 
 
 id to the 
 
 
 ^"3 I'ds. 
 
 
 seconds, 
 
 
 le other 
 
 
 velocities 
 
 
 What 
 
 
 Dtion of 
 
 
 motion 
 
 
 ertically 
 
 
 ity with 
 
 
 jsistance 
 
 
 enerates 
 
 
 enerates 
 
 
 3S of the 
 
 
 •est and 
 at accel- 
 
 i 
 
 ible the 
 
 
 EXAMPLES IX MECHANICS. 41 
 
 29. If two bodies propelled from rest by equal unifc.nn 
 pressures describe the same space, the one in half the time that 
 the other does, conpare their final velocities mu.I momenta. 
 
 30. A force /'generates in a body in one minute from rest •. 
 velocity „f 1,300 miles per hour; which is the greater force /' 
 or gravity ? ' 
 
 31. In the equation ir = mg, explain the meaning of each 
 symbol. What is the unit by which ]Y is measured ? 
 
 32. A shot of mass m is fired from a gun, whicii is free to 
 move and whose n.ass is J/, with a velocity n, relative to the 
 
 gun ; shew that tiie actual velocity of the shot is - -^^''- and 
 
 m -f M ' 
 
 that of the gun is . 
 
 m + M 
 
 FORCES ACTING SIMULTANEOUSLY. 
 
 1. ^ .Sliew that the principle of " the physical independence of 
 torces follows from Newton's Second Law of Motion. 
 
 2. Exi)lain the Tarallelogram of Forces. 
 
 3. Three constant forces, each equal to 8 poundals, act on a 
 partHe, the second at an angle of 120= with the first, and the 
 third at an angle of 120° with the second, and towards the same 
 parts as the first and second. Draw a diagram, and find the 
 mag'iitude of the resultant. 
 
 4. Explain the Triangle and Polygon of Forces ; and describe 
 an experimental proof. 
 
 5. If the third force in Ex. 3 wer.. towards oi>posite parts 
 apply Ex. 4 to shew that there would be equilibrium 
 
 6 
 
r 
 
 42 
 
 EXAMPLES IN MECHANICS, 
 
 6. Two forces act on a particle, and tlieir greatest and least 
 resultants are 72 pds, and oG pds. Find the forces. 
 
 7. Find the resultant of two forces, 12 pds., 35 pds., 
 acting at right angles on a particle. 
 
 8. Two forces, whose magnitudes are as 3 is to 4, acting on a 
 particle at right angles to each other, {produce a resultant of 15 
 pds. Find the forces. 
 
 9. Two forces of 8 pds. and 10 pels., respectively, act 
 upon a particle at an angle of GO'"'. Find the resultant. 
 
 10. Forces of 10 pds. and 12 v^a pds., respectively, act 
 at a point at an angle of 150". Find their resultant. 
 
 11. Two forces, 5 pds. and 15 pds., are kept in equili- 
 brium by a force of 18 pds. Find the angle between the 
 forces. 
 
 12. The resultant of two forces, acting iit an angle of GO'^, is 
 21 pds. ; one of the components is 9 [)ds. Find the other. 
 
 13. Shew that if the angle at whicii two forces are inclined 
 is increased, their resultant is diminished. 
 
 14. How can forces of 13 pds. and G5 pds, be a])plied to 
 a particle so that the resultant is 22 pds. ] 
 
 15. Two strings, whose lengths are G inches and 8 inches, res- 
 pectively, have their ends fastened at two points distant 10 inches 
 from each other ; their other ends are tied together, and they are 
 strained tight by a force at the knot equivalent to 5 pds. acting 
 perpendicularly to the straight line joining the })omts. Find the 
 tension of each string. 
 
 16. Two forces are represented in magnitude and direction 
 by two chords of a circle, drawn from a point on the circumfer- 
 ence at right angles to each other. Shew that the resultant is 
 represented in magnitude and direction by the diameter which 
 passes through the point. 
 
I 
 
 and least 
 
 35 pds., 
 
 ting on a 
 ^nt of 15 
 
 I'ely, act 
 
 ely, act 
 
 n equili- 
 ,'een the 
 
 f GO'', is 
 
 V. 
 
 inclined 
 >plied to 
 
 ;hes, res- 
 inches 
 they are 
 s. acting 
 Find the 
 
 lirection 
 rcumfer- 
 iiltant is 
 ;v which 
 
 
 EXAMPLES IX MEfHANlca. 43 
 
 17. A wul li are fixed points ; at a point M forces of aiven 
 ".«gn,t,Kle act along MA an.l MB ; if their resultant is of con- 
 stant n.agnit.ule, sUew that M lies on one or other of two eqnal 
 arcs described on AB as chord. 
 
 18. State the conditions for eqiiilibrium when co-planar forces 
 act on a particle. 
 
 Express these conditions analytically. 
 
 19. AllCD is a sqnare; three forces, 1 pd., 2 pds .3 
 pds respe^ctively, act on a particle : their directions are paralld 
 to AD, AB, CB. Find their resultant. 
 
 20 Three forces, represented by the nund.ers 1, 2, 3, act on 
 a P.rt,cle in directions parallel to the si.les of n . qu latm" 
 tnangle taken in order. Determine their resultant. 
 
 21 Can a particle be kept at rest by three forces whose 
 magnitudes are as the numbers 3, 4, 7 ? 
 
 22. The circmiference of a circle is divided into any number 
 ot equal parts ; forces are represented in magnitude and direction 
 by straight I.nes drawn from the centre of the circle to the 
 pomts of d,v,s,on. Shew that the..- forces are in equilibrium. 
 
 23. The circumference of a circle is divided into a given odd 
 
 B .a,ght hnes are drawn to the ,-est. Find the magnitude and 
 direction of the resultant 
 
 24. A body of 8 lbs. mass is on a smooth plane inclined to the 
 
 angle of 45 to the plane, will keep the body in equilibrium. 
 
 25. Forces; of 3 poundals and 4 poundals, at right angles to 
 each other, act upon a mass of 5 lbs. Find the acceW io,! 
 pn^cluced. How long will the mass take to move .8 l^lZ 
 
 26 Two horses pull at a block of stone : one with a hori- 
 ■=ontal force of 100 pds, the other with a horizontal force of 
 
44 
 
 EXAMPLES IX MECHANICS. 
 
 130 pds., the forces being inclined at an angle cos ~ ^ \f^. 
 Wiiat is the force which can keep tlie stone at rest? 
 
 27. bV)ices of 40 pels., 41 pels, and 9 pds., acting at a point, 
 are in eipiilibrium. Sliew that two of them are at right angles. 
 
 GRAVITATION. 
 
 1. State Newton's Law of Gravitation. 
 
 2. Three ecpial masses are placed at three successive points 
 of a regular hexagon of 6-inch side. How far from the centre 
 of the polygon must be placed a body whose mass is eight times 
 each of these masses, in order to counteract their attractions on a 
 particle at the centre 1 
 
 3. Apply the Integral Calculus to prove that the resultant 
 attraction of a spherical .shell of uniform density on a particle is 
 the same as if the shell were condensed to a point at the centre. 
 
 4. From the law for m shell of uniform density, deduce the 
 law for a sphere. 
 
 5. A sphere of 1 foot radius is placed with its centre 5 feet 
 from a particle, and from the centre of this sphere another of half . 
 its radius is cut. Find the radius of a third sphere which, when 
 placed 5 feet from the ])article, will attract it as strongly as the 
 part that is left. 
 
 6. Explain why a body weighs (apparently) more at the pole 
 than at the equator. 
 
 7. Shew that if the day were about 1 hour 25 minutes long, 
 bodies at the equator would appear to be without weight. 
 
 8. Explain hov>^ to find the value of y. 
 
 9. At the equator the value of y is 32-09, and in London 
 32 2. If a merchant weie to buy tea at the equator at a shilling 
 
t 
 
 
 £XA.MI'r.Es IX irECIIAXIfS. 
 
 40 
 
 Pe'- II.., unci sell it i„ Loiulon, «t «l„.t mto p,.f lb (,. uent^ 
 
 must 1,0 s,.„ s,, U,ut 1. „„.y „eitl,e.. ,.i„ L ,o«V i 
 
 tl-t l,e ,,«. the .s„,„e .,„,„,, ,,..,.,„.„. ,,,,,„, ^,.„„^,^^ ;.^,;^ ^'"8 
 
 ^_^ia W„,„,, H,i. .liHiculty „,.,. „,t,, ,„.. eo,„,.„„. ,.e„„. 
 
 fi onn '^;'''''"'? "" "'"■"' •''"■""^ """' if« 'li'""eter was only 
 itant ? I l,e dmnieter ,s ,,|,,„oxi,„„tely 8,000 mites. 
 
 M tl '"■ h''''°T •"'" """' "•■ "'" "'" '° '» 300,000 time, 
 
 14. A small Jiole runs to the contrp nf fi,^ .u 
 -.,• to t„e centre a particle is pi, e^ 'Lw f .Ih t 'f;;;'' 'f 
 fii«t second considerincr the nff,. .• ^^'^ it tall ,n the 
 
 tances? ^' " ^'''^^'^ou constant for small dis- 
 
 15. The moon's mass is 136 v 1021 iko i 
 "X IO«feet; t„e mass of the ean./i^ H J ^""lo.™ t^' 
 and Its rudius 21 X 10« feet P,n,l 1, f JO-' ll>.s., 
 
 .-.■face would falH„ a secoL I e , 7 f '' " ""'" '^ ''"' "'"""'' 
 tion of the e.,,h bei^ Tegleld "'' '"' '° "'<' '^"'- 
 
msb~^ii:n3uSSS3S 
 
 46 
 
 EXAMPLES IN MECHANICS, 
 
 ! 
 
 WORK AND KNKHdY. 
 
 1. Explain the four common units of work, 
 
 2. Shew tlmt tlio woi'lc done in piisliiii;^ a heavy hody alon^ 
 a smooth inclined plane is ccmal to the work done in raising the 
 same body through the correspond in,i( vertical height. 
 
 3. A force of 4 pds. Jicts through 2 feet ; then it is 
 suddenly changed into a force of 5 pds. and acts through 2 
 feet ; and then it is chang(Ml into a force of 7 pds. and acts 
 through 3 feet. Find the whole work, and represent grajihically. 
 
 4. Define the terms Jlorae-jnywer and Watt, 
 
 5. A force of 18 pds. acts against a resistance, and in 
 pas.sing through 20 feet falls uniformly to zero. Find the work 
 done by the force. 
 
 6. Find how many cubic feet of water an engine of 40 horse- 
 power will raise in an hour from a mine 80 fathoms deep, sup- 
 posi'ig 1 cubic foot of water to weigh 1,000 o ^ces. 
 
 7. A cubical stone of 6-foot edge wei^ !60 lbs. to the 
 cubic foot. Find the work done in turning it to an adjacent face. 
 
 8. l^'iud the accumulated work (or kinetic energy) in a body 
 whi(.h weighs iiOO lbs. and has a velocity of G4 feet per second. 
 
 9. Calculate the kinetic energy possessed by a 20-lb. 
 cannon ball moving with a velocity of 512 feet |)er second. 
 
 10. Ciilculate the horse-jiower of a steam engine which will 
 raise 30 cubic feet of water per minute from a mine 440 feet deep. 
 
 11. Shew that when weights are raised vertically through 
 various heights the whole work is the same as that of raising a 
 weight ecpial to the sum of the weights vertically from the first 
 position of the centre of gravity of the system to the last. 
 
 12. A well is to be made 20 feet deep, and 4 feet in diameter ; 
 find the work in raising the material, supposing that a cubic 
 foot of it weighs 140 lbs. 
 
t:XAMPLES IX MECIIAMCS. 
 
 47 
 
 h. 
 
 a 
 
 1st 
 
 I'; 
 
 ic 
 
 i 
 
 13. A Hliuft (I feet in depth is full of water, Fin<l the depth 
 of the suifjice of the water wlieii one quarter of tlie work 
 required to empty the nliaft has been done. 
 
 14. If H pit 10 feet deep and with an area of 4 sqiuire feet 
 he excavated, and the eartli thrown up, how nuich work will 
 have been done, snpposini^ 1 cubic foot of earth to weigh 0() 
 pounds { 
 
 15. A 4-ounce bullet is j)rojeeted vertically upwards with a 
 velocity of tSQO feet per second. What is its potential energy 
 when it has ascended to its inaxinnun height? 
 
 16. What are the units of energy in the Frencii and English 
 absolute systems when tlie kinetic energy is defined to be equal 
 to i mv"l 
 
 17. Compare the amount of kinetic energy in (1) a boulder 
 of 112 lbs., which has fallen for a .second from rest, and (2) 
 a 1-lb. jnojectile moving with a velocity of 8(>0 feet per second. 
 
 18. What is the work done in displacing, through an angle of 
 GO', a spherical bob of G lbs. suspended by a cord 4 feet 
 long '\ 
 
 19. A cricket ball weighing 5 ounces is given, by a blow, a 
 velocity of G5 feet jier second. Measure the work done. 
 
 20. An engine is drawing a train of 120 tons up a (smooth) 
 inclined plane, 1 in GO, at the rate of 24 miles per hour. How 
 much work is being done per second { 
 
 21. Supposing that a man of 12 stone in walking raises his 
 whole weight a distance of 1 inch eveiy stej), and that the length 
 of the step is '2h feet ; find how much work the man does in this 
 way in walking a mile. 
 
 22. A man has to raise a cwt. of bricks 8 feet ; he throws 
 them up so that they arrive at a i)oint 8 feet high with a velocity 
 of 5 feet a second. Compare the necessary work with the super- 
 fluous work done. 
 
 23. Reduce a kilogrammetre to foot-pds. and reciprocally. 
 
 i^ 
 
48 
 
 EXAMPLES IN MECHANIC'S. 
 
 24. Compare the amounts of kinetic energy in a pillow of 20 
 lbs. which has fallen through 1 foot vertically, and that of an 
 ounce bullet moving at 200 feet per second. 
 
 25. A ball weigliing 5 ounces, and moving with a velocity of 
 1,000 feet per second, strikes a shield, and after jnercing it moves 
 on with a velocity of 400 feet per second. How much energy 
 has been expended in piercing the sliield ? 
 
 26. Calculate the kinetic energy of a hammer of one ton let*^ 
 fall half-a-foot. « 
 
 27. The bob of a simple pendulum is pulled through an arc 
 of 60^ and let go. Compare the kinetic enei-gy after describing 
 an arc of 30' with its energy at its lowest point. 
 
 28. A train of 120 tons runs on a level road, and the resist- 
 ance ;o be overcome is 16 pds. per ton. How many units of 
 work must be px[)ended in making a run of 40 miles 1 
 
 29. A body is projected with a velocity n at an angle a with 
 the horizontal. By the method of energy find the hei<'ht to 
 which it will rise. 
 
 30. A shot travelling at the rate of 200 metres per second is 
 just able to pierce a ])lank 4 cm. thick. What velocity is required 
 to i)ierce a plank 12 cm. thick, assuming the resistance propor- 
 tional to the thickness of the planks ? 
 
 31. If a bullet, with a velocity of 150 metres per second, can 
 penetrate 2 cm. into a block of wood, through what distance 
 would it penetrate when moving at the rate of 450 metres per 
 second ? 
 
 32. A shot travelling at the rate of 300 metres per second can 
 just penetrate a plank 3 cm. thick ; it is tired through a plank 5 
 cm. thick, with a velocity of 600 metres per second. Find the 
 velocitv with which it emer<;es. 
 
EXAMPLE>S ly MECHAXICS, 
 
 49 
 
 CENTRE OF JNIASS. 
 
 1. Five masses, 1 lb., 2 lbs., 3 lbs., 4 lbs., 5 lbs., .are situated 
 at distances fr-oni the horizontal plane 2 feet, 4 feet, 6 feet 8 
 feet, 9 feet, respectively. Find the height of the cent're of m.ass 
 above the horizontal phme. 
 
 2. Find the centre of mass of two masses of 18 lbs. and 
 24 lbs., resi)ectively, situated 21 inches apart. 
 
 3. Two masses, T) lbs. and 7 lbs., move uniformly in the 
 same straight line with velocities 8 feet aad 10 feet per second 
 respectively. Prove that their centre of n.ass moves uniforndy! 
 and find its velocity. 
 
 4. Masses 2 lbs. and 3 lbs. connnence simultaneously and 
 move unifoimly in parallel stiaight lines at right angles to 
 the Ime joining their first positions; the smaller n.ass has a 
 velocity of 2 feet per second, and the larger a velocity of 1 foot 
 VV.V second. Shew that the centre of mass moves uniformly in a 
 straight \nn\ and find its velo 'fty. 
 
 5. Two bodies of masses, m and m' lbs., move uniformly in 
 a straight line ; the fo.-mer with a velocity of v feet per second. 
 J^ind the velocity with which the latter mass must move that the 
 centre of mass may remain fixed. 
 
50 
 
 EXAMPLES IN MECHANICS. 
 
 COLLISION. 
 
 1. An inelastic, heavy particle of mass m falls from a height 
 li upon a horizontal plane. Find the velocity with which it strikes 
 the plane, and the momentum destroyed. 
 
 2. If the cootHcient of restitution between the particle and 
 the plane be e, find the height to which the particle will rise 
 on first lebound, and the whole change of momentum immedi- 
 ately after rebounding. 
 
 3. If a lb. mass fall from a height of 50 feet to the 
 ground, what is the impulse of the pressure which it will exert, 
 supposing it inelastic ? 
 
 4. A ball falls from a height of 25 feet \\\)0\\ the floor. Find 
 the time occn})ied until it strikes the floor the third time, and 
 the height to which it will rise after that impact {y = 32, e = \). 
 
 5. If a piano is moving witii a velocity of 2 feet per second, 
 and a sphere of mass 4 ll)S. be moving in the same direction 
 with a velocity of 8 feet per second ; find the velocity of the 
 sphere after impact, and the whole change of momentum, sup- 
 posing the coefficient of restitution is |, and the motion of the 
 plane is unchanged. 
 
 6. If the elasticity were jierfect, what would be the velocity 
 of the sphere 1 
 
 7. Shew that if two perfectly elastic balls of equal mass, 
 moving in the same straight line, impinge upon one another, they 
 will exchange their velocities. 
 
 8. A sphere of mass 8 lbs. strikes a smooth horizontal 
 surface at an angle to the normal of GO"^. Determine the ande 
 of reflexion, snpposing the coellicient of restitution to be |. 
 
 9. ivxplam why the force of -ravity nerd not be taken into 
 account in di>;eus>ing (pu-.stions of im])act. 
 
 
 I 
 
 i 
 
EXAMPLES IN MECHANICS. 
 
 61 
 
 / 
 
 I 
 
 10. From Newton's Third Law deduco, and write down the 
 equation stating, that the al_yrl)iMii':d siuu of the niojnenta of two 
 balls is the same after impact as ix'i'oic. 
 
 11. An inelastic body impinges on another of twice its mass 
 and at rest. Shew tliat th(^ iinpingini;' body loses two-thirds of its 
 velocity. 
 
 12. A body of 5 lbs., moving with a velocity of 14 feet 
 per second, impinges on a body of 3 lbs., moving with a velo- 
 city of S feet per second. Find the velocities after impact, sup- 
 posing e = §. 
 
 13. A ball weighing 8 lbs., and moving with a velocity of 
 12 feet per second, strikes directly a ball of 12 lbs., moving 
 with a velocity of 8 feet pei' second in the opoosite direction, 
 the coefficient of restitution being \. Find the velocity of each 
 after impact, and tiie impulse of the pressure. 
 
 14. A ball falls IVom rest at a height of 20 feet above a fixed 
 horizontal table. Find the height to which it will rebound, e being 
 ^ and g 32. 
 
 15. Two bodies arc; nioving in the .same direction with velo- 
 cities 7 and 5, and after impact their velocities are 5 and G. Find 
 the coefficient of restitution and the ratio of the masses. 
 
 16. A impinges on B at rest, and is itself reduced to rest by 
 the impact. Find the ratio between the masses, if e = }j. 
 
 17. A, J) and C are the masses of three bodies, which are 
 formed of the same substance ; the first impinges on the second 
 at i'(>st, and the second impinges on the thii'd at rest. Find the 
 vahie of e in order that the velocity communicated to C may be 
 the same as if .1 impinged directly on C. 
 
 18. A particle is projected horizontally with a velocity of 40 
 feet per second from a point 30 feet above a fixed horizontal 
 j»lane. Find the height to which it will rise, and its range after 
 the first rebound ; the coefficient of restitution being J. 
 
^mm mmm 
 
 52 
 
 EXAMPLES n^ MECHANICS. 
 
 19. A shot of 000 lbs., nii'l movinj; with w velocity of 
 1,200 feet per second, enters the side of a sliip of 6,000 tons, ai»d 
 remains imbedded in it. Find the velocity which it communi- 
 cates to the ship, neglecting the resistance of the water. 
 
 20. A ball of elasticity e is dropped from a heiijh'. h upon a 
 horizontal plane. Shew that the whole distance tlirou^'h which 
 it moves before coming to rest is 
 
 21. A ball impinging on a smooth plane surface has its direc- 
 tion turned through a riglit angle. Find the angle of incidence, e 
 being the coefficient of restitution. 
 
 22. A BC is a horizontal circle ; a Ijall projected from A is 
 reflected at B and G and returns to A. Shew that the ratio of 
 the time from A to B to that from C to -4 is equal to the coeffi- 
 cient of restitution. 
 
 23. A ball falls from a given height above an elastii; smooth 
 plane. Prove that the time of hopping is the same iov all incli- 
 nations of the plane. 
 
 24. A smooth circular ring rests on a smooth horizontal 
 table, and a small spherical mass is projected from the centre of 
 the circle, with velocity v. Shew that the whole time which 
 elapses before the nth impact is 
 
 a 2 — e»-i — e" 
 V e"~^ — e" 
 
 25. A smooth ring is fixed horizontally on a smooth table, 
 and from a point of the ring a particle is projected along the 
 surface of the table. If e be the coefficient of elasticity between 
 the ring and the particle, shew that the latter will, after three 
 rebounds, return to the point of projection if the initial direction 
 makes with the normal to the ring an angle tan ~ ef. 
 
EXAMPLES IN MECHANICS, 
 
 &3 
 
 EQUILIBRIUM OF RIGID BODIES. 
 1. Explain the terms strain and stress. 
 
 suppose soIkIs to be perfectly . igi<l. 
 
 3. Give two general divisions of external forces, furnishing 
 examples ; and explain how the two kinds are to be n.eusured. " 
 
 4. Explain, on mechanical principles, why a sharp point will 
 so easily penetrate a body. i i ^ ^i" 
 
 5. Which is the greater : 30 ounces per square inch, or 1 \ 
 tons per square yard ? ' ^2 
 
 6. If water weighs 1,000 ounces per cubic foot, how high 
 must the water be in a tube in order to exert a pressure of 15 
 pds. per square inch on the bottom ? 
 
 7. Reduce 1 pd. per square foot, and 1 pd. per square inch, to 
 dynes per square cm. (^ = 981). 
 
 8. The average weight of a man is 12 stone, and on an aver- 
 age 5 men occupy 7 square feet. Find the pressure per square 
 toot due to a dense crowd on a bridge. 
 
 9. What is the pressure, on a sluice-gate, 12 feet broad 
 against which the water rises 5 feet ? 
 
 10. Define moment of a force, and explain the meaning of 
 positive and negative moments. 
 
 11. The algebraical sum of the moments of two forces ronnrl 
 a point in the plane containing the forces is equal to the moment 
 ot their resultant. 
 
 12. The algebraical sum of the moments of two forces which 
 torm a couple is constant round any point in the ,,lane of the 
 couple. ' 
 
54 
 
 EXAMPLES IN MECHANICS. 
 
 13. P and Q are fixed points on the circumference of a circle ; 
 QA and QB'avq any two chords at right angles to each other, on 
 opposite sides of QP. If QA and QB denote forces, shew tliat 
 the difference of their moments witli respect to P is constant. 
 
 14. Forces are represented in position, magnitude? and sense 
 by the sifles of a closed polygon taken the same way i-ound. Prove 
 that the sum of the moments of these forces about any i)oint in 
 their ])lane is numerically equal to twice the area of the polygon. 
 
 15. State the conditions for equilibriutii when a number of 
 co-planRr forces act on a rigid body. 
 
 16. A uniform rod weighing 10 lbs. is hinged to a verti- 
 cal wall, and can turn in a vertical plane. It is held in a hori- 
 zontal position by a string attached to the end and fastened to a 
 point in the wall above the hinge, a distance the length of the 
 rod. Find the tension of the strinjr. 
 
 17. A ladder who.se weight (99 lbs.) acts at a point one- 
 third of its length from the foot, is made to rest against a smooth 
 vertical wall, inclined to it at an angle of 30'', by a force applied 
 horizontally at the foot. Find the force. 
 
 18. Find the true weight of a body which is found to weigh 
 8 ounces and 9 ounces when placed in the right or left scale-pans 
 of a false balance, respectively. 
 
 19. A weight of 12 lbs. is suspended from a tixed hook by 
 a string ; a .second string is tied to the weight, and by pulling it 
 horizontally the tirst string is caused to make an angle with the 
 vertical whose cosine is % Find the forces apj)lied by the 
 strings. 
 
 20. A picture, the weight of which is 4 lbs, is suspended 
 from a nail by a flexible cord ; the top of the picture is hori- 
 zontal, and the angle between the two parts of the cord is 30°. 
 Find the tension of the cord. 
 
 i 
 
 J 
 
EXAMPLES IX MECHANICS. 
 
 55 
 
 21. The arms of a lever are 2 feet and 3 feot lespectively. 
 What force acting at an angle of 30' to the longer arm will balance 
 a force of 30 pels, acting at right angles to the shorter arm ? 
 
 22. A lever, with a fulcrum at one end, has arms sucli that 
 one is 3 feet longer than the other. If the power is 10 times the 
 weight, what is the length of the lever? 
 
 23. A uniform straight rod, 2 feet long and weighing 2 
 lbs., rests in a horizont.J position between two fixed pegs 
 placed at a distance of 3 inches apart, one of tlie pegs being at 
 the end of the rod ; a weight of 5 lbs. is suspended at the 
 other end of tije rod. Find the pressure on each of the pegs. 
 
 24. A heavy uniform beam of weight W is supported in 
 a horizontal position by two men, one at each end ; and a weight 
 Q is placed at a distance ? of the length of the beam from one 
 end. Find the weight supported by each man. 
 
 25. ABC DEF is a regular hexagonal lamina. Prove that it 
 will be kept in equilibrium by the following seven forces : 2 
 pounds along AB, CD, DL\ FA and AD; 5 pounds along GB, 
 and 3 pounds along FF, 
 
 i 
 
 -■ 
 
 I 
 
 J 
 
56 
 
 EXAMPLES IiV MECHANICS. 
 
 PARALLEL FORCES. 
 
 1. A hody is acted on l)y two parallel forces '2P and oP 
 applied in the same sense, their lines of action being G inches 
 a[)art. Determine the magnitude of a third force which will be 
 such as to keep the body at rest. 
 
 2. Tf the forces (Ex. 1) act in opposite senses, find w!<at 
 force is necessary. 
 
 3. Like parallel forces, 1, 2 and 3 pds., act on a bar at 
 distances 4, 6 and 7 inches, respectively, from one end. Find their 
 centre. 
 
 4. Any number of parallel forces act at points in a body. 
 Shew that the magnitude of their resultant is unchanged 
 when the directions of all the forces are turned through the same 
 angle. Use this result in the following examples : 
 
 5. Equal like parallel forces act at 5 of the angular points of 
 a regular hexagon. Determine the centre of the parallel forces. 
 
 6. Forces of 4 pds., 5 pds., 6 pds. act at the points A, B, C 
 of a square of 6-inch side. Find the centre of the system. 
 
 *7, Parallel forces P, Q, R act at the angular points A, B, C 
 of a triangle. Shew that the perpendicular distance of their centre 
 from the side BC is 
 
 P 2 area of triangle 
 
 P+Q + R^^ BC 
 8. ABCD is a square whose side is 17 inches, and E the 
 intersection of the diagonal; like parallel forces, proportional to 
 3, 8, 7, 6 and 10, act at the points A, B, C, D, E, respectively. 
 Prove that the distances ot their centre from AB and AD are 9 
 inches and 10 inches, respectively. 
 
 0. Find the centre of like parallel forces 7, 2, 8, 4, 6 pds. 
 which act in order at equal distances apart along a straight line. 
 
 10. Find the centre of equal like parallel forces acting at 7 
 of the angular points of a cube. 
 
EXAMPLES ly MECHANICS. 
 
 57 
 
 CENTRE OF GRAVITY 
 
 1. Explain what is the C. G. of a body, and deduce the 
 formula for finding its position from the pi'incii>le of moments. 
 
 2. Give an experimental method of determining the C, G. of 
 a plate. 
 
 3. Explain the terms stability and instability of equilibrium, 
 giving examples. 
 
 4. State when a body, which can turn freely about an axis 
 which is not vertical, is in stable or unstable equilibrium. 
 
 5. Explain what are Balancing Couples. 
 
 6. Describe the Balance, and state the requisites of a good 
 one. 
 
 7. Explain the method of Double weighing. 
 
 8. Find the C. G. of a triangle, and shew that it coincides 
 with that of three equal heavy pai-ticles placed at its angular 
 points. 
 
 9. Find the C. G. of any rectilinear figure. 
 
 10. Shew how to find the C. G. of a wire bent in the form of 
 a triangle. 
 
 11. Find the 0. G. of a pyramid, and of a cone. 
 
 12. A rod 3 feet long and weighing 4 lbs., has a weight 
 of 2 lbs. attached to one end. Find where it must be suspended 
 in order to rest horizontal. 
 
 13. Find the C. G. of a uniilorm circular disc out of which 
 another circular disc has been cut, a diameter of the latter being 
 a radius of the former. 
 
 14. If three men support a heavy triangular board at its three 
 corners, compare the forces exerted by each. 
 
 8 
 
 I 
 
68 
 
 EXAMPLES IX MECHAXIGS. 
 
 15. A heavy Imr 14 fee t long is hcnt into a lii^lit angle, so 
 that the lengths of tlie sides are 8 t'eet and feet, tesjtoctively. 
 Shew that tlie C. G. of the bar so bent from the ])oint wliich was 
 the C. G, when tlie bar was straight is ^ v^ 2 feet. 
 
 16. The sit es of a triangle are 3, 4 and 5 feet. Find the dis- 
 tance of the C G. from each side. 
 
 IV. Find the 0. G. of a tigure consisting of an equilateral 
 triangle and a square, the base of the triangle coinciding with 
 one of the sides of the square. 
 
 18. A rod of uniform thickness is made up of equal lengths 
 of three substances, whose densities, taken in order, are in the 
 proportion of 1, 2, 3. Find the C. G. of the rod. 
 
 19. A square stands on a horizontal table. If equal [)ortions 
 be removed from two opi)Osite corners by straight lines parallel 
 to a diagonal, find the least portion which can be left so as not 
 to topple over. 
 
 20. A circular tower, whose diameter is 20 feet, is being 
 built, and for every foot it rises it inclines 1 inch from the ver- 
 tical. What is the greatest height it can reach without falling ? 
 
 21. A uniform square plate whose side is 6v^2~inches, and 
 which weighs 5 lbs., has a 25-lb. weight attached to one 
 corner. At what point must a string be attached that the plate 
 may hang horizontal 1 
 
 22. Find the C. G. of the remainder of a square out of which 
 one of the triangles formed by the diagonals has been removed. 
 
 23. Explain why in ascending a hill we appear to lean for* 
 wards ; in descending to lean backwards. 
 
 24. Why does a person rising from a chair bend his body 
 forwards, and his legs backwards ? 
 
 25* What is the use of a rope-dancer's pole 1 
 
 i 
 
EXAMPLES IX MKCHAXICS. 
 
 1 
 
 
 59 
 
 26. A cylind'T, the diiirneU'i- of wliiuli is 10 feet and lieicht 
 60 feet, rests on anotiiei- <;ylind«'r, tlic^ dlinuottn- of wliioli is 18 
 feet and height G feet, and tlioir axes coincide. Find their com- 
 mon C. G. 
 
 27. Find the C^ G. of seven e(iual heavy particles placed at 
 the angular points of a rei^iilar octagon. 
 
 28. Find the C. G of a quadrilateral, two of whose sides are 
 l)!irallel to one another, and resj)ectively G inches and 1 4 inches 
 while the other sides are each 8 inches long. 
 
 29. Find the C. G. of the frustum of a cone, when the radii 
 of the faces are 4 inches and 8 inches, respectively, and the dis- 
 tance between them 7 inches. 
 
 30. A circular table weighing 20 lbs. rests on three legs 
 which are symmetrically in the circumference. Find the greatest 
 weight that can be placed on any part of the table without 
 uj)setting it. 
 
 f 
 
T 
 
 60 
 
 EXAMPLES IN MECHANICS. 
 
 FIUCTION. 
 
 1. Explain ciiii'fully tlio tcniis n)i<ile (if repose and coofficieiit 
 of friction, juid fiml tlio relution Ix'twcoii then). 
 
 2. P^nnnc'iato tlu^ laws of limiting friction. 
 
 3. If tlio smallest force which will njove a given l>lock 
 weighing .3 lbs. along a horizontal plane is v^ 3 pds., find 
 the greatest atigle at which tlu; j)lane may he inclined before 
 sliding commences. What is the coefficient of friction? 
 
 4. Find the work done in dragging a 2.5-lb. block 20 feet 
 along a rough horizontal [)lane who.se coefficient of friction is ^. 
 
 5. What is the coefficient of friction if a weight ju.st rests on 
 a rough plane inclined at 45° to the horizon? 
 
 6. A weight of 10 lbs. rests on a rough plane inclined to 
 the horizon at an angle of 30°. Find the pressure at right angles 
 to the plane, and the friction. 
 
 7. A body rests on a horizontal plane and is acted on by a 
 force of 10 pds. in the direction making 60° with the plane. 
 What amount of frictici' is called into play? 
 
 8. A body weighing 40 lbs. rests on an inclined plane 
 which makes with the horizon an angle of 30°. What amount of 
 friction is acting between the body and the plane ? 
 
 0. A uniform ladder rests with one end on a horizontal 
 Stone pavement, the other leaning against a vertical brick wall. 
 Find the limiting position of equilibrium, the coefficients being ^ 
 and |, respectively. 
 
 10. A body placed on a liorizontal plane is on the point of 
 moving when acted on by a force equal to its own weight, inclined 
 to the horizon at an angle of 60'. Find the coefficient of 
 friction. 
 
 

 EXAMPLES IN MErHAXICS. 
 
 61 
 
 11. A weight of U Ihs., wlicii plac'd on a rough piano 
 inclined to the horizon at an uni,'le of (iO", slides down unless a 
 force of at least 7 pds. acts on it up tlie piano. What is the 
 coefficient of friction ? 
 
 12. How much work is done by an engine weighing 10 tons 
 in moving half-a-mile on a horizontal road if the resistance is 12 
 pds. per ton ? 
 
 13. A body weighing 40 lbs. is projecterl along a rough 
 horizontal i)lane with a velocity of 150 feet per second; the 
 coefficient of friction is J. Find the work done against friction in 
 5 seconds. 
 
 14. If the height of a rough inclined plane be to the length 
 as 3 is to .5, and a weight, of 10 lbs. can just be supported^by 
 friction alone, «hew that it will just be on the point of being 
 drawn up by a force of 12 pds. along the plane. 
 
 15. Find the work done in di-agging a weight W up a rough 
 plane inclined to the horizon at an angle a, through a space s, fx 
 being the coefficient of friction. 
 
 16. What is the work done in dragging the body down the 
 plane, which is supposed to be too rough to allow sliding of 
 itself? 
 
 17. Find the H.-P. of a locomotive which is to move at the 
 rate of 30 miles an hour, the weight of the engine and load being 
 50 tons, and the total resistance from friction, etc., 16 pds. 
 per ton. 
 
 18. Find the H.-P. of an engine which is to move at the rate 
 of 20 miles an hour up an incline wiiicli rises I in 100, the weight 
 of the engine and load being GO tons, and the resistance from 
 friction being 12 pds. per ton. 
 
 19. Find at what rate an engine of .30 H.-P. could draw a 
 train weighing 50 tons up an incline of 1 in 280, the resistance 
 of friction being 7 pds. per ton. 
 
62 
 
 EXAMPLES IX MECHANICS. 
 
 20. A body slides down a rough plane the coefficient of 
 friction beini? - — :. ; the inclination to the horizon is 30°. Find 
 the space passed over in 8 seconds. 
 
 21. If a weight of 4 lbs. is just on the point of slipping 
 down a rough plaue inclined at 45 when a force of '2, pds. 
 acts up the plane, tind the least force which will move the 
 weight up the plane when the inclination is 30° to the horizon. 
 
 22. Weights of 4 lbs. and 5 lbs., respectively, connected 
 by a light rigid lod, are placed on a rough inclined plane 
 with the rod parallel to the line of greatest slope. If the coeffi- 
 cient of friction between the 4-lb. weight and the plane is 'G, and 
 that between the other weight and the plane is '42, find the 
 greatest inclination of the plane to the horizon consistent with 
 equilibrium. 
 
 23. Two equal heavy rings hang on a rough horizontal rod, 
 and are connected by a string of length c which supports an 
 equal Ijeavy ring. Find the greatest possible distance between the 
 first two rings. 
 
 24. A heavy uniform rod 2 feet long rests between two pegs 
 in a vertical board, 3 inches a[)art, one of which is at the 
 end of the rod. The coefficient of friction between the rod and 
 the pegs is ^. What is the greatest angle through which the 
 board may be turned in the vertical plane before the stick begins 
 to slip ? 
 
 25. A uniform pole leans .xg.iinst a smooth wall at an angle 
 of 45°, the lower end being on a rough horizontal plane. Shew 
 that the amount of friction required to prevent sliding is half 
 the weight of the pole. 
 
 I 
 
 4 
 
 1 
 
EXAMPLES IN MECIIANICS. 
 
 63 
 
 MISCELLANEOUS EXERCISES. 
 
 1. Draw a diagram to represent the velocities of a swinging 
 pendulum ; the velocity-lines being horizontal, and the timedint 
 vertical. 
 
 2. A point goes k feet in t seconds. How long does it take 
 to go m miles with tlie same velocity? 
 
 3. A train goes n miles in /. hours. How i\u- iocs U go in t 
 seconds ? 
 
 4. The velocity of a train is 30 miles an hour ; 
 
 (1) How long will it take to tiavcrse 100 yards ? 
 
 (2) How many seconds will it take to go 150 feet 1 
 
 5. A velocity of a yards per second is k times one of 70 feet 
 per minute. What is /I- ^ 
 
 6. A man 6 feet high walks in a straight linr^ awav from a 
 lamj, post which is 10 feet high. Supposing the man 'to start 
 from the post and walk at the rate of 4 mile.s per hour, find the 
 rate at which the end of the shadow travels and also the late at 
 which his shadow lengthens. 
 
 7. One point moves uniforndy round the circumference of a 
 circle while another is moving uniformly across the diameter. 
 Compare the velocities. 
 
 8. One point deseiibes the circumference of a circle of a feet 
 radius in h minutes ; and another describes the circumference of 
 a circle of b feet in a nanutcs. Compare the velocities. 
 
 9. If the unit of (ime Ije a minute, and a foot be the unit of 
 si'ace, what is ti.e numeiical value for ihe velocity of 40 miles 
 per hour 1 
 
 10. How many minutes will a body take to go a mile witji 
 a velocity 5, the standard velocity being 25 feet per 3 minutos ? 
 
 1 
 
r 
 
 64 
 
 EXAAfPLES /iV MECHANICS. 
 
 11. If the velocity of 30 feet per second be represented by 
 5, what will be the measure of the velocity of 7 yards per 2 
 minutes ? 
 
 12. Given, that a certain line 11 inclies long represents a 
 velocity of 3 miles [)er hour to the east, how would you repre- 
 sent a velocity of 100 yards per minute to the north-east ? 
 
 13. Find the velocity in metres per second, arising from tlio 
 rotation of the earth, of a point in latitude 60^. 
 
 14. A point passes over h feet in 2 seconds ; after an interval 
 of t seconds it is observed to be displaced k feet in 2 seconds ; 
 find the acceleration. Total time is < -f 4 seconds. 
 
 15. Represent graphically the space passed over by a particle 
 which moves with a variable velocity. 
 
 16. A point is observed to move 48 feet from rest in GO 
 seconds, what is its acceleration, supposed constant 'I 
 
 17. A j)oint, having a constant acceleration, is displaced 72 
 feet while its velocity increases from 16 to 20 feet per second. 
 What is its acceleration ? 
 
 18. A train's acceleration is 5 feet a second per second. How 
 long will it take to acquire a velocity of 100 yards per minute 
 from rest ? 
 
 19. Shew that the acceleration of 360 feet a second per hour 
 is double that of 1 yard a minute per minute. 
 
 20. Compare the acceleration of m. feet a second per minute 
 with the acceleration n feet a minute per second. 
 
 21. A body falls freely from the top of a tower, and during 
 the last second of its flight falls |^ of the whole distance. Find 
 the height of the tower. 
 
 22. A particle, under the action of srravitv, passes a <^iven 
 moving downwards with a velocity of 50 metres per second. 
 long before this was it moving upwards at the same rate ] 
 
 ))oint 
 How 
 
EXAMPLES IN MECHANICS, 
 
 65 
 
 L 
 
 23. A balloon ascends with a uniform acceleration of 4 foot- 
 second units ; at the end of half-a-minute from leaving the ground 
 a body is released from it. Find t.he time that elapses before it 
 reaches the ground. 
 
 24. A body slides down chords of a vertical circle ending in 
 its lowest point. Shew that the velocity on reaching the lowest 
 point varies as the length of the chord. 
 
 25. Find the straight line of quickost descent from a given 
 point to a given circle in the same vertical plane. 
 
 26. Find the position of a point on the circumfenmce of a 
 vertical circle, in order that the time of rectilinear descent from 
 it to the centre may be the same as that to the lowest ])oint. 
 
 27. Find the line of quickest descent from the focus to a 
 parabola whose axis is vertical and vertex upwards, and shew 
 that its length is equal to that of the latus rectum. 
 
 28. A body starts with velocity ?fc and moves with uniform 
 acceleration ; if a, h, c are the spaces described in the p">, q"' and 
 r"» seconds, respectively, prove that 
 
 a{q — r)-\-h (r — p) + c (p — q) = 0. 
 
 29. A particle starts from rest with acceleration /; at the 
 end of time t the acceleration becouKJs 2/; '.]/' at the end '2t, 
 and so on. Find the velocity at the end of time nf, and she.v 
 that the space described is 
 
 12 
 
 ')i 
 
 {n+\){2n-\-l)/r^ 
 
 30. Shew that the highest point of a wheel rolling on a hori- 
 zontal plane moves twice as fast as a t>oint on the rim whose 
 distance from the ground is half tiie radius. 
 
 31. Bodies slide down smooth faces of a ])yrami(l, starting 
 from rest at the vertex. Shew that at any time f they all lie on 
 a si)here whose radius is |v/<-. 
 
 32. One particle describes the diameter A/i of a circle with 
 uniform velocity, and another the semi-circumference AB with 
 
66 
 
 EXAMPLES IN MECHANICS, 
 
 uniform tangential acceleration ; tbey start together from A and 
 arrive together at B. Shew that the velocities at B are as 1 : ;:. 
 
 33. A body falls through a feet and acrpiires a velocity v, 
 with a uniform acceleration/ in t seconds. What are the units 
 of time and length? 
 
 34. ^ is a fixed i)oint on a circle upon which the point P 
 moves with uniform velocity. Shew that the apparent angular 
 velocity of P about A is constant, and is equal to one-haff its 
 angular velocity about the centre of the circle. 
 
 35. Explain why a man, walking in the rain, holds his 
 umbrella a little in front of him, though there is no wind. 
 
 36. A ])ressure of a kilogram acts on a body continuously for 
 10 seconds, and causes it to describe 10 metres in that time 
 Find the mass of the body. 
 
 37. AVci-lits w, and m^ lbs. are attached to the ends of a 
 string and hung over a pulley ; if the tension of the string is 
 M pds. , ]uove that M is a harmonic mean between m and m, . 
 
 38. A mass P is drawn up a smooth ])lane inclined to the 
 horizon at an angle of 30°, by a mass Q attached to a stri.ig 
 passing over a pulley at the top of the plane ; if the acceleration 
 of the system be one-fourth that of a freely falling body, find the 
 ratio of Q to P. 
 
 39. P hangs veitically and is lbs. ; Q is a mass of G 
 lbs., on a smooth i)]ane inclined to the horizon at 30°. Shew 
 that if connected P will drag Q up the whole length of the plane 
 in half the time that Q, hanging vertically, will drag P up the 
 plane. 
 
 40. A mass of 2 lbs. is struck and starts off with a velocity 
 of 10 feet i)er second. If the time duiing which the blow 
 lasts be -^-i ,-, of a second, find the average value of the force acting 
 on the mass. 
 
EXAMPLES IN MECHANICS. 
 
 67 
 
 41. A shot of 
 
 ojected witli 
 
 L 
 
 mass 1 ounco is 
 1,000 feet per second from a gun whose mass is 10 Ihs. Find 
 the velocity with which the latter begins to recoil. 
 
 42. A shot, whose mass is 800 lbs., is discharged from an 
 81-ton gun with a velocity of 1,400 feet per second. Find the 
 steady pressure which, acting on the gun, would stop it after a 
 recoil of 5 feet. 
 
 43. A 13-ton gun recoils, on being discharged, with a velocity 
 of 10 feet per second, and is brought to rest by a uniform friction 
 of 4yig tons. How far does it recoil 1 
 
 44. The weights of an eight-day clock are together 11 lbs., 
 and when the clock is wound up they are raised a yard. How 
 many such clocks could an engine of 1 horse-power drive ] 
 
 45. Explain each symbol in the equation W =: yi>V, f^ivin^^ 
 the units in each case (in the C. G. S. system). 
 
 46. Tf a second be the unit of time and an acre be repre.sented 
 by 10, what will be the numeiical value of a velocitv of 45 miles 
 per hour ? 
 
 47. The density of water being the unit of density, and 10 
 lbs. the unit of mass, find the unit of length -^it being given 
 that a cubic foot of water contains 1,000 ounces. 
 
 48. If the unit of time be .5 minutes, and the unit of length 
 5 yards, find the value of g. 
 
 49. If the area of a 10-acre field lie represented by 100, and 
 the acceleration of a falling body by 58g, find the unit of time. 
 
 50. A shot is fired at an elevation of 30° so as to strike an 
 object at a distance of 2,500 feet, an<l on an ascent of 1 in 40. 
 Find the velocity of projection, neglecting the r«'sistance of the 
 air. 
 
 51. Neglecting the resistance of the air, the greatest range 
 of a rifle bullet on level ground is 20,000 feet. Find its initial 
 velocity, and its maximum range up an incline of 30^. 
 
r 
 
 68 
 
 EXAMPLES IN MECHANICS, 
 
 52. Tf two circles, one within the other, touch each other at 
 their highest or lowest points, and a straight line be drawn 
 through this point, the time of falling from rest down the straight 
 line intercepted between the circumferences is constant. 
 
 53. Given that the mass of the earth is 614 x 10^7 grams, 
 and its radius 6-37 X lO^cm., and^ = 981. Shew that if 2 
 spheres of mass 3,928 grams each, be placed with their centres 
 1 cm apart, the force of attraction between them is 1 dyne. 
 
 54. A particle is projected with velocity 2 v^a^ .so that it just 
 clears two walls, of equal height a, which are at a distance 2a 
 from each other. Shew that the latus rectum of the path is 2a, 
 
 and that the time of passing between the walls is 2 /— {g being 
 the acceleration of gravity). 
 
 55. A l)all is projected from a given point, at a given inclina 
 tion, towards a vertical wall at a distance c. Determine the 
 velocity of projection so that, after striking the wall, the ball 
 may return to the point of projection ; e being the coefficient of 
 restitution. 
 
 56. An imperfectly elastic ball is thrown from a given point 
 against a vertical wall. Find the direction in which it must be 
 projected with the least velocity so as to return to the point of 
 projection ; e being the coefficient of restitution. 
 
 57. The masses of five l)a]ls at rest in a straight line form a 
 geometrical progression whose ratio is 2, and their coefficients of 
 restitution are each j^. If the first ball V)e started towards the 
 second, shew that the velocity communicated to the fifth is (|)*w. 
 
 58. A particle slides down the arc of a vertical circle. Shew 
 that its velocity at the lowest point varies as the cliord of the 
 arc of descent. 
 
 59. A force acting uniformly during one-tenth of a second 
 produces in a given body thn velocity «>f a mile a minute. Com- 
 pare this force with the weigiit of the body. 
 
EXAMPLES IN MECHANIC!^. 
 
 m 
 
 60. Of two equal and perfectly elastic ball^ one is projected 
 so as to describe a parabola, and the otlier is dropped from the 
 directrix so as just to fall upon the tirst when at its highest 
 point. Determine the position of the vertex of the new parabola. 
 
 61. A particle is projected with a given velocity at a given 
 inclination to the horizon from a point in an inclined plane. Find 
 the whole time which elapses before the particle ceases to hop. 
 
 62. At what angle of inclination to a level road should the 
 traces be attached to a sleigh that it may be drawn along with the 
 least exertion ? 
 
 63. Find the C. G. of n equal particles arranged at equal 
 intervals along a circular arc. 
 
 64. A cone whose height is equal to four times the radius of 
 its base is hung from a point in the circumference of its base. 
 Find the position in which it will rest. 
 
 65. If 6' be the C. G. of a triangle ABC, shew that 
 3 (^6'2 + BG^ + CG'^) ^ AB^ -{■ BG^ + CA^. 
 
 66. A stream of water falls from rest at a height of :iO feet 
 above a horizontal inelastic plane at the rate of 100 gallons per 
 minute. Find the pressure on the plane, supposing the water to 
 flow freely off" it. 
 
 67. A jet of water, the area of whose transverse section is 
 one square inch, impinges directly upon a wall with a velocity of 
 128 feet per second. Find the pressure on the wall, the water 
 spreading freely over it. 
 
 68. A stone is thrown in such a manner that it would just 
 hit a bird at the top of a tree, and afterwards reach a height 
 double that of the tree. If, at the moment of throwing the stone, 
 the bird flios away horizontally, prove that the stone will, not- 
 withstanding, hit the bird, if its horizontal velocity is to that of 
 the bird as v/2 + 1 : 2. 
 
 69. Find the work done in drawing up a Venetian blind. 
 
70 
 
 EXAMPLES IN MECHANICS. 
 
 rise at t 
 ill cease. 
 
 70. A ball of elasticity J falls from a height of ( 
 a horizontal plane. Find the height to which it wil 
 first rebound, and the time at which the rebounding 
 
 71. Find the direction of the straight line of quickest descent 
 between two given parallel straight lines in a vertical plane. 
 
 72. A triangular lamina is acted on by three forces which are 
 represented by the lines drawn through each angle and bisecting 
 the opposite side. Prove that the lamina is in equilibrium. 
 
 73. Find the centre of the parallel forces 1, 2, 3 acting at 
 the angles of an equilateral triangle. 
 
 74. Two equal rods AB, BC are firmly joined together at B 
 at right angles. If they were suspended from A so as to be 
 capable of turning freely about that point, in what position will 
 they hang? Could you make them hang with one side vertical 
 by attaching a heavy weight &t B? 
 
 75. A circular disc of 1 foot radius has a circular hole of 
 3-inch radius cut out of it ; the centre of the hole being at a 
 distance of 2 inches from the centre of the disc. Find the C. G. 
 of the disc. 
 
 76 A uniform rod 4 inches long is placed with one end inside 
 a ismooth hemispherical bowl, of which the axis is vertical and 
 radius is ^3 inches long. Shew that one-fourth of the rod will 
 l)roject over the rim of the bowl. 
 
 77. A force can just move a given weight up a plane of 30", 
 and can just prevent a weight twice as great from moving down 
 a plane of 60°. Prove that the coefficient of friction, which is 
 the same for both planes, is -26 nearly. 
 
 78. Explain why it is easier to drag a wheelbarrow after you 
 over a log than to push it before you. 
 
 79. A heavy particle is placed on the top of a smooth sphere. 
 Prove that it will leave the sphere when it has descended a ver- 
 tical distance ec^^ual to one-third of the radius. 
 
EXAMPLES IN MECHANICS. 
 
 71 
 
 80. A ball, falling from the top of a tower, had descended a 
 feet when another was dropped from a point distant h from the 
 top of the tower. If they reach the ground together, prove that 
 the height of the tower is {a -\- b)^ / 4a feet. 
 
 81. Two small smooth unequal spheres are placed in a fixed 
 smooth hemispherical bowl. When in equilibrium under gravity, 
 find the inclination to the horizon of the line joining their 
 centres. 
 
 82. The altitude of a right cone is h, and the diameter of the 
 base is b ; a string is fastened to the vertex and to a point in the 
 circumference of the base, and is then put over a smooth peg. 
 Shew that if the cone rests with axis horizontal, the length of the 
 string is y/ (^2 -f. ^2^. 
 
 83. A uniform stick 6 feet long lies on a table Nvith one end 
 projecting 2 feet over the edge; the greatest weight that can be 
 suspended from the end of the projecting ])ortion without des- 
 troying the equilibrium is 1 lb. Find the weight of the 
 stick. 
 
 84. Find the C. G. of a cube from one corner of which a cul)e 
 whose edge is one-half the edge of the first has been removed. 
 
 85. Two equal smooth spheres are stiung on a thread which 
 is then suspended by its extremities so that its upper portions 
 are parallel. Find the pressure between the spheres, the holes 
 being smooth. 
 
 86. Explain how it is that an ice-boat can travel faster than 
 the wind. 
 
 87. Two uniform beams of equal weight, but of unequal 
 length, are placed with their lower ends in contact on a smooth 
 horizontal plane, and their upper ends against smooth vertical 
 planes. Shew that in the position of equilibrium the two beams 
 are equally inclined to the horizon. 
 
 88. Shew that the time of descent from any point on a 
 cycloid to the corresponding jioint on its evolute is the same. 
 
f9 
 
 EX AMP /,£!'! /A^ MEflfANICS. 
 
 89. When a particle starts from the cusp of an inverted 
 cycloid, the vertical velocity of the particle is greatest when it 
 has completed half its vertical descent. 
 
 90. A particle descends from the cusp of an invertod cycloid, 
 and P is any point on the cycloid. Shew that when passing P 
 the pressure on tlie curve is twice what it would be if the 
 particle started from P. 
 
 91. An imperfectly elastic particle falls down an inclined 
 plane of given length, and at the foot impinges on a horizontal 
 plane. Shew that the range on this plane will be greatest when 
 the angle of elevation of the inclined plane is tan~-^ v''^. 
 
 92. An imperfectly elastic ball is dropped into a hemis- 
 pherical bowl from a height n times the radius of the bowl 
 above the point of impact, so that it strikes the bowl at a point 
 30'^ from its lowest point, and just rebounds over the edge of the 
 bowl. Shew that the coefficient of restitution is ^v/3. n"~i. 
 
 93. A beam wei^lis ^^^0 lbs., and is 28 feet long. A 
 boy lifts one end, and a man with a lever 4 feet Ioul? raises the 
 other. The fulcrum is G inches fron^ the beam, and the pressure 
 exerted by the man on his end of the lever is twice as great as 
 that by the boy on the end of the beam. Find how much the 
 boy lifts, and the point where the lever presses the beam. 
 
 94. On the moon there seems to be no atmosphere, and 
 gravity is about ^ as great as on the earth. What space of 
 country can be commanded by a lunar fort able to project shot 
 at 1,600 feet jjcr .second 1 
 
 95. A, B, C are three equal balls situated at the angular 
 points. A, B, C oi an equilateral triangle, and connected by 
 fine inolistic strings A B, BC. The ball B receives an impulse 
 in a (liioftion at right angles to AC, and in the plane ^1 B C. 
 Prove that the velocity produced thereby in B is | of what it 
 would have been if Ji had been free. 
 
 I 
 
EXAMPLES IN MECHANICS. 
 
 — o 
 
 « •> 
 
 98. A rod AB slulos with its oxtroinitii'S always on two 
 straight linos. Find thn instantaneous centre at any time. 
 
 97. A number of ei^ual heavy particles are fastened at equal 
 distances «, on an inelastic string, and placed in contact on tlie 
 edge of a table. Shew that if the lowest be then iill<>wed to fall 
 freely, the velocity with which the ni\\ begins to move is equal 
 
 to 
 
 \ 
 
 / 07 
 
 Jl|J2u-.J). 
 
 98. Shew that in the limit, wlicii <t is iiidetniitt'ly sjuall and 
 na finite, tiie chain (or heavy th'.\il>l(' rope) will descend with a 
 uniform acceleration ^ij. 
 
 99. A hollow spherictd shell has a small hole at its lowest 
 point, and any number of particles start down chords from the 
 interior surface at the same instant, pass through the hole, and 
 tiien move freely. Shew that befoi-e and aftei* passing thrinigh 
 the hole, they lie on the surface of a sphere, and determine its 
 radiu.-s and j)Ositiou at any instant. 
 
 100. Prove that, on account of the rotation of the earth, the 
 apparent weight at the equator of a body is less than its weight 
 at the pole by about .jjpth of the latter ; and at latitude /, ap- 
 proximately ^|g cos -/. 
 
 10 
 
ANSWERS. 
 
 I" 
 
 sec. 
 
 Velocity, p. 27. 
 2. 15 mis. per hr.; 20 mis. per hr. 8. 10 J, 10, 11 ft. per 
 
 11. 112 ft. per sec. 12. 575 ft. 13. 40 sec. previously. 
 14. 2^2 mis. per hr. each. 15. 5^3 ft. per sec. IG. v cos (f, 
 V sin 0. 
 
 17. 8- 06 knots per hr. at about 60° N.E. by N. 18. 9^3 yds. 
 per sec. 
 
 19. (a) 20 ft. per sec. (b) 1^2 ^C^f ft. per sec. 
 20. 28 -28.... ft. per sec. ^^ 
 
 21. (a) 26-4. ... ft. per sec. (b) G-7 . . . . ft. per sec. (c) 
 
 1-551 .... ft. per sec. 22. 33-8 .... ft. per sec. at angle 30°. 
 
 23. 35-5 ft. from point of stoppage. 24. 28 ft. per sec. 
 
 S. 60° W. 25. 100 ft. 26. 8-7 hmi. nearly ; up stream 30o. 
 27. 21 ft. per sec. nearly. 
 
 Acceleration, p. 31. 
 
 2. 16 ft. a sec. per sec. 3. 2 ft. a sec. per sec. ; 29 ft. per 
 sec, 4. 102 ft., 208 ft., 27,600 ft. 
 
 5. 277 ft. per. sec. ; 3 ft. a sec. per min. 6. 15 sec ; 15 sec. 
 more. 7. 11 sec. 8. G5 ft. per sec. 9. 8 ft. a sec. per sec. 10 
 115,200. 11. The same. 
 
 12. 16 ft. 15. 4 ft. a sec. per sec. 10. 48 J ft. 18. 15 cms. 
 
 19. 210. ... ft. 20. 2661-5. . . . ft. 21. 715'5. . . . ft. per 
 
 sec. 
 
 22. I sec. 23. 35 ft. l>G. Draw a vortical through 7', and 
 • lescribe a circle with centre in this line and touching the given 
 straight line. Join P to the point of contact. 27. 504 ft. 
 
I 
 
 78 
 
 ANSWERS. 
 
 Projectiles, p. 34. 
 5. 56i ft. above point of projection; 56J ft. 6. 1860.75 
 
 ft. per sec. ; tan-^ (2^)- ^' ^'^^^^* ^'^'' ^' ^^ ^^' 
 9. 8,954-82 ft. 10. Elevation, tan-i ^ (20 i x/sTl) 
 
 12. (Distance)^ = { w^ + y'^ - 2uv cos (a — ,3^) }<2 . 14 
 320 ^/3 it. from the clift'. 
 
 15. 7,812Jft. ; l,953Jft. 
 
 Mass, Density, etc., p. 36. 
 
 1. G2J. 2. 1 c.c; 1. 3. 13-6 x 62J = 850 ; 1. 
 4. 1-58.... ; no. 5. 656.25 ; 1754. .. .lbs. 
 
 6. 7,8 12J; 1 lb. mass moving with velocity, 1 ft. per sec. 
 
 7. 3,060. 8. //=. 2,250; V ^ 18 (125 ^/J- 96); resul- 
 tant = 18 x 14211 or 2,557-98. " ^ ' 
 
 9. 80:7. 10. Equal. 
 
 Force, p. 37. 
 
 2. 48 units ; 16 units. 3. 4 ft. a sec. per sec. 5. 18 units 
 per second. 6. 1 ft. per sec. in opposite direction. 7. 36 ft. 
 
 per sec. 
 
 8. IJ units, 4 units; 3:8; lOJ ft. per sec; 28 ft. per sec. 
 
 9. 12 units; 1 J units per sec. 10. (a) The same. (6) 1 : 
 3: 1000. 
 
 Units of Force, p. 38. 
 
 1. (a) 1 ft. per sec. (b) 1 ft. a sec. per sec. (c) 1 lb. mov- 
 ing with a vel. of 1 ft. per sec. (d) Unit force in 1 sec. will 
 generate in 1 lb. mass a velocity of 1 ft. per sec. 
 
 1^ ^LM 
 
I 
 
 ANSWEJiS. 
 
 1- 
 
 ts 
 :t. 
 
 I : 
 
 79 
 
 2. 32 ft. per sec; 1 ft. per sec; 32 : 1. 3. 32 poumlals = 1 
 pd. 6. 1 oz. = 27,869-32 dynes nearly (7 =981). 
 
 7. 5 ft. per sec 8. 153 6 ft. i)er sec. 9. 150 ft. per sec; 
 900. 10. -016 ft. a sec i^-- sr..; 9'G ft. per sec 
 
 11. lOcms. persec 12. 25,000 : 327. 13. 7,357,500 C. G. 8. 
 units. 
 
 14. 224:G75. 15. ij| cm. per sec ; IG ; 2^i cm. 
 
 16. 4; 50 ft. 17. 3 grams. 18. 72 ft. 19. 192 ft. 
 
 20. 480 ft. per sec; 96,000 units. 21. GG5 ft. 22. 32 ft. 
 23. 24v^3 ft. per sec. ; 3G^/3 ft. 24. 1 : 2. 25. Sufficient to 
 generate momentum 6^/3 . 26. ^{v~ -f 2ylt). 27. 27 : 40 ; 
 27 : 100. 
 
 28. 20 ; 10. 29. 2:1; 1:2. 30. Gravity, 57G : 575. 
 
 31. Force units, i.e. poundals or dynes. 
 
 Forces acting Simultaneously, p. 41. 
 
 3. 16pdls. G. G4pds., 8pds. 7. 37 pds. 8. 9 pd.s., 12pds. 
 
 9. 1562. ...pds. 10. 1311.... pds. 11. Almos. 60°. 12. 
 15 pds. 
 
 14. In a straight line. 15. Shorter, 4 pds.; longer, 3 pds. 
 
 17. Angle J MB is constant. 19. 2^2 ikIs. parallel to DB. 
 
 20. Equivalent to 1, 2 at an angle of 120°. 21. Yes, all in 
 a straight line. 23. 71 times the radius. 24. 4^/2 pds. 
 25. 1 ft. a sec. per sec; 6 sec. 26. 10v^509 pds. 
 
 V- 
 
 ill 
 
 Gravitation, p. 44. 
 
 2. 3^/0 feet. 5. Gy 7 inches. 9. ^Uf, shillings per lb. 
 11. Increased, 1G:9. 
 
 12. 30 pds. 13. 4 ft. 14. 8 ft. 15. 2-5 ft. nearly. 
 
80 
 
 AXSWEIiS. 
 
 Work and Energy, p. 4G. 
 
 3. 39 ft.-pck 5. 180 ft.-p.ls. G. 2,G40. 7. 42,945.7 
 . . . . ft.-ptls. 8. 19,200 ft. -ptk 9. 81,920 ft.-p(l«. 
 
 10. 25 h.-p. 12. 112,000 x tt. 13. hi. U. 1^,000 
 ft.-pds. 
 
 15. 2,500 ft.-pds. IG. foot-pouiulals, or ergs. 17. 112 : G2.j. 
 
 18. 12 ft.-pds. 19. 20G3 ft.-pds. approx. 20. 140,800 
 ft.-pds. 
 
 21. 29,5G8 ft.-pds. 22. 25:512. 23. 7-23308 ft.-pds. 
 •138254 kgins. 24. 64 : 125. 25. 131,250 ft.-pdls., or 4,101 ^J^^ 
 ft.-pds. 26. 1,000 ft.-pds. 27. y/^-\ : 1. 
 
 28. 405,504,000 ft.-pds. 30. 346-4. . . .metres per sec. 
 
 31. 18 cms. 32. 458-25. . . . m. per sec. 
 
 Centre of Mass., p. 49. 
 
 1. 7 ft. 2. 1 ft. fi-om smaller. 3. [)}. ft. per sec. 
 
 4. 1| ft. per sec. 5 r m/in it. per sec. in opposite din-cticjii. 
 
 COLLISI P. 50. 
 
 e^ v^ 
 
 2. I--; change = mv (1 -f e). 3. 40^2. 4. 4ji- in. ; 3J 
 "9 
 
 sec. 
 
 5. 2 ft. per sec. in opposite direction ; change, 16 units. 6. 
 
 4 ft. per sec. in opjiosite direction. 8. cot '^ -— -- • 10. Ml\ 
 
 + mi*j — AfV— mv = 0. 
 
 12. 11, 13 ft. per sec. 13. W'l. of each is reversed and re 
 duced one-half; pressure = change of momentum = 144 units. 
 
 14. lU ft. 15. e=i; in = 2m. lO. />' = 3^1. 17. 
 
 ' - B{A +~C) • 
 
 18. 7-5 ft.; 54-772.... ft. l!V y'^^L n ,>er sec. 21. 
 
ANSWERS. 
 EQUitiBRifiM OF Kir.rD Bodies, p 53. 
 
 81 
 
 0. The Ifittor, 81 : 100. (5. ;U-5() ft. 7. 179 X 10*, GO x 10*, 
 persf|. cm. H. 120 pds. U. 'J,;}7r) pd.s. 
 
 U). T)^ 2" pels. 17. ll^/M |.(Is. 18. Ov/2 oz. 19. 20 pels., 
 1<5 pds. 
 
 20. 4^/2:1:71 pels. 21. 40p(k 22. 31 ft. 23. 41 pds., 
 48 pds. 
 
 24. iir+ ?c>; i^F+ 3^. 
 
 Parallel Forces, p. 56. 
 
 3. 0^ in. from the end. 5. At a distance from the centre 
 of the hexagon equal to | of a side. G. 2^, in. from Ali ; 1|^ in^ 
 from /JC. 9. At the point where 8 i)ds. acts. 
 
 10. On the diagonal through the point where no force acts, at 
 4 of the diagonal from this point. 
 
 21 
 
 Centre of Gravity, p. 57. 
 
 4. Stable when C. G. is in lowest position ; unstal^le in high- 
 est. In both cases the C. G. and the axis are in same plane. 
 
 8. Join A to n)id-point of /iC : the C. G. is in this line :j 
 of distance froni .1. 9. Divide into trianj-h'S. 
 
 10, It is the centre of inscril)ed circle. 
 
 1 1 . Join the vertex to the C. G. of the base, and take f of the 
 distance from the vertex. 12. 1 ft. from the end. 
 
 13. At a distance from the centre of tlie large circle erjual to 
 jl of its radi'js. 14. All equal. 10. ^,1, I ft. 
 
 17. At Ji distance from the base of the triangle equal to 
 o 
 ~Q~T^.ijx times the base. 18. From the <lensest end, 7^ of 
 
 whole length. 19. ^ of tlie s(pi;iic. 2u. 240 ft. 
 II 
 
82 
 
 ANSWERS. 
 
 21. In the diagonal 5 in. from tho centre. 
 
 22. Join (J tlie centre of squiire to M he centre of the o\)p 
 sitle to removed tri.ingh;, the C G. is >] C^iJ from C. 
 
 20. 27:|0'f ft. from the base. 
 
 27. If A is uiM)L'Cii|»ied ci)rn<'r, and (f the centre of the 
 octagon, the rcMjiiired point is in A() produced through a 
 distance \ A<>. !28. fl\/'.\ iu. from longer side. 
 
 29. \\ in. from centre of smaUer face. 30. 20 lbs. 
 
 J 
 1 
 
 Friction, p. 60. 
 
 3. 30"; •o77..,. 4. 250 ft.-pds. .5. 1. 6. 5^/.s pds. ; 
 /) pds. 
 
 7. 5 pds. S. i>0 jkIs. !». tan -' :-; to horizon. 10. 2 ± x/3 . 
 Can the doubh; sign bt' uHowed ? 
 
 11. -73 12. 310, SCO ft.-pds. 13. 3,500 ft.-pds. 
 
 15. W (sin a -j- //. cos a) s. 10. W {.'i COS a — sin a) s. 17. 
 04 h.-p. 
 
 18. 1024 1i.-p. 10. 1,400 ft. per min. 20. 250 ft. 
 
 21. 301 . . . .pds. 22. tan-i J. 23. 3 //c V (1 + 9/z2). 
 
 24. Stick 45"^ to horizon. 
 
 Miscellaneous Exercises, p. 63. 
 
 2. 22m/ 15^ hrs. 3. 22/i^, 45A; yds. 4. /'^- min.; S^Hj sec. 
 5. 7|. lOndsperhr.; nds. per hr. 7. r : 1. 
 8. a2:6-. 9. 3,5:>0. 10. 120-72. II. r.j^, . 
 
 12. By a line I2r^ iu. Ion-- a( b"»^ to first. 13. 231 48 
 
 metres per sec. 14. (/ — //) (2/ -f 4). 10. A fp. ff. |j?^. per 
 sec. 17. 1 ft. a sec. per sec. 18. One seco^^.t). 
 20. mm. 21. 100 ft. 22. i(f4 sec. H 15 sec. 
 
ANSWERS. 
 
 So 
 
 
 26. The radius through it is 60^ from vertical. 
 
 33. -J sees. ; -/ft. 36. 49-05 kilos. 38. They are equal. 
 
 40. 2,000 pdls. = m pds. (approx.) 41. 6J ft per sec. 
 
 42. 'J ^ (loi)g) tons. Use principle of energy. 43. 5 ft. 
 
 44.11,520,000. 46.1. 47. 6-014. .. .inches. 48.192,000. 
 41). 1 1 sec 
 
 50. 310-6 ft. per sec. 51. 800 ft. per sec; § of level 
 
 range. 55. v'^ sin 2a = (jrc I 1 +-V 56. ^^ = 45°; see No. 55. 
 
 59. 271 60. In old directrix. 61. Time is 2u sin 
 (^^ — /^) / f/ (1 — e) cos ;i . 62. tan-i p.. 63. Distant 
 
 64. Base and axis equally 
 
 7A- 1 
 
 (I . na , . 
 
 rrom centra - sin / sni 
 
 n n — 1 ' 
 
 iiiclined to vertical. 
 
 66. 22-82 ...pds. 
 
 67. 222| pds. 70. 16 ft. ; 6 sec. 73. Bisection of line join- 
 ing " 3 " to that point of trisection of opposite side nearer to " 2." 
 
 74. Vertical through A trisects BC ; no. 
 
 .■) 
 
 75. If C be the centre of large disc, CG = — in. 
 
 15 
 
 83. 2 Ib.s. 85. The weight of one sphere. 
 
 93. 50 lbs. ; 1 ft. from middle. 94. Nearly 26,000 square 
 miles.