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Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, ate, peuvent dtre filmds i des taux de reduction diff6rents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de Tangle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 6 6 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.