44. A mass of 15 lbs. lying on a smooth flat table acted upon by a force of 60 poundals: how far will move in 6 seconds?

45. A body of mass 10 is connected with anothe body of mass 6 by a string passing over a frictionles pulley: find the acceleration and the distance move through in 4 seconds. Show how such an arrangemen could be employed for finding the value of g, and explai why the method would be better than that of experi menting with a freely falling body.

46. Weights of 14 and 21 lbs. are hung on the end of a rope passing over a pulley: find the tension in the rope in pounds weight and in poundals.

47. Two masses of 100 and 120 grammes are attached to the extremities of a string passing over a smooth pulley if the value of g is 975, what will be the velocity after 8 seconds?

48. Two unequal masses are attached to the ends o a string passing over a smooth peg: find the ratic between them in order that each may move through 16 feet in 2 seconds, starting from rest.

49. Two buckets, each weighing 28 lbs., are suspended from the ends of a rope passing over a windlass; a gallon (10 lbs.) of water is poured into one of the buckets: find how far it will descend in 10 seconds, neglecting friction.

50. Describe Atwood's machine, and explain how it may be used to prove

(a) That when different forces act upon the same mass the accelerations observed are proportional to the forces.

(b) That when the force is constant the accelerations are inversely proportional to the masses. (c) That the space described in n seconds from rest is proportional to n2.

What other experimental method has been devised for testing the last proposition?

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51. The sum of the two weights in an Atwood's machine is 2 lbs., and the difference between them is an ounce; find the acceleration and the space described in the first second.

52. In an experiment with Atwood's machine the masses were 520 and 480 grammes; in 2 seconds from rest the heavier mass descended 76 centimetres. What value does this give for the acceleration of gravity? If your result differs from the usual value, suggest any cause for the difference.

53. The two equal masses in an Atwood's machine are each 100 grammes; what excess weight must be placed upon one of them in order that, at the end of 3 seconds, it may be descending with a velocity of 2 metres per second?

54. By means of an Atwood's machine a force equal to the weight of 10 grammes was made to act upon a mass of 500 grammes, and it was found that an acceleration of 19.6 cm. per second was produced. Find the value of g.

55. A train starts from rest on a level line and moves through 1200 feet in the first minute. It then begins to ascend an uniform incline, up which it is found to run with uniform velocity: find the inclination of this portion of the line on the supposition that the engine exerts a constant pull.

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56. A body of mass m moves with uniform velocity v in a circle of radius r. Prove that a force is required to keep it in its circular path, and that this force is directed along the radius and towards the centre.

57. A body of mass 2 lbs. is attached to the end of a string a yard long, and is whirled round at an uniform rate, making twenty revolutions in a minute: what is the tension in the string?

58. A mass of a kilogramme is connected to a fixed unint h... string and matre in length and whirls round

in a circle once a second: find the tension of the strin n terms of the weight of a gramme.

59. Assuming that there are 86,164 seconds in sidereal day, and that the earth's mean equatorial radiu is 3962 miles; calculate (in feet per second) the accele ation of a point on the equator.

60. Starting from the result of the preceding problem discuss the effect of the earth's rotation upon a spring balance which is used to weigh the same body (1) at th pole, (2) at the equator; and show that if the eart revolved about seventeen times as fast as it now does, body on the equator would have no apparent weight. 61. Prove, by any method, that the time of a complet oscillation of a simple pendulum is 2π when th

62. Find the value of g at a place where the length o the seconds pendulum is 0.994 metre.

[N.B.-A seconds pendulum is one which makes half a complet oscillation in a second.]

63. A pendulum 10 feet in length makes ten complet oscillations in 35 seconds: what is the value of g at the place?

64. Supposing a pendulum to be constructed to bea seconds at a place where g=950; how would its length have to be altered in order to make it beat seconds or the surface of the moon, where g=150?

65. Show that a pendulum of one metre in length would beat seconds if the value of g were 987.

66. What is the value of g at Greenwich, where the length of the seconds pendulum is found to be 39.14 inches, and what is the length of a pendulum which loses 10 minutes a day at this place?

67. The bob of a pendulum can be raised by means of a screw which has thirty threads to the inch; if the pendulum loses 5 minutes a day, how many turns of the head of the screw must be made in order to correct it?

(Assume that the pendulum keeps correct time when its length is 39 in.)

68. Enunciate the law of universal gravitation, and give an account of the method of measuring the attraction between two spheres, devised by Mitchell, and carried out by Cavendish.

69. How did Newton prove that the weight of a body is proportional to its mass? Describe the nature of his experiment, and explain how he deduced his conclusions.

70. Assuming the preceding proposition, and the third law of motion, show that it follows immediately that the attraction between two gravitating masses is directly proportional to the product of these masses.

71. Prove that a spherical shell exerts no attraction upon a particle placed within it. You may assume (a) That the area of a transverse section of a cone of small aperture varies as the square of the distance from the vertex.

(b) That a transverse section has a smaller area than an oblique section at the same distance, in the proportion of the cosine of the angle between them.

72. Prove that a uniform spherical shell attracts an external particle as if its whole mass were condensed at its centre.

Work. When the point of application of a force F moves through a distance s in the direction of the force, the work (W) done is

W=Fs.

If the force is measured in dynes (see § 3) and the distance in centimetres, the work will be expressed in ergs.

If the force is measured in poundals and the distance in feet, the work will be expressed in foot

roundals 14 poundel or British absoluto unit of foun

is that force which, acting upon a mass of one p generates in it an acceleration of one foot per seco a second.)

When the unit of force is one which dep upon the intensity of gravity, the work is expresse gravitation-units, whose value varies from place to p By way of distinction, the erg and the foot-pounda called absolute units. The engineer's unit of workfoot-pound-is a gravitation-unit; it represents the done at any place in raising a pound weight vertic through a distance of one foot at that particular pl A foot-pound is equal to g foot-poundals; or, tal g=32,

I foot-pound = 32 foot-poundals.

Since half an ounce is of a pound, the foot-poun about corresponds to the work done in raising half ounce through a vertical distance of one foot.

The kilogramme-metre (which is the French gineer's unit of work) is the work done in raising weight of a kilogramme through a vertical distance one metre against the force of gravity. It is open to same objections as the foot-pound, viz. that its value var from place to place and from level to level.

The gramme - centimetre is the work done raising a gramme weight through a vertical distance one centimetre; at a place where g=981, the weight a gramme is 981 dynes, and a gramme-centimetre equal to 981 ergs.

73. Find the work done by a force of 50 dynes acti through a distance of 2 metres.