16 inches: if the radii were each shortened by 4 inches, find what Weight would be supported by the same Power. 5. If the radii of the Wheel and Axle be as 8 is to 3, and two weights of 6 lbs. and 15 lbs. respectively be suspended from their circumferences, find which weight will descend. Supposing that the weight which tends to descend is supported by a prop, find the pressure on the prop and on the fixed supports of the Wheel and Axle. 6. The radius of the axle of a capstan is 2 feet, and six men push each with a force of one cwt. on spokes 5 feet long: find the Weight they will be just able to raise. 7. The difference of the diameters of a Wheel and Axle is 2 feet 6 inches; and the Weight is equal to six times the Power: find the radii of the Wheel and the Axle. 8. The radius of the Wheel being three times that of the Axle, and the string on the Wheel being only strong enough to support a tension of 36 lbs., find the greatest Weight which can be raised. 9. If the string to which the Weight is attached be coiled in the usual manner round the Axle, but the string by which the Power is applied be nailed to a point in the rim of the Wheel, find the position of equilibrium, the Power and the Weight being equal. 10. In the Wheel and Axle if the two ropes were coiled each on itself, and their thickness not neglected, find whether the ratio of the Power to the Weight would be increased or diminished as the Weight was raised, supposing the ropes of the same thickness. XIV. The Pully. 190. The Pully consists of a small circular plate or wheel which can turn round an axis passing through the centres of its faces, and having its ends supported by a framework which is called the Block. The circular plate has a groove cut in its edge to prevent a string from slipping off when it is put round the Pully. 191. Let A denote a Pully the Block of which is fixed; and suppose a Weight attached to the end of a string passing round the Pully. If the string be pulled at the other end by a Power equal to the Weight there will be equilibrium. Thus a fixed Pully is an instrument by which we change the direction of a force without changing its magnitude. We have already adverted to this in Art. 28. ત W As we proceed with the present Chapter it will be seen that by the use of a moveable Pully we can gain mechanical advantage. Theoretically the fact that the Pully can turn round its axis is not important; but practically it is very important. When the Pully can turn round it is found that the tension of the string is almost exactly the same on both sides of the Pully in the condition of equilibrium. But when the Pully cannot turn round it is found that there may be considerable difference between the tensions of the two parts of the string: this is owing to Friction, which we shall consider hereafter. In all that follows we shall assume that the tension of a string is not changed when the string passes round a Pully. We shall always neglect the weight of the strings; and also the weight of the Pullies unless the contrary be stated. 192. In a single moveable Pully with the strings parallel when there is equilibrium the Weight is twice the Power. Let a string pass round the Pully A, have one end fixed, as at K and be pulled vertically upwards by a Power, P, at the other end. Let a Weight, W, be attached to the Block of the Pully. The tension of the string is the same throughout. Hence we may regard the Pully as acted on by two parallel forces, each equal to P, up W wards, and by the force W downwards. Therefore W =2P. It may be observed that the line of action of W must be equally distant from the two parts of the string; that is, it must pass through the centre of the Pully. The pressure on the fixed point K is equal to P, that is, to W. 193. The preceding Article will probably present no difficulty to the student; but perhaps the following remarks should be made. The Wheel and the Block of the Pully are really two distinct bodies; but when there is equilibrium we shall not disturb it by rigidly connecting the two bodies: thus we obtain one rigid body, and the condition of equilibrium follows by Art. 62. Sometimes the principle of the lever is employed in obtaining this condition of equilibrium; the strict mode of employing the principle is as follows: The Wheel of the Pully is capable of turning round its axis, and for equilibrium the moments of the forces round this axis must be equal; this condition is satisfied if the axis be equidistant from the two parts of the string. The pressure on the axis is equal to the sum of the two forces; and this pressure is supported by the Block. Thus the Block is acted on by 2P upwards, and by W downwards. Therefore W=2P. 194. To find the ratio of the Power to the Weight in the single moveable Pully with the parts of the string not parallel. Let a string pass round the Pully, A, have one, end fixed, as at K, and be pulled by a Power, P, at the other end. Let a Weight, W, be attached to the Block of the Pully. The tension of the string which passes round the Pully is the same throughout. Hence we may regard K the Pully as acted on by two forces, each equal to P, and a force W. Therefore the line of action of W must bisect the angle formed by the lines of action of the two forces P; that is, the two parts of the string must be equally inclined to the vertical. Suppose them each to make an angle a with the vertical. Then W is equal and opposite to the resultant of two equal forces P, which are inclined at an angle 2a. Thus the ratio of P to W is known by the Parallelogram of Forces. By Art. 30, we have W=2P cosa. 195. We now pass on to investigate the conditions of equilibrium of various combinations of Pullies. 196. In the system of Pullies in which each Pully hangs by a separate string and all the Pullies are parallel, when there is equilibrium the Weight is equal to the Power multiplied by 2", where n is the number of Pullies. In this system the string which passes round any Pully except the highest has one end attached to a fixed point, and the other end to the Block of the next higher Pully; the string which passes round the highest Pully has one end attached to a fixed point, and the other end supported by the Power. Suppose there are four moveable Pullies. Let W denote the weight, which is suspended from the block of the W KLMN W that is ; the tension of the string which passes under the next Pully is half of this, that is W 24 This last tension must be equal to the Power which acts at the end of the Similarly, if there be or W=24P. any number of moveable Pullies and n denote this number, W=2′′ P. This system of Pullies is sometimes called the First System of Pullies. 197. Let K, L, M, N denote the points at which the ends of the strings are fixed in the system of Pullies considered in the preceding Article. Then the pressure at W Kis the pressure at Z is the 22, pressure at M is W W 24. W 23 the pressure at N is Hence the sum of these pressures is W + 1 1 22+2+); by summing the Geometrical Pro 23 24 gression, we find that this is W(1 Thus the sum of these pressures together with the Power is equal to the whole Weight. |