(37) Two heavy bars AC, BC, fig. (97), of uniform material and of the same thickness, are attached by two small hooks to two small fixed rings, A and B, in the same horizontal line, and their other extremities are also hooked together at C; find the magnitude and direction of the force exerted at C; and prove that its direction will fall without or within the triangle ABC, accordingly as the perpendicular from Con AB falls within or without the triangle.

Let XY be the line of action of the force at C: let

▲ BAC=α, ▲ ABC=ß, ▲ ACB=7, 4 ACX=0, ‹ BCY=0; let P, Q, represent the respective weights of the bars AC, BC, and T the magnitude of the required force. Then the magnitude and direction of the required force are given by the equa

(38) Two weights of similar material, connected by a fine smooth string, rest on a rough vertical circular circumference, with which the string is in contact: prove that, if μ be the coefficient of friction, the angle subtended at the centre by the distance between the limiting positions of either weight, is equal to 2 tanμ.

(39) A horizontal rod is moveable freely round its middle point, which is fixed at the vertex of an inclined plane: a weight P hangs freely from the extremity of the rod which is not directly above the inclined plane, and to the other extremity one end of a string, the length of which is equal to half that of the rod, is fastened: to the free end of this string is attached a weight W, resting on the plane: prove that, if a be the inclination of the plane to the horizon,

P=2W sin2 a.

If the plane were rough, and the coefficient of friction such

that W would just rest on it without any support, prove that the equilibrium would not be disturbed if P were increased by a weight bearing to P the ratio of cos 2a: 1.

(40) A series of n equal balls, of different substances, connected together by strings, are placed on a rough plane inclined to the horizon at an angle a, so as to form a line perpendicular to the intersection of the plane with the horizon: prove that, if μ, be the coefficient of friction between the 7th ball and the plane, and the system be on the point of slipping,

CHAPTER IX.

MECHANICAL POWERS.

SECT. 1. Wheel and Axle.

(1) A MOUSE, of weight W, clings to the lower circumference of the wheel, in a wheel and axle, and so just supports a weight 5 W, the ratio of the radii of the wheel and axle being 10 to 1; to find the inclination, to the vertical, of the radius of the wheel which passes through the position of the mouse; and to shew that the mouse is in a position of stable equilibrium, but that, if it were on the upper surface of the wheel, at a point vertically above its present position, its equilibrium would be unstable.

Let 0, fig. (98), be the common axis of the wheel and axle, OA being the horizontal radius of the axle from the end A of which the weight hangs, and OB the radius of the axle at the end B of which the mouse clings. Let be the inclination of OB to the vertical. Taking moments about O, for the equilibrium of the system, we have

W. OB. sin 0 = 5W. OA,

and therefore, since OB = 100A,

sin0=†, 0=7.·

6

Suppose the mouse to be placed a little above its present position: then, the moment of W about O being increased, the mouse will descend towards its position of rest, raising 5 W. Again, suppose the mouse to be placed a little below its present position: then, the moment of W about O being diminished, the weight 5 W will descend, raising the mouse towards its position of rest. Thus we see that the position of equilibrium is stable.

If the mouse cling to the wheel at a point vertically above B, it will be in a position of equilibrium; but, as may readily be seen by reasoning like the above, it will recede further from its position of rest, if slightly displaced either way. Its latter position of equilibrium will therefore be unstable.

(2) What weight, suspended from the axle, can be supported by 14lbs., suspended from the wheel, if the radius of the axle is 1ft., and the radius of the wheel is 3 feet?

The required weight = 3lbs. 4oz.

(3) Two men, who can exercise forces of 200 lbs. and 248lbs. each, work at an axle to which two wheels are attached, of 5 feet and 4 feet diameter respectively, the diameter of the axle being 20 inches: find the greatest weight the men can raise by it.

The required weight is 1195 lbs.

(4) If the difference between the radii of a wheel and axle be eight inches, and the power and weight be as 6 to 7, find the radii.

The radii of the wheel and axle are respectively 4ft. 8 in. and 4ft.

(5) If a weight W be kept from sliding down an inclined plane, of inclination a, by a string, which is parallel to the plane, and which passes round a wheel of radius r, find the weight which must hang from an axle of radius r', having a common axis with the wheel, that there may be equilibrium.

The required weight is equal to. W sin a.

(6) If there be a system of wheels and axles, such that the rope, which is wound round the axle of the first, passes over the wheel of the second, that round the axle of the second over the wheel of the third, and so on; prove that the power applied at the first wheel is to the weight supported at the last axle as the product of the radii of the axles to the product of the radii of the wheels.

SECT. 2. Toothed Wheels.

Two toothed wheels work against each other: shew that, if the number of the teeth in one be prime to that in the other, before two teeth, which have been in contact once, come into contact again, every tooth of the one wheel will have been in contact with every tooth of the other.

SECT. 3. Single Moveable Pully.

(1) A rope passes over a pully; one end is attached to a man, who grasps the other end with both hands; to find the proportion of his weight sustained by each arm, when he exerts the same stress on both.

Let W denote the weight of the man, and let P be the force exerted with each hand: then the tension of the string will be 2P: but the whole weight of the man is sustained by the sum of the tensions of the two portions of the string: hence

2P+2P=W, P=1W;

hence a quarter of the man's weight is sustained by each arm.

(2) A weight W, fig. (99), is suspended from a single moveable pully, which is supported by a weight P hanging over a fixed pully, the strings being parallel: prove that, in whatever position they hang, the position of their centre of gravity is the same.

(3) An endless string hangs at rest, over two pegs in the same horizontal plane, with a heavy pully in each festoon of the string if the weight of one pully be double that of the other, prove that the angle between the portions of the upper festoon must be greater than 120°.

SECT. 4. First System of Pullies.

(1) In a system of three moveable pullies, where each pully hangs by a separate string, a weight of W pounds is suspended