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 5W, 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 O, 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 = 5 W. OA,

and therefore, since OB 100A,

sin 0 = 3, 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?

(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 1195lbs.

(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, 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 120o.

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

at the first pully and one of 3 W pounds at the second: to find the power.

Let T1, T2, T, be the tensions of the strings under the pullies C1, C2, C,, fig. (100), respectively: then

But P= T: hence the power P is equal to W.

(2) To find the ratio of the power to the weight of each pully in the system of pullies, in which each pully hangs by a separate string, the pullies being of equal weight, (1) when there is no mechanical advantage, (2) when the power just supports the pullies.

Let P represent the power, W the weight, A the weight of each pully, and n the number of pullies. Then, as is proved in systematic treatises on Statics,

2" (P-A) = W– A.

(1) When there is no mechanical advantage, W is not greater than P: but our equation shews that it cannot be less than P: hence WP, and therefore

2′′ (P— A) = P− A,

and therefore P= A, or the ratio of the power to the weight of each pully is a ratio of equality.

(2) When the power just supports the pullies, W=0; and therefore

(3) If a weight of 160 pounds be supported on an inclined plane by means of a string passing under a fixed pully at the top, and carried vertically upwards to the lowest pully of the system where each pully hangs by a separate string; to find the

number of pullies, when 6 pounds will maintain equilibrium. The height of the plane is 2 yards, and its length 10 feet.

The tension of the string is equal to the component of the weight taken along the inclined plane, that is, to

Then, putting P=6 and W=96 in the formula 2"P=W, where n denotes the number of the moveable pullies, we have

2" × 6 = 96, 2" =16, n=4.

(4) To determine the relation between the radii of the pullies of the system, in which each pully hangs by a separate string, and the strings are parallel, in order that, if their centres be at any time in a straight line, they may always continue to be so.

Let c, c1, C2, C3,... be the depths of the centres of the pullies, beginning with the highest, below a horizontal plane, and let ~, 71, 72, 7 ̧‚....... be the respective radii of the pullies. Then, supposing the centres to be in a straight line, we must have

where n denotes any one of the natural numbers.

.(1),

α α a

Suppose c to be augmented by a length a; then c1, c2g, C3,... will be respectively augmented by lengths, 2, 3,..... Hence, supposing the relation (1) still to hold good, we must have

a result which shews that, r, being the radius of the second pully, the radii of the rest must be