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their work is regarded as positive; when they tend to diminish it, their work is regarded as negative. It is the aggregate of all the work expended, both positive and negative, that is measured by the quantity, Įmv2.

If, at any instant, a body whose mass is m, has a velocity v, and, at any subsequent instant, its velocity has become v', we shall have, for the accumulated work at these two instants,

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and, for the aggregate quantity of work expended in the interval,

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When the motive forces, during the interval, perform a greater quantity of work than the resistances, the value of v' will be greater than that of v, and there will be an accumulation of work in the interval. When the work of the resistances exceeds that of the motive forces, the value of v will exceed that of v', Q" will be negative, and there will be a loss of living force, which is absorbed by the resistances.

Living Force of Revolving Bodies.

149. Denote the angular velocity of a body which is restrained by an axis, by ; denote the masses of its elementary particles by m, m', &c., and their distances from the axis of rotation, by r, r', &c. Their velocities will be re, r', &c., and their living forces will be mrs, m'r'242, &c. Denoting the entire living force of the body, by L, we shall have, by summation, and recollecting that 42 is the same for all the terms,

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But (mr) is the expression for the moment of inertia of the body, taken with respect to the axis of rotation. De

noting the entire mass by M, its radius of gyration, with respect to the axis of rotation, by k, we shall have,

L = Mk22.

If, at any subsequent instant, the angular velocity has become ', we shall, at that instant, have,

L' = Mk2012 ;

and, for the loss or gain of living force in the interval, we

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which shows that the moment of inertia of a body, with respect to an axis, is equal to the living force lost or gained whilst the body is experiencing a change in the square of its angular velocity equal to 1.

The principle of living forces is extensively applied in discussing the circumstances of motion of machines. When the motive power performs a quantity of work greater than that necessary to overcome the resistances, the velocities of the parts become accelerated, a quantity of work is stored up, to be again given out when the resistances offered require a greater quantity of work to overcome them than is furnished by the motor.

In many machines, pieces are expressly introduced to equalize the motion, and this is particularly the case when either the motive power or the resistance to be overcome, is, in its nature, variable. Such pieces are called fly-wheels.

Fly-Wheels.

150. A fly-wheel is a heavy wheel, usually of iron, mounted upon an axis, near the point of application of the

force which it is destined to regulate. It is generally com

posed of a heavy rim, connected with the axis by means of radial arms. Sometimes it consists of radiating bars, carrying heavy spheres of metal at their outer extremity. In either case, we see, from Equation 139, that, for a given quantity of work absorbed, 'the value of 8'2 62 will be less as M

Fig. 129.

and are greater; that is, the change of angular velocity will be less, as the mass of the fly-wheel and its radius of gyration increase. It is for this reason that the peculiar form of fly-wheel indicated above, is adopted, it being the form that most nearly realizes the conditions pointed out. The principal objection to large fly-wheels in machinery, is the great amount of hurtful resistance which they create, such as friction on the axle, &c. Thus, a fly-wheel of 42000 lbs. would create a force of friction of 4200 lbs., the coefficient of friction being but; and, if the diameter of the axle were 8 inches, and the number of revolutions 30 per minute, this resistance alone would be equal to 8 horse powers.

EXAMPLES.

1. The weight of the ram of a pile-driver is 400 lbs., and it strikes the head of a pile with a velocity of 20 feet. What is the amount of work stored up in it?

SOLUTION.

The height due to the velocity, 20 feet, is equal to

400

6.22 ft., nearly.

641

Hence, the stored up work is equal to

400 lbs. × 6.22 ft. = 2488 lbs. ft.;

or, the stored up work, equal to half the living force, is equal to

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2. A train, weighing 60 tons, has a velocity of 40 miles per hour when the steam is shut off. How far will it travel, if no break be applied, before the velocity is reduced to 10 miles per hour, the resistance to motion being estimated at 10 lbs. per ton. Ans. 16834 ft.

Composition of Rotations.

FH

Fig. 130.

other at O, and let P be

151. Let a body ACBD, that is free to move, be acted upon by a force which, of itself, would cause the body to revolve for the infinitely small time dt, about the line AB, with an angular velocity v; and at the same instant, let the body be acted upon by a second force, which would of itself cause the body to revolve about CD, for the time dt, with an angular velocity v'. Suppose the axes to intersect each any point in the plane of the axes. Draw PF and PG respectively perpendicular to OB and OC, denoting the former, by x, and the latter, by y. Then will the velocity of P due to the first force, be equal to va, and its velocity due to the second force will be equal to v'y. Suppose the rotation to take place in such a manner, that the tendency of the rotation about one of the axes, shall be to depress the point below the plane, whilst that about the other is to elevate it above the plane; then will the effective velocity of P be equal to ve v'y. If this effective velocity is 0, the point P will remain at rest. Placing the expression just deduced equal to 0, and transposing, we have,

vx

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-To determine the position of P, lay off OH equal to v, OI equal to v', and regard these lines as the representatives of two forces; we have, from the equation, the moment of v, with respect to the point P, equal to the moment of v', with respect to the same point. Hence, the point P must be somewhere upon the diagonal OK, of the parallelogram described on v, and v'. But P may be anywhere on this line; hence, every point of the diagonal OK, remains at rest during the time dt, and is, consequently, the resultant axis of rotation. We have, therefore, the following principles: If a body be acted upon simultaneously by two forces, each tending to impart a motion of rotation about a separate axis, the resultant motion will be one of rotation about a third axis lying in the plane of the other two, and passing through their common point of intersection.

The direction of the resultant axis coincides with the diagonal of a parallelogram, whose adjacent sides are the component axes, and whose lengths are proportional to the impressed angular velocities.

I

K

H

Fig. 131.

Let OH and OI represent, as before, the angular velocities v and v', and OK the diagonal of the parallelogram constructed on these lines. as sides. Take any point I, on the second axis, and let fall a perpendicular on OH and OK; denote the former by r, and the latter, by "; denote, also, the resultant angular velocity, by v". Since the actual space passed over by I, during the time t, depends only upon the first force, it will be the same whether we regard the revolution as taking place about the axis OH, or about the axis OK. If we suppose the rotation to take place about OH, the space passed over in the time dt, will be equal to rudt; if we suppose the rotation to take place about OK, the space passed over in the same time will be equal to r''v'dt. Placing these expressions equal to each other, we have, after reduction,

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