may mutually influence one another, if forces equal to the effective forces were applied in directions opposite to their actual directions, these would be in equilibrium with the impressed forces, which is the principle of D'Alembert.

104. The work accumulated in a moving body through any space is equal to the work which must be done upon it, in an opposite direction, to overcome the effective force upon it through that space.

This is evident from Arts. 68. and 69., since the effective force is the unbalanced pressure upon the body.

If the work of the effective force be said to be done upon the body, then the work of the effective force upon it is equal to the work or power accumulated in it, and this work of the effective force may be all said to be actually accumulated in the body as in a reservoir.

MOTION OF TRANSLATION.

DEFINITION. When a body moves forward in space, without at the same time revolving, so that all its parts move with the same velocity and in parallel directions, it is said to move with a motion of translation only.

105. In order that a body may move with a motion of translation only, the resultant of the forces impressed upon it must have its direction through the centre of gravity of the body.

For let w,, w,, w,, &c. represent the weights of the parts or elements of the body, and let f represent the additional velocity per second, which any element receives or would receive if its motion were at any instant to become uniformly accelerated. Since the motion is one of translation only, the value of f is evidently the same in respect to every other element. The effective forces P,, P2, P., &c. on the different elements of the body are therefore represented by

This cannot perhaps be correctly said, since work supposes resistance.

Now the forces P,, P., P,, &c. are evidently parallel pres sures. Let X be the distance of the centre (see Art. 17.) of these parallel pressures from any given plane; and let x,,„, ,, &c. be the perpendicular distances of the elements w1, w, w, &c. that is, of the points of application of P,, P1, P„, &c. from the same plane. Therefore (by equation 18),

But this is the expression (Art. 19.) for the distance of the centre of gravity from the given plane; and this being true of any plane, it follows that the centre of the parallel pressures P, P, P., &c. which are the effective forces of the system, coincides with the centre of gravity of the system, and therefore that the resultant of the effective forces passes through the centre of gravity. Now the resultant of the effective pressures must coincide in direction with the resultant of the impressed pressures, since the effective pressures when applied in an opposite direction are in equilibrium with the impressed pressures (by D'Alembert's principle). The resultant of the impressed pressures must therefore have its direction through the centre of gravity. Therefore, &c.

MOTION OF ROTATION ABOUT A FIXED Axis.

106. Let a rigid body or system be capable of motion about the axis A. Let m,, m,, m,, &c. represent the volumes of elements of this body, and the weight of each unit of volume. Also let f, ff, &c. represent the increments of velocity per second, communicated to these elements respectively by the action of the forces impressed upon the system. Let P, P,, P., &c. represent these impressed forces, and P, P., &c. the perpendicular distances from the axis at which they are respectively applied.

Now since um,, um, um, &c. are the weights of the ele ments, and f,f,, &c. the increments of velocity they receive

the effective forces upon them (Art. 103.). Let P1, P2, P2, &c. represent the distances of these elements respectively from the axis of revolution, then since their effective forces are in directions perpendicular to these distances, the moments of these effective forces about the axis are

g

ffa, &c. Also P11, P2P2, P2P1, &c. are the moments of the impressed forces of the system about the axis. Now the impressed forces P, P., P., &c., together with the resistance of the axis, which is indeed one of the impressed forces, are in equilibrium with the effective forces by D'Alembert's principle. Taking then the axis as the point from which the moments are measured, the sum of the moments of P1, P,, &c. must equal the sum of the moments of the effective forces, or

μm, fils + uma falat • =P11+P2P2+

g

....

Now let f represent that value of f,f,, &c. which corresponds to a distance unity from the axis. Since the system is rigid, and f, f, fa, &c. represent arcs described about it in the same time at the different distances 1, P., P2, &c. it follows that these arcs are as their distances, and therefore that f=fffff=f1, &c. Substituting these values in the preceding equation, we have

" m, fp,2 + " m2 fl22 + . . . . =P‚μ‚+P‚μ‚+ . . . .;

g

g

where I represents the moment of inertia of the mass about its axis of revolution.*

* If a represent the angular velocity, or the velocity of an element at dis

da

tance unity, then by equation (72), ƒ=±dt'

da ...α = dt

Ppa;

G

107. If the impressed forces P be the weights of the parts of the body and be, in any position of the body, the inclination to the vertical Ay of the line AG, drawn from A to the centre of gravity G, then since the sum of the moments of the weights of the parts is equal to the moment of the weight of the whole mass collected in its centre of gravity (Art. 17.), we have, representing

AG by G,

ΣPp=Mμ . GG,=Mμ . G. sin. ◊

MG

therefore (equation 78), f=gsin. . . . . .

(79).

108. To find the resultant of the effective forces on a body which revolves about a fixed axis.

The resultant of the effective forces upon a body which revolves about a fixed axis, is evidently equal to that single force which would just be in equilibrium with these if there were no resistance of the axis. Let R be that single force, then the moment of R about any point must equal the sum of the moments of the effective forces about that point.

Take a point in the axis for the point

about which the moments are measured, and let L be the perpendicular distance from A of the resultant R. Now, as in Art. 106. it appears that the sum of the moments of the effective forces about A is

Now

pa is the velocity of a point at distance p, therefore Ppa is the work.

(Art. 50.) of the force P per second; therefore Ppadt is the work of P

0

(equation 40) in the time t, which is represented by U, therefore a1-a,' 2gU which corresponds with the result already obtained. See equation

+

με

To determine the value of R let it be observed that the effective force" fm,p, on any particle m, acting in a direcforce-fmp,

tion n,m,, perpendicular to the distance Am, from the axis A, may be resolved into two others, parallel to the two rectangular axes Ay and Ax, each of which is equal to the product of this effective force, whose direction is n,m,, and the cosine of the inclination of nm, to the corresponding axis. Now the inclination of mn, to Ax is the same as the inclination of Am, to Ay, since these two last lines are perpendicular to the two former. The cosine of this inclination

equals therefore AN, or, if AN,y,. Similarly the cosine

1

AM.

Am1 P1 of the inclination of n,m, to Ay equals: or, if AM, = ̧.

Am, Pi

The resolved parts in the directions of Ax and Ay of the effective force fmp, are therefore fm,,, and fm,

9

g

Similarly the resolved parts in the directions of Ax and

The sums X and Y of the resolved forces in the directions of Ax and Ay respectively (Art. 11.) are therefore

and f{m,x,+ m ̧x, + m ̧x, + . . . . . } = Y.

Now let G, and G, represent the distances G,G and G,G of the centre of gravity of the body from Ay and Ax respectively, and let the whole volume of the body be represented by M,