number of bodies in motion, and P1, P2, P3, &c. the moving forces (Art. 92.) upon these bodies at any given instant of the motion (i. e. the unbalanced pressures, or the pressures which are wholly employed in producing their motion, and pressures equal to which, applied in opposite directions, would bring them to rest, or to a state of uniform motion.

W

W

Then (Art. 95.), P1='f1, P2= 2f2, P3

g

W 3=3f3, &c. where f1f2f3, &c. represent the additions of velocity which the bodies would receive in each second of time, if the moving force upon each were to become, at the instant at which it is measured, an uniform moving force. Suppose these bodies, whose weights are W1, W2, W3, &c. to form a system of bodies united together by any conceivable mechanical connection, on which system are impressed, in any way, certain forces, whence result the unbalanced pressures P1, P2, P3, &c. on the moving points of the system. Now conceive that to these moving points of the system there are applied pressures respectively equal to P1, P2, P3, &c. but each in a direction opposite to that in which the motion of the corresponding point is accelerated or retarded. Then will the motion of each particular point evidently pass into a state of uniform motion, or of rest (Art. 92.). The whole system of bodies being thus then in a state of uniform motion, or of rest, the forces applied to its different elements must be forces in equilibrium.

Whatever, therefore, were the forces originally impressed upon the system, and causing its motion, they must, together with the pressures P1, P2, P3, &c. thus applied, produce a state of equilibrium in the system; so that these forces (originally impressed upon the system, and known in Dynamics as the IMPRESSED FORCES) have to the forces P1, P2, P3, &c. when applied in directions opposite to the motions of their several points of application, the relation of forces in equilibrium. The forces P1, P2, P3, &c. are known in Dynamics as the EFFECTIVE FORCES. Thus in any system of bodies mechanically connected in any way, so that their motions 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 w1, w, w3, &c. represent the weights of the parts or elements of the body, and let ƒ 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 ƒ is evidently the same in respect to every other element. The effective forces P1, P2, P3, &c. on the different W1 elements of the body are therefore represented by ff, 9

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

sures.

Now the forces P1, P2, P3, &c. are evidently parallel presLet X be the distance of the centre (see Art. 17.) of these parallel pressures from any given plane; and let x1, x2, x3, &c. be the perpendicular distances of the elements w1, w, w3, &c. that is, of the points of application of P1, P2, P3, &c. from the same plane. Therefore (by equation 18),

X39

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 P1, P2, P3, &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, m2, m3, &c. represent the volumes of elements of this body, and the weight of each unit of volume. Also let f1, f, fa, &c. represent the increments of velocity per second, communicated to these elements respectively by the action of the forces impressed upon the system. Let P1, P2, P3, &c. represent these impressed forces, and P1, P2, &c. the perpendicular distances from the axis at which they are respectively applied.

Now since μm, μm,, μm3, &c. are the weights of the elements, and f1, f2, &c. the increments of velocity they receive

per second, it follows that if, mf, umf, &c. are the

9

effective forces upon them (Art. 103.). Let P1, P2, P3, &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

2P29

umfp3, &c. Also P1P1, P2P2, P3P2, &c. are the moments of

9

the impressed forces of the system about the axis. Now the impressed forces P1, P2, P3, &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, P2, &c. must equal the sum of the moments of the effective forces, or

Now let ƒ represent that value of fi, f2, &c. which corresponds to a distance unity from the axis. Since the system is rigid, and ƒ, ƒ1, ƒ1⁄2, &c. represent arcs described about it in the same time at the different distances 1, P1, P2, &c. it follows that these arcs are as their distances, and therefore that f1 =ƒP19f2=f?29 £3=f3, &c. Substituting these values in the preceding equation, we have

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

C

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. GG2=Mμ. G. sin. §;

MG

therefore (equation 78), f=g sin.

I

(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.

* If a represent the angular velocity, or the velocity of an element at distance unity, then by equation (72), f=±, ... =+pa;

da

da

dt

Now

ρα

is the velocity of a point at distance p, therefore Ppa is the work (Art. 50.) of the force P per second; therefore padt is the work of

0

P (equation 40) in the time t, which is represented by U, therefore

2gU
9
μΙ

a12—a2=+ which corresponds with the result already obtained. See equation (51).