CHAPTER V

THE MOTION OF A SOLID BODY OF ANY FORM WHATEVER.

169. Ir a solid body receives an impulse in a direction passing through its centre of gravity, all its parts will move with an equal velocity; but if the direction of the impulse passes on one side of that centre, the different parts of the body will have unequal velocities, and from this inequality results a motion of rotation in the body round its centre of gravity, at the same time that the centre is moved forward, or translated with the same velocity it would have taken, had the impulse passed through it. Thus the double motions of rotation and translation are produced by one impulse.

170. If a body rotates about its centre of gravity, or about an axis, and is at the same time carried forward in space; and if an equal and contrary impulse be given to the centre of gravity, so as to stop its progressive motion, the rotation will go on as before it received the impulse.

171. If a body revolves about a fixed axis, each of its particles. will describe a circle, whose plane is perpendicular to that axis, and its radius is the distance of the particle from the axis. It is evident, that every point of the solid will describe an arc of the same number of degrees in the same time; hence, if the velocity of each particle be divided by its radius or distance from the axis, the quotient will be the same for every particle of the body. This is called the angular velocity of the solid.

172. The axis of rotation may change at every instant, the angu lar velocity is therefore the same for every particle of the solid for any one instant, but it may vary from one instant to another.

173. The general equations of the motion of a solid body are the same with those of a system of bodies, provided we assume the bodies m, m', m", &c. to be a system of particles, infinite in number, and united into a solid mass by their mutual attraction.

Let x, y, z, be the co-ordinates of dm, a particle of a solid body

urged by the forces X, Y, Z, parallel to the axes of the co-ordinates; then if S the sign of ordinary integrals be put for E, and dm for m, the general equations of the motion of a system of bodies in article 158 become

which are the general equations of the motion of a solid, of which m is the mass.

Determination of the Equations of the Motion of the Centre of Gravity of a Solid in Space.

174. Let +x', y+y', z+z', be put for x, y, z, in equations (28)

in which x, y, z, are the co-ordinates of o the moveable centre of gravity of the solid referred to P a fixed point, and x'y' z' are the co-ordinates of dm referred to o, fig. 47. Now the co-ordinates of the centre of gravity being the same for all the particles of the solid,

And, with regard to the centre of gravity,

S. x'dm0

S. y'dm=0

S. z'dm=0

which denote the sum of the particles of the body into their respec

tive distances from the origin; therefore their differentials are

These three equations determine the motion of the centre of gravity of the body in space, and are similar to those which give the motion of the centre of gravity of a system of bodies.

The solid therefore moves in space as if its mass were united in its centre of gravity, and all the forces that urge the body applied to that point.

175. If the same substitution be made in the first of equations (29), and if it be observed that as a, y, z, are the same for all the particles

also

yd x)

S (Td'y yd2) dm = m (7d'y S (TYyX) dma.S. Ydmy.S.Xdm; S (x'd2y - y'd2ñ + ïd2y' — ÿd2x') dm = dy.S.x'dmdx.S.y'dm + x.S.dy'.dmy.S.d'x'.dm = 0, because x', y', z', are referred to the centre of gravity as the origin of the co-ordinates; consequently the co-ordinates 7, y, z, and their differentials vanish from the equation, which therefore retains its original form. Similar results will be obtained for the areas on the

other two co-ordinate planes, and thus equations (29) retain the same forms, whether the centre of gravity be in motion or at rest, proving the motions of rotation and translation to be independent of one another.

The integrals of equations (29), with regard to the time, will be

These equations, which express the properties of areas, determine the rotation of the solid;-equations (31) give the motion of its centre of gravity in space. S expresses the sum of the particles of the body, and relates to the time alone.

177. Impetus is the mass into the square of the velocity, but the velocity of rotation depends on the distance from the axis, the angle being the same; hence the impetus of a revolving body is the sum of the products of each particle, multiplied by the square of its distance from the axis of rotation. Suppose oA, oB, oC, fig. 10, to be the co-ordinates of a particle dm, situate in m, and let them be represented by x, y, z; then because mA = Ro, mBQo, mC = Po, the of the distances of dm from the three axes ox, oy, oz, squares are respectively

(mA) = y + z2, (mB)2 = x2 + z2, (mC)2 = x2 + y2. Hence if A', B', C', be the impetus or moments of inertia of a solid with regard to the axes ox, oy, oz, then

A'S. dm (y2 + z2)

B'S. dm (x2 + z2)

C'S. dm (x2 + y2).

(33)

178. If an impulse be given to a sphere of uniform density, in a

*

direction which does not pass through its centre of gravity, it will revolve about an axis perpendicular to the plane passing through the centre of the sphere and the direction of the force; and it will continue to rotate about the same axis even if new forces act on the sphere, provided they act equally on all its particles; and the areas which each of its particles describes will be constant.

179. If the solid be not a sphere, it may change its axis of rotation at every instant; it is therefore of importance, to ascertain if any axes exist in the solid, about which it would rotate permanently.

180. If a body rotates permanently about an axis, the rotatory pressures arising from the centrifugal forces of the solid are equal and contrary in each point of the axis, so that their sum is zero, and the areas described by every particle in the solid are proportional to the time; but if foreign forces disturb the rotation, the rotatory pressures on the axis of rotation are unequal, which causes a perpetual change of axis, and a variation in the areas described by the particles of the body, so that they are no longer proportional to the time. Thus the inconstancy of areas becomes a test of disturbing forces. In this disturbed rotation the body may be considered to have a permanent rotation during an instant only.

181. When three axes of a solid body are permanent axes of rotation, the rotatory pressures on them are zero; this is expressed by the equations S.xydm = 0; S.xzdm = 0; S.yzdm = 0; which characterize such axes. To show this, it is necessary to prove that when these equations hold, the rotation of the body round any one axis causes no twisting effort to displace that axis; for example, that the centrifugal forces developed by rotation round z, produce no rotatory pressure round y and r; and so for the other, and vice versa.

Demonstration.-Let r = √x2 + y2 be the distance of a particle dm from z the axis of rotation, and let w be the angular velocity of v2 the particle. By article 171 w = therefore w2.r=

v

2 is the

centrifugal force arising from rotation round z, and acting in the direction r. When resolved in the direction a, and multiplied by