be the radius vector and the vectorial

Now let r and angle of an element of mass of a rigid body, which can only

do dt

rotate round a fixed axis. is the rate of change of the

angle between a line of particles in the body and a line fixed

in space. Therefore and are the same for

dt

dt

being in fact the angular velocity and angular acceleration of the body. Denote the former by w. Also during the motion and for the same element dm, r does not change.

If the change of motion be sudden,

Σồm {x (v' — v) — y (u' — u)} = (w' — w) Σdm r2.

These are the moments of the effective forces round the fixed axis. The single resultant through the axis is the same as through any other point, and is therefore the resultant effective force of the whole mass collected at the centre of inertia.

If there is no fixed axis the centre of inertia is taken as the point, the motion of which and the motion round which determine the circumstances. The resultant effective

force is that of the whole mass collected there and moving with its motion. And the motion round it is the same as if it were a fixed point. The forces therefore reduce to

acting at the centre of inertia, and a couple (w' – w) Σômr2; or, if the motion be accelerated, to

centre of inertia).

Em r2, (r being the distance of ôm from the

If the system consists of rigid bodies not rigidly connected these forces may be reduced to one at the common centre of mass, but the couples must be taken for the separate bodies.

6. The expression Σồm (xv — yu) or Σôm (1o de)

is called the angular momentum about the origin. It has sometimes been called the moment of the momentum. might with advantage be kept separate.

If there is a fixed axis of rotation,

These names

Σồm (xv-yu) = M (xv — ÿu) + w Σdm r2,

in which r is the distance of the element Sm from the axis through the centre of inertia. Let the term “moment of momentum" be kept for the whole mass collected at the centre of inertia, i.e. for the expression M (xv — yu), and let the term wom r2 be called the "quantity of rotational motion." Round a fixed axis or round the centre of inertia, angular momentum is then the same as quantity of rotation.

It is important to notice that the quantity of rotational motion of an element is measured by the square of the distance from the axis. When Newton attacked the problem of precession, he proved that if a rotating ring communi

cated motion to a mass attached to it, the whole quantity of motion would remain the same. This is right; but Newton measured the quantity of motion by the sum of the linear motions of the elements, which is wrong. In Rigid Dynamics we introduce a new kind of motionrotation-caused by another kind of force (viz. a couple). Now a couple is measured by the moment of a simple force; quantity of rotation therefore is measured by the moment of a momentum.

Imagine a fly-wheel whose mass is m and radius a, rotating about its centre with velocity w. Every point is moving with linear velocity aw. And therefore in one sense (excluding the idea of direction) the whole quantity of linear momentum is maw. The impulse which applied at a point on the wheel will stop the motion is maw. There is sometimes an advantage in considering the motion thus; but our knowledge of the Geometry of motion indicates distinctly that the only complete way of treating problems of motion will be to consider a body as animated by a directional translation and a rotation. The whole linear momentum, in any direction, of the wheel mentioned above is zero; the angular momentum is ma3w. In this view the force which stops the motion is a couple, and there is also a single force,-a pressure on the axle.

7. Our present results applied to those of Lesson III. enable us to assert that the impressed forces are in equi

du
dt

dv

librum with the forces M and M acting at the centre

dt

do

of inertia, reversed, and the couple Em r2 reversed. If

at

the forces are impulsive they are in equilibrium with M (u' — ù), M (v' — î) and (w' — w) Edm r2 all reversed.

8. The reduction of the expressions for the effective forces is now theoretically complete. They have been shewn to be equivalent, when the motion is continuous, to a resultant force M

du

dt,

dv
dt

M acting at the centre of inertia and to a

resultant couple. But practically the solutions of problems may be much simplified by a proper choice of co-ordinates. Thus if we use the above expressions or those in x, y when the motion is referable to a fixed point, it is obvious that we shall have for each problem to work through those differentid'x dy

ations which shew that and parallel to the axes are

2

(de)* along and

dt

angles to the radius vector.

It might be convenient to employ the expressions for the accelerations along the tangent and normal to the path of the centre of inertia. Or, yet again, if the motion of G is best defined by reference to a point A which is itself moving, we can use the proposition that the acceleration (or velocity) of G in any direction is equal to that of G relatively to A (supposing A fixed) in that direction, together with the acceleration (or velocity) resolved in the same direction.

In general in every analytical solution of a motion of a rigid system equations are required connecting the velocities of different parts. These are called the geometrical equations, and may often be simplified or reduced in number by a proper choice of variables.

9. A system consisting of two masses A and B rigidly connected by a straight massless wire is moving without rotation with velocity V. A point O of the wire between A and the centre of inertia suddenly becomes fixed, and the system proceeds to rotate about O with angular velocity w. It is required to find the resultant impulsive forces which must have caused this change of motion.

The momentum of translation has suffered a change,

B. OB.-A. OA.w-(A+B) V.

This therefore is the measure of the force which acting at G at right angles to AB would cause the change.

The angular momentum about G has been changed from

zero to

(A. AG2+B.BG3) w.

This expression is therefore the measure of the couple which must have caused the change.

10. Given that a circular hoop of radius a is rolling with uniform velocity v along a road. What are the resultant effective forces (1) on the whole, (2) on a given part?

(1) As the centre of inertia is moving uniformly in a straight line, the resultant force is zero.

To find the angular velocity (w), consider the motion of the point in contact with the ground.

This is carried forwards with velocity v by the motion of translation. It is carried backwards by the rotation round the centre with velocity aw, and its resultant velocity is zero; for it is the instantaneous centre. Hence vaw, or the rotation is also uniform. Hence the couple effective in producing rotation is zero also.

(2) Let m be the mass and G the centre of inertia of the part AB, the effective forces on which are to be found.

A

The acceleration of G in any direction is equal to that of C in that direction, together with that of G relatively to C measured in the same direction. Now C moves uniformly and G moves uniformly about C. Hence the only acceleration of G is towards Cand is w2. CG. The resultant effec