If a moment is acting on a pivoted laxly such as a door, its immediate effect is to produce angular acceleration; just as the effect of a force in translation is to produce linear acceleration. It is important to determine the connection be tween the moment of the force and the resulting angular acceleration. The simplest case is that of a particle of matter joined to an axis by a massless rigid rod, and a force acting in the particle at right angles to the rod. if the rod has a length r, and the particle has a mass in, the moment of the force F around the axis is Fr. and the linear acceleration of the particle in the direction of the force is Therefore, the angular acceleration (a) is --; and if the mo ntr ment of the force is called L, The voetlicient of a, air', is called the 'moment of inertia' of the particle around the axis. If, now, the rotating body is of any shape or size, it may be shown that the angular acceleration (a) resulting from a moment ( L) is given by the formula a niir' is the sum of the products of the mass of each particle of the body by the square of its distance from the axis. Mott' is called the moment of inertia of the whole body around the axis and is commonly written I. Hence a formula for rotation of a rigid hotly around an axis whose position is fixed, which corre sponds perfectly with the fernitilai I = nun for translation. In the same way, therefore. that in measures the inertia of a body so far as trans lation is concerned, I measures its inertia for rot at ion.
A simple illustration is that of a body pivoted about a horizontal axis so that it can make oscil lations under the action of gravity. like a eeni mon clock's pendulum. Take a plane section of the body at right angles te the axis of rotation (at 0) and passing through the centre of in ertia (C), to describe the rotation choose the line fixed in the body as the one joining the centre of inertia of the body and the point where the axis meets the plane (0C), and as the line fixed in space the one where OC' comes when the body is hanging at rest (0.1). As the body Vibrates, it will occupy in tnrn different posi tions which are completely described by the angle (0) between OC and OA. The problem is to find the angular acceleration. There are two forces acting on the body: one is the supporting force of the pivot, and its mo ment about the axis is zero because it passes through 0; the other is the weight of the body, which is my. where m is the mass of Wily and g is the linear acceleration of a hotly falling freely, and its line of action is vertically down through the centre of inertia— both of which facts will be ex plained later. Calling the length of the line OC, 1, the moment of the force my tlur axis through 0 is mglsing ; therefore the angular acceleration is and it is in such a direction around the axis as to produce angular motion tending to bring 00 to coincide with ()A. If the amplitude of the vibration is small the sine may be replaced by 0; and the angular acceleration is Consequently the motion is simple harmonic; and the period of one 00111 plete vibration is 27A Such an oscillating S body is called a 'compound pendulum,' and it has ninny interesting properties. (See CENTRE OF GYRATION : CENTRE OF OSCILLATION,) A simple pendulum is ‘? special case of a eompottnil one in it 1 = ml' and so the period becomes, as before, —.
9 Since L = la, if the angular velocity around the axis is called w, this equation may be written — L_ t where fit —64 is the change in the angular veloc ity in t seconds. The product Lt is called the `impulse' of the moment of the force, or the moment of the impulse of the force. As a result. of au impulsive moment, the produet 1w—called the 'angular momenturn'—is The time taken for a moment L to change the angular velocity from to w is evidently The angle through which the body turns while this change is going on is given by the formula of kinematics 6 ; and as a = L/I, the angle — o 2 = Or The product L is called 'work*: and the work is said to he done fn the moment if w is in creasing. and against the moment if w is de creasing. is called the 'kinetic energy of rotation of the body whose moment of inertia about. a given axis is I and whose angular speed is cd.
IN ( iENERAL. If the rigid body is not pivoted around a fixed axis. hut is free to move in any direction 'or mantwr, it will receive, in general, both linear and angular acceleration under the influence of a force. e.g. if a body is thrown in the air. ( [mkt' the action of gravity alone there is. however. only linear acceleration.
for reasons to be given i aliately.) It has been shown that the linear ameleration of the centre of inertia of a body acted on by any forces is the same as that whieh a partielc having a mass equal to that of the holly would have under the action of the same forces. force in general does not have a line of action passing through the centre I if inertia: imagine a plane section of the body through the line of action and the centre of inertia; the force will then in general have a moment about an axis through the (-entre of inertia perpendicular to this plane. Sillvt.
translation of the centre of inertia of the body under the action of the force is quite independent of the rotation, the rotation will he as if the above axis is fixed, i.e. if ta is the total mass of the body, I its moment of inertia about this particular axis, F the force, and L its moment about the axis, the linear acceleration of the centre of inertia will be — and the an ti, L gular 'o, if the force has its line of action through the centre of inertia, there will be no angular acceleration, e.g. the action of gravity.
If an impulsive force. whose impulse is K and whose lever arm with reference to an axis through the centre of inertia is k, arts 111)011 the body. the of the centre of inertia in the dinagion of the force will change accord.
ing to the formula r =1:7»i. and the lar veloeity about the axis through the centre of inertia will be given by the formula = If the body is originally at rest, its centre of inertia will move instantly in the direction of the force with a velocity I:/m, and it will in stuffily rotate With an angular velocity it the line of the fence is through the eent re ertia fr = It. and there is no angular motion. This fact furnishes on experimental method for the determination of the centre of inertia (q.v.).
If the linear velocity of the centre of inertia :it any instant is r, and if the angular velocity is entire kinetic energy is where n is the total mass and I is the mo ment of inertia of the body about the axis of rotation thron_di the centre of inertia.