MECHANICS. 249 MECHANICS. forces acting on it. If such a liody is pivotc'tl on an axis wliose position is fixed, e.g. a dcjor, a giindstone. etc., it is self-evident that the an- gular motion jiroduced in it by a force such as a push or pull depends not alone on the amount of the force and its direction, but also on its point of applieation. Thus if the force is at ri!,'ht angles to the door and near the hinges, there is only a slight effect ; if it is applied near the edge of' the door, it is much greater; and if the line of action of the force passes through the axis of rotation, there is no eil'ect so far as rotation is concerned. If a plane section be imagined in the body, at right angles to the axis, it is evident that a force jjerpendicular to this plane, i.e. parallel to the axis, has no effect on the angular motion; while a force lying in this jdane has an efl'ect which depends upon both the force and the perpendicular distance from the point where the axis cuts the ]ilaiie to the line of action of the force. This i)er|)cn<licular dis- tance is called the 'lever-arm' of the force with reference to the .axis ; and the product of the numerical value of the force and its lever-arm is called the 'moment of the force' around the axis. A 'moment' such that the resilting effect of the force is to produce rotation in one direction is called positive ; while if its effect is to produce the opposite rotation, it is called negative. A moment is then a rotor. It can be shown by thcnrefical considerations that the 'moment of a force' about an axis is the proper numerical value to give the rotational effect of the force ; and this is in accordance with experience, for, if a body pivoted on an axis is kept from turn- reg under the opposing actions of two forces dif- ferently placed, it is found that the moments of the two forces about the axis are equal and opposite. If a moment is acting on a pivoted body 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 on the particle at right angles to the rod. If the rod has a length r, and the particle has a mass m, the mnment of the force F around the axis is Fr, and the linear acceleration of the particle in the F direction of the force is — . Therefore, the m F aiiqiihir acceleration (a) is — ; and if the mo- mr ment of the force is called L, L = Fr = HI r a. The coefficient of o, nir', is called the 'moment of inertia' of the particle aroiuid the axis. If, now, the rotating Ijody is of any shape or size, it may be .shown that the angular acceleration (a) resulting from a moment (T^) is given by the fornnila a = — ^, where Son-' is the sum Svnr- ef the products of the mass of each particle of the body by the square of its dislancte from the axis. t,mr' is called the moment of inertia of the whole body around the .Txis and is commonly written I. Hence L = lo, a formula for rotation of a rigid body aroind an a.xis whose position is fixed, which corre- sponds perfectly with the fornuihi F = tiiti for translation. In the same way, therefore, that m measures the inertia of a body so far as trans- lation is concerned, I measures its inertia for rotation. 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 com- mon clock's pendulum. Take a i)lane section of the body at right angles to the axis of rotati(ui (at 0) and passing through the centre of in- ertia (C), to describe the rotation clioose 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 (OC), and as the line fixed in space the one where OC comes when the bodj' is hanging at rest ( OA ) . As the body vibrates, it will occupy in turn different posi- tions which are completely described by the angle (0) between OC and OA. The prolilem 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 ; the other is the weight of the body, which is miy, where m is the nuiss of "the liody and g is the linear acceleration of a body falling freely, and its line of action is vertically down through the centre of inertia — both of which facts will be ex- Calling the length of the line OC, I, the moment of the force mr/ about thfTaxis through O is mylsind ; therefore the angular mglaine plained later. acceleration is I id it such a direction around the axis as to produce angular motion tending to bring OC to coincide with OA. If the amplitude of the vibration is .small the sinS may be replaced by 0; and the angular acceleration is —f-9. Consequently the motion is simple harmonic; and the period of one com- . Such an oscillating mgl body is called a 'compound pendulum,' and it has many interesting properties. (See Centre of Gyration; Centre of Oscillation.) A simple pendulum is -^ special case of a compound one; in it I = ml- and so the period becomes, as before. plete vibration is 27r- -V!- Since L = la, if the angular velocity around the axis is called w, this equation may be written lu — I(J„ L = t where a — u„ is the change in the angular veloc- ity in * seconds. The product Lf is called the 'impulse' of the moment of the force, or the moment of the impulse of the force. As a resjilt of an impulsive moment, the product lu — called the 'angular momentum' — is changed. The time taken for a moment L to change the angular velocity from u<, to u is evidently ^_ la — l(j„
