TRAL FORCLS.
fl has been shown that, if there are no external influences, the centre of inertia of a system of particles or of a large body continues. if in mo tion, to move in a straight line with a constant speed. This is owing to the fact that the action and reaction of each pair of particles arc equal and opposite. If, however, there are external forces, the acceleration of the centre of inertia in any direction is the sum of the components of these forces in this direction divided by the mass of the whole system. This is equivalent to saying that the motion of the centre of inertia of a sys tem of particles is exactly as if a single particle of the mass of the system were under the in fluence of the given forces. Thus if an iron beam falls from a building (without touching any thing as it falls) the motion of its centre of inertia is like that of a falling particle—verticat —however the beam revolves. If a hammer is thrown at random into the air, its centre of inertia will describe a parabola, because that is the path of a projected particle. See PROJECTILE.
Many forces are not constant and some are abrupt, like the blow of a hammer; and in these cases it is impossible to measure them. Their effect is evidently to produce a sudden change in velocity; and it is measured by the total change in the linear momentum. Force itself is the rate of change of linear momentum; so if a force F acting on a particle produces a change of momentum front to my in an interval of time t, F= and thus the total change of momentum equals the product of the force and the interval of time. This product Ft is called the 'impulse' of the force, and may be measured even if both F and are unknown. Similarly, if an impulsive force acts on a large body. the velocity of its centre of inertia will be changed front to r in the direction of the force. In other words, the change of velocity of the centre of inertia, equals the amount of the impulse divided by the mass of the body, entirely regardless of the point of application of the force. The time required for a force F to change the velocity front to v is — The distance required for this same force F to produce this change in velocity from to t' in its direction is found by the of kine matics, which show that under a constant ac celeration a, the distance traversed while the speed changes from to s is such that 2a.r = se. Therefore, in this case, since a = In, x= The product Fs, that is, the force multiplied by a distance in its line of action, is called the 'work'; the quantity is called the energy' of translation of the body whose mass is at when it has the speed s. This formula is
expressed in words by saying that the 'work (lone by the joie,' I III t he body equals the M c/rose in its kinetic energy of translation, pro vided the speed is increasing, e.g. a train being set in motion. If the speed is decreasing, e.g. a train slowing up by virtue of its brakes and the resulting friction, it is said that the body limes an amount of kinetic energy of translation equal to the work it does in overcoming friction or 'against the fore(' F.
IZoTATioN. 'rigid body' is defined as one which is not deformed in any way under the forces acting on it. If such a lady- is plated on an axis whose position is fixed, e.g. a door, a grindstone, etc., it is self-evident that the an gular motion produced 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 application. Thus if the force is at right 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 effect so far as rotation is concerned. If a plane section he imagined in the body, at right angles to the axis, it is evident that a force perpendicular to this plane, i.e. parallel to the axis, has no effect in the angular motion; while a force lying in this plane has an effect which depends upon both the force and the perpendicular distance from the point where the axis cuts the plane to the line of action of the force. This perpendicular 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 resulting 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 theoretical 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 hotly 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.