Page:Elementary Text-book of Physics (Anthony, 1897).djvu/64

50 the plane, is the same. If $$\phi$$ be the angle between the direction of the force $$W$$ and the normal to the surface of the plane, and $$s$$ the distance traversed by the weight, this condition is fulfilled when $$Ps - Ws \sin{\phi} = 0.$$ Hence the condition of equilibrium is $$P = W \sin{\phi}.$$

The principle of work is of special value in all such cases as those illustrated by the combination of pulleys, in which, however complicated the arrangement of the parts of the system which transfer the action of the applied force or power to the point of application of the other force, we know that the forces equivalent to the reactions between those parts occur in pairs, of which the members are equal and opposite, so that the work done during any displacement of the system by each of these pairs is zero; for in any such case equilibrium obtains when the work done by the one force equals the work done by the other.

47. Motion of a Rigid Body in Three Dimensions.— The motion of a rigid body which is not under the restriction hitherto imposed upon it, but which is free to move in all directions, is in many respects analogous to the motion already studied, though the details are necessarily more complicated. We will attempt no demonstration of the laws of the motion of a rigid body in the general case, but will limit ourselves to a short description of them.

Any displacement of a rigid body may always be replaced by a translation and a rotation about some axis; this may readily be seen by considering any simple example. By the use of an example, it will also appear that the direction of the axis does not in general coincide with the direction of the translation; it is, however, always possible to find a direction such that translation in that direction and rotation about an axis in that direction will produce the displacement required. An infinitesimal displacement may be produced, therefore, by an infinitesimal translation and a rotation about an axis in the direction of the translation, that is, by a motion resembling that of a screw when driven forward. The axis of rotation in this case, which will in general change its direction and position in space as the body traverses its path, is called the instantaneous