Page:Elementary Text-book of Physics (Anthony, 1897).djvu/65

§ 47] axis, and any infinitesimal motion of a body is a screw motion around the instantaneous axis.

The rotation of a rigid body free to move and acted on by no forces will be about an axis passing through the centre of mass. The kinetic energy of the rotating body depends on its moment of inertia about its axis of rotation, and this may be shown to depend upon the moments of inertia of the body about three axes in it at right angles to each other; these axes, called the principal axes of inertia, are such that the moment of inertia about one of them is the greatest, and that about another the least, that the body can have. If a body be set rotating around either of these two axes, it will continue to rotate about that axis forever, and its condition is stable; that is, if an infinitesimal change be made in the direction of its axis of rotation, this deviation will never become large. If it be set rotating about the third or mean axis, it will continue to rotate about that axis forever, but its condition is unstable; that is, if an infinitesimal change be made in the direction of the axis of rotation, this deviation will tend continually to increase, and will become finite. If the body be set rotating about any axis which is not coincident with one of the principal axes, the direction of the axis of rotation in the body changes continually. In the case of real bodies set in rotation and acted on by friction and other such forces, the tendency is for the body to rotate with increasing exactness around the axis of greatest moment of inertia.

In the study of the angular velocity of a rotating body we represent the axis of rotation by a line and the amount of the angular velocity by a length measured on that line. If we conceive of two angular velocities about intersecting axes, it may be shown that they are equivalent to a single angular velocity about another axis passing through the point of intersection; the amount of this angular velocity and the direction of the axis are determined by the parallelogram law. Manifestly this law may be applied to the composition and resolution of any number of angular velocities about axes which intersect at one point.

The resolution of angular velocities into their components is