MOMENT of a dynamical quantity is the importance of that quantity in regard to its dynamical effect relatively to a given point or axis. The moment of a force about a point is the product of its amount into its perpen dicular distance from the point. The tendency of the action of such a force is to cause rota tion about an axis perpendicular to the plane passing through the point and containing the force. Thus, in the case of a pendulum, the effectiveness of the force in causing rotation is measured by the moment Wl, where W is the weight of the pendulum, and 1 is the distance of the line of action of the force W from the centre of rotation C, or the distance of the cen tre of mass G from the vertical line through C.
The term moment enters into several other phrases, all of which relate either directly or indirectly to rotation. Thus, there is the mo ment of momentum, or angular momentum, whose rate of change is the measure of the moment of the force producing the change. To obtain it for any given body rotating with angu lar speed 6. about an axis, we first imagine the body broken up into a great many small portions of masses mi,m3,ms, etc., at distances etc., from the axis, multiply the momen tum (mr6)) of each mass by its distance, and then take the sum of all these products. The angular speed w being the same in every ex pression, the moment of momentum takes the form u (me, , etc.), which it is usual to write in the symbolic form (JE me. The quantity me, which is the sum of the prod ucts of each mass into the square of its dis tance from the axis, is called the moment of inertia about that axis. It is the factor in the
moment of momentum which depends upon the distribution of matter in the body. It enters into all question's of mechanics in which rota tion is involved, from the spinning of a top or the action of an engine governor to the stability of a ship. By an obvious extension, the word moment is also used in such combinations as moment of a velocity and moment of an accel eration. Such phrases correspond to nothing truly dynamic, unless we regard velocity as meaning the momentum of unit mass and acceleration as the rate of change of that momentum.
If the mass of every small portion of mat ter in a body be multiplied by the square of its perpendicular distance from a straight line, the sum of all such products is called the moment of inertia of the body about the line regarded as an axis. The radius of gyration of the body is the distance from the axis at which all the matter of the body might be concentrated with out altering the moment of inertia. Thus, if 1 is the moment of inertia of the body, M its whole mass and k its radius of gyration, 1= M k'. We see that the moment of inertia of a body about a line is found by adding a great number of products of small masses and squares of distances; if the body can be defined mathematically as to shape, size and density, finding its moment of inertia is a problem of the integral calculus.