whence, integrating with respect to t,
1⁄2M (ẋ2 + ẏ2) + 1⁄2Iθ̇2 = ∫ (X dx + Y dy + N dθ) + const.
(13)
The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.
| Fig. 75. |
The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose mass-centre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knife-edge is replaced by a cylindrical pin. If α be the radius of the cylinder, h the distance of G from its axis (O), κ the radius of gyration about a longitudinal axis through G, and θ the inclination of OG to the vertical, the kinetic energy is 1⁄2Mκ2θ̇2 + 1⁄2M·CG2·thetȧ2, by § 3, since the body is turning about the line of contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is therefore
1⁄2M (κ2 + α2 + h2 − 2 ah cos θ) θ̇2 − Mgh cos θ − const.
(14)
Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to q̇, and the total kinetic energy may be expressed in the form 1⁄2Aq̇2, where A is in general a function of q [cf. equation (14)]. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-ordinate q.
| Fig. 76. |