36 PRINCIPLES OF ELEMENTARY DYNAMICS
the power curve and the axis, for any time interval, is proportional to the work done during the given interval. An area above the axis represents positive work, that is work done by the electromotive force or by the torque; an area below the axis represents negative work, that is work done against the electromotive force or against the applied torque. When the two harmonic curves are of equal period and in quadrature, that is, when they
differ in phase by 2 radians or 90°, Fig. 24, the total work done
during one cycle is zero.
The quotient obtained by dividing the area under the power curve for one-half cycle, by the distance between the intersections of this curve with the axis of the curve, equals the mean value of the power during the half cycle. When the two curves are in quadrature, Fig. 24, the mean power during a complete cycle is zero.
The phase difference between the current and the electromotive force in an alternating current circuit can be altered by changing either the capacitance or the inductance of the circuit. If the current and the electromotive force in a circuit containing an alternating current generator and motor are in the same phase, then the generator is putting electric energy into the line and the motor is transforming a maximum fraction of it into mechanical work. If the current and electromotive force differ in phase by one-quarter period, then we have a " wattless current " and the work done during one complete cycle is zero. If the current and the electromotive force are in opposite phase, then the motor is operating as a generator, opposing the electromotive force of the generator.
26. Oscillations of a Coupled System. Resonance. - The vibrations executed by a body which has been displaced from its position of equilibrium and then released are called free vibrations. The vibrations executed under the action of an external periodically varying force are called forced vibrations.
If the point of support of a vibrating pendulum is in periodic motion, as when a pendulum is attached to another pendulum, or when a pendulum is on a ship which is rolling with a constant period, the pendulum will acquire a compound periodic motion having two components of different periods. These periods will be different from the periods of the freely vibrating pendulum or of the freely vibrating support, except when the inertia of the pendulum is negligible compared with the inertia of the support.