above, given amplitude or energy, phase of motion is determined by the initial position or initial velocity. . A combination of two simple harmonic motions with arbitrary amplitudes and phases is not necessarily periodic. It is periodic only if frequency of one motion is an integral multiple of the other’s frequency.
However, a periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate amplitudes. . The period of SHM does not depend on amplitude or energy or the phase constant. Contrast this with the periods of planetary orbits under gravitation (Kepler’s third law).
. The motion of a simple pendulum is simple harmonic for small angular displacement. . For motion of a particle to be simple harmonic, its displacement x must be expressible in either of the following forms : x = A cos ω t + B sin ω t x = A cos ( ω t + α ), x = B sin ( ω t + β ) The three forms are completely equivalent (any one can be expressed in terms of any other two forms).
Thus, damped simple harmonic motion is not strictly simple harmonic. It is approximately so only for time intervals much less than 2m/b where b is the damping constant. Exercises . Which of the following examples represent periodic motion?
A swimmer completing one (return) trip from one bank of a river to the other and back. A freely suspended bar magnet displaced from its N-S direction and released. A hydrogen molecule rotating about its centre of mass. (d) An arrow released from a bow.
. Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? the rotation of earth about its axis. motion of an oscillating mercury column in a U-tube.
motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point. (d) general vibrations of a polyatomic molecule about its equilibrium position. . Fig.
. depicts four x-t plots for linear motion of