m = ω A and acceleration amplitude a m = ω A , respectively. . The force acting in a simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion. .
A particle executing simple harmonic motion has, at any time, kinetic energy K = ½ mv and potential energy U = ½ kx . If no friction is present the mechanical energy of the system, E = K + U always remains constant even though K and U change with time. . A particle of mass m oscillating under the influence of Hooke’s law restoring force given by F = – k x exhibits simple harmonic motion with k (angular frequency) k (period) Such a system is also called a linear oscillator.
. The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by, L g POINTS TO PONDER . The period T is the least time after which motion repeats itself.
Thus, motion repeats itself after nT where n is an integer. . Every periodic motion is not simple harmonic motion. Only that periodic motion governed by the force law F = – k x is simple harmonic.
. Circular motion can arise due to an inverse-square law force (as in planetary motion) as well as due to simple harmonic force in two dimensions equal to: – m ω r . In the latter case, the phases of motion, in two perpendicular directions ( x and y ) must differ by π / . Thus, for example, a particle subject to a force – m ω r with initial position ( , A ) and velocity ( ω A , ) will move uniformly in a circle of radius A .
. For linear simple harmonic motion with a given ω, two initial conditions are necessary and sufficient to determine the motion completely. The initial conditions may be (i) initial position and initial velocity or (ii) amplitude and phase or (iii) energy and phase. .