Consider a particle free to move on x-axis, is being acted upon by a force given by,
F = – kxn
Here, k is a positive constant.
Now, following cases are possible depending on the value of n.
(i) If n is an even integer (0, 2, 4, ..... etc), force is always along negative x-axis, whether x is positive or negative. Hence, the motion of the particle is not oscillatory. If the particle is released fro many position on the x-axis (except at x= 0) a force in negative direction of x-axis acts on it and it moves rectilinearly along negative x-axis.
(ii) If n is an odd integer (1, 3, 5, ..... etc), force is along negative x-axis for x > 0, along positive x-axis for x < 0 and zero for x= 0. Thus, the particle will oscillate about stable equilibrium position, x= 0. The force in this case is called the restoring force. Of these, if n = 1, i.e., F = – kx the motion is said to be SHM.
