Question 6: Derive equations for the K.E and P.E of a body a body of mass m executing S.H.M.

Answer

P.E of mass m executing S.M.H

Consider a body of mass ‘m’ attached with a spring of spring constant ‘k’ as shown in the figure. When the body is at rest at point ‘O’, no force acts on it. F_{i} = 0. Let the spring is compressed to point ‘A’ by a force F_{f} such that OA = x_{0}. Then F_{f} = kx_{0}.

Now the average force is,

Average work done, W = Fd, is given by

This work is stored in the spring as potential energy. Therefore,

At any point, however,

K.E of mass m executing S.H.M

Similarly, the velocity in case of SHM at any displacement is,

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