Show that for a particle in linear S.H.M., the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Explanation

To show that for a particle in linear simple harmonic motion (S.H.M.), the average kinetic energy over a period of oscillation equals the average potential energy over the same period, we can follow these steps:

### Step 1: Define the position and velocity of the particle in S.H.M.
The position \( x \) of a particle in simple harmonic motion can be expressed as:
\[ x(t) = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.
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