Question

Derive expressions for average energy of a body executing SHM.

Answer 1

Answer on Question #61576-Physics-Mechanics

Derive expressions for average energy of a body executing SHM.

Solution

$$U = \frac{1}{2} k x^{2} = \frac{1}{2} k A^{2} \cos^{2}(\omega t + \phi)$$ $$K = \frac{1}{2} m v^{2} = \frac{1}{2} m \omega^{2} A^{2} \sin^{2}(\omega t + \phi) = \frac{1}{2} k A^{2} \sin^{2}(\omega t + \phi)$$

Since $k = m\omega^2$.

$$E = K + U = \frac{1}{2} k A^{2} \cos^{2}(\omega t + \phi) + \frac{1}{2} k A^{2} \sin^{2}(\omega t + \phi) = \frac{1}{2} k A^{2}$$

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