For a body executing S.H.M. :(a) Potential en | vedclass

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(A) In $S.H.M.$,the total mechanical energy $(E = K.E. + P.E.)$ remains constant at any point in time,which validates statement $(c)$.The average kine

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$(c)$ and $(d)$

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(A) In $S.H.M.$,the total mechanical energy $(E = K.E. + P.E.)$ remains constant at any point in time,which validates statement $(c)$.The average kinetic energy over one complete time period $T$ is given by $\langle K.E. \rangle = \frac{1}{T} \int_{0}^{T} \frac{1}{2} k A^2 \cos^2(\omega t + \phi) dt = \frac{1}{4} k A^2$.The average potential energy over one complete time period $T$ is given by $\langle P.E. \rangle = \frac{1}{T} \int_{0}^{T} \frac{1}{2} k A^2 \sin^2(\omega t + \phi) dt = \frac{1}{4} k A^2$.Since $\langle K.E. \rangle = \langle P.E. \rangle$ over one time period,statement $(d)$ is correct.Statement $(b)$ is incorrect because the average energies are only equal over a full time period or half-integer multiples thereof,not over 'any' given time interval.Therefore,statements $(c)$ and $(d)$ are correct.

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