Starting from the origin a body oscillates simple harmonically with a period of $2\ s$. After what time will its kinetic energy be $75\%$ of the total energy?
AIEEE 2006, Diffcult
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$K.E.$ of a body undergoing $SHM$ is given by,
$K . E .=\frac{1}{2} m a^{2} \omega^{2} \cos ^{2} \omega t, \quad T . E .=\frac{1}{2} m a^{2} \omega^{2}$
Given $\mathrm{K} . \mathrm{E}=0.75 \mathrm{T.E.}$
$\Rightarrow 0.75=\cos ^{2} \omega t \Rightarrow \omega t=\frac{\pi}{6}$
$\Rightarrow t=\frac{\pi}{6 \times \omega} \Rightarrow t=\frac{\pi \times 2}{6 \times 2 \pi} \Rightarrow t=\frac{1}{6} s$
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