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  4. A particle in simple harmonic motion starts from it's extreme position of oscillation at time t=0 and has a time period T=6 s. Then the earliest time when it's potential energy equals three times it's kinetic energy will be?

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Q. A particle in simple harmonic motion starts from it's extreme position of oscillation at time $t=0$ and has a time period $T=6\, s$. Then the earliest time when it's potential energy equals three times it's kinetic energy will be?

Oscillations

Solution:

Since the SHM starts from extreme position, $x=A \cos (\omega t)$,
Hence Potential Energy and Kinetic Energies are such that
$U \propto \cos ^{2}(\omega t)$ and $K E \propto \sin ^{2}(\omega t)$,
Therefore for $U=3 K E$,
$\omega t=\pi / 6 $
$\Rightarrow t=T / 12$