At what time is the potential of a simple harmonic and kinetic en…

Question

At what time is the potential of a simple harmonic and kinetic energy oscillator equal?

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Answer

Potential energy of simple harmonic oscillator :
$U_{t}=\frac{1}{2} k A^2 \sin ^2 \omega t$
Kinetic energy $K _{ t }=\frac{1}{2} k A^2 \cos ^2 \omega t$
$\begin{aligned}\frac{1}{2} k A^2 \sin ^2 \omega t & =\frac{1}{2} k A^2 \cos ^2 \omega t \\\sin ^2 \omega t & =\cos ^2 \omega t \\\sin ^2 \omega t & =1-\sin ^2 \omega t \\2 \sin ^2 \omega t & =1\end{aligned}$
$\begin{aligned}\sin ^2 \omega t & =\frac{1}{2} \\\sin \omega t & = \pm \frac{1}{\sqrt{2}}=\sin \frac{\pi}{4} \text { or } \sin \frac{3 \pi}{4} \\\omega t & =\frac{\pi}{4} \text { or } \frac{3 \pi}{4} \\t & =\frac{\pi}{4 \omega}=\frac{\pi}{4 \times \frac{2 \pi}{T}} \\t & =\frac{T}{8} \\t & =\frac{3 \pi}{4 \omega}=\frac{3 \pi}{4 \times \frac{2 \pi}{T}}=\frac{3 T}{8}\end{aligned}$

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