An object of mass $0.5\, {kg}$ is executing simple harmonic motion. Its amplitude is $5\, {cm}$ and time period (T) is $0.2\, {s} .$ What will be the potential energy of the object at an instant $t=\frac{T}{4}$ s starting from mean position. Assume that the initial phase of the oscillation is zero. (In ${J}$)
JEE MAIN 2021, Medium
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by using formula of time period
$T=2 \pi \sqrt{\frac{m}{k}}$
$0.2=2 \pi \sqrt{\frac{0.5}{k}}$
$k=50 \pi^{2}$
$\approx 500$
$x=A\, \sin (\omega t+\phi)$
$=5\, c m\, \sin \left(\frac{\omega t}{4}+0\right)$
$=5\, {cm} \sin \left(\frac{\pi}{2}\right)$
$=5 \,{cm}$
$P E=\frac{1}{2} k x^{2}$
$=\frac{1}{2}(500)\left(\frac{5}{100}\right)^{2}$
$=0.6255$
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