In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic?
AIPMT 1995, Medium
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Displacement $(x)=\frac{a}{2}$.
Total energy $=\frac{1}{2} m \omega^2 a^2$ and
kinetic energy when displacement is $(x)$
$K=\frac{1}{2} m \omega^2\left(a^2-x^2\right)$
$=\frac{1}{2} m \omega^2\left(a^2-\left(\frac{a}{2}\right)^2\right)=\frac{3}{4}\left(\frac{1}{2} m \omega^2 a^2\right)$
Therefore fraction of the total energy at $x$ is
$=\frac{\frac{3}{4}\left(\frac{1}{2} m \omega^2 a^2\right)}{\frac{1}{2} m \omega^2 a^2}=\frac{3}{4}$
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