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