In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the t

Question

In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic?\

Professor PC Thomas Classes · Answer

(b) : Displacement $(x)=\frac{a}{2}$ Total energy $=\frac{1}{2} m \omega^{2} a^{2}$ and\ kinetic energy when displacement is $(x)$\ $=\frac{1}{2} m \omega^{2}\left(a^{2}-x^{2}ight)=\frac{1}{2} m \omega^{2}\left(a^{2}-\frac{a^{2}}{4}ight)=\frac{3}{4}\left(\frac{1}{2} m \omega^{2} a^{2}ight)$\ Therefore fraction of the total energy at $x$, $$ =\frac{\frac{3}{4}\left(\frac{1}{2} m \omega^{2} a^{2}ight)}{\frac{1}{2} m \omega^{2} a^{2}}=\frac{3}{4} . $$