Question

A particle is executing linear simple harmonic motion of amplitude A. What fraction of the total energy is kinetic when the displacement is half the amplitude?

• A. $\frac{1}{4}$
• B. $\frac{1}{2\sqrt{2}}$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{3}{4}$

Kinetic energy $\left(KE\right)=\frac{1}{2}m{\omega }^2(A^2-x^2)$
Potential energy $\left(PE\right)=\frac{1}{2}m{\omega }^2{x}^2$
Total energy $\left(E\right)=\frac{1}{2}m{\omega }^2{A}^{2}Whenx=\frac{A}{2}$
KE = $\frac{1}{2}m{\omega }^2(A^2-\frac{A^2}{4})=\frac{3}{8}m{\omega }^2{A}^{2}E=\frac{1}{2}m{\omega }^2{A}^2\therefore \frac{KE}{E}=\frac{3}{4}$