Question

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

• A. 1/2
• B. 3/4
• C. zero
• D. 1/4

Answer 1

Answer: (B)

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