Question

A particle of mass $m$ performs SHM along a straight line with frequency $f$ and $U=0$ at $x=0$ amplitude $A$.

• A. The average kinetic energy of the particle is zero
• B. The average potential energy is $m{\pi }^2{f}^2{A}^2$
• C. The frequency of oscillation of kinetic energy is $2f$
• D. Velocity function leads acceleration by $\frac{\pi }{2}$

Answer 1

Answer: (B), (C)

The average potential energy is $m{\pi }^2{f}^2{A}^2$
The frequency of oscillation of kinetic energy is $2f$

if $x=Asin\omega t\Rightarrow v=A\omega cos\omega t\Rightarrow v^2=A^2\times 4{\pi }^2{f}^{2}co{s}^2\omega t=4{\pi }^2{f}^2{A}^2(1+cos2\omega t)/2=2{\pi }^2{f}^2{A}^2(1+cos2\omega t)$

The average potential energy is equal to average kinetic energy is equal to average of $m{v}^2/2=\frac{1}{2}m\times 2{\pi }^2{f}^2{A}^2(1+cos2\omega t)-\left(1\right)$

$\Rightarrow m{\pi }^2{f}^2{A}^2$

From $\left(1\right)$, angular frequency of oscillation of kinetic energy is $2\omega$ i.e., frequency of oscillation of kinetic energy is $2f$