Question

Assertion: Average kinetic energy in one oscillation during $\text{SHM}$ of a body is $\frac{1}{4}m{\omega }^2{A}^2$.
Reason: Maximum kinetic energy is $\frac{1}{2}m{\omega }^2{A}^2$.

• A. If both assertion and reason are correct and reason is the correct explanation of assertion.
• B. If both assertion and reason are correct but reason is not the correct explanation of assertion.
• C. Assertion is true but reason is false.
• D. Assertion is false but reason is true.

Answer 1

The correct option is B If both assertion and reason are correct but reason is not the correct explanation of assertion.

If $v=A\omega \cos \omega t$
Then, $v_{max}=A\omega$
$\therefore K{E}_{max}=\frac{1}{2}m{v}_{max}^2=\frac{1}{2}m{\omega }^2{A}^2$
Further, Average kinetic energy$<KE>=<\frac{1}{2}m{\omega }^2{A}^2{\cos }^2\omega t>$
$=\frac{1}{2}m{\omega }^2{A}^2<{\cos }^2\omega t>$
Average value of ${\cos }^2\omega t$ for one oscillation $<{\cos }^2\omega t>=\frac{1}{2}$
$\therefore <KE>=\frac{1}{4}m{\omega }^2{A}^2$
Here both assertion and reason are true but reason cannot explain the assertion correctly.
Hence, option $\left(b\right)$ is correct.