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 $v=A\omega \sin \omega t$
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$<KE>=<\frac{1}{2}m{\omega }^2{A}^2{\sin }^2\omega t>$
$=\frac{1}{2}m{\omega }^2{A}^2<{\sin }^2\omega t>$
Average value of ${\sin }^2\omega t$ for one oscillation $<{\sin }^2\omega t>=\frac{1}{2}$
$\therefore <KE>=\frac{1}{4}m{\omega }^2{A}^2$