Question

A simple harmonic oscillator has amplitude $A$, angular velocity $ \Omega $ , and mass $m$. Then, average energy in one time period will be :

• A. $ \frac{1}{4}m\omega ^{2}A^{2}$
• B. $ \frac{1}{4}m^{2}\omega ^{2}A^{2}$
• C. $\omega^{2}A^{2}$
• D. zero

Answer 1

Answer: (A)

Average energy $=\frac{\int\limits_{0}^{T} U d t}{\int\limits_{0}^{T} d t}=\frac{1}{T} \int\limits_{0}^{T} U d t$
$=\frac{1}{2 T} \int\limits_{0}^{T} m \omega^{2} A^{2} \cos ^{2}(\omega t+\phi) d t$
$=\frac{1}{4} m \omega^{2} A^{2}$