Question

A pendulum has maximum kinetic energy $K_{1}$ . If its length is doubled keeping amplitude same then maximum kinetic energy becomes $K_{2}$ . Then relation between $K_{1}$ and $K_{2}$ is:

• A. $K_{2}=2K_{1}$
• B. $K_{1}=2K_{2}$
• C. $K_{2}=K_{1}$
• D. $K_{1}=4K_{2} \, $

Answer 1

Answer: (A)

Maximum kinetic energy $=\frac{1}{2}m\omega ^{2}A^{2}$
$\omega =\sqrt{\frac{g}{L}}$
$A=L\theta $
$KE=\frac{1}{2} \, m\frac{g}{L}\times L^{2}\theta ^{2} \, $
$KE=\frac{1}{2}mgL\theta ^{2} \, $
If length is doubled
$K_2=\frac{1}{2} m g(2 L) \theta^2$ [Here we are assuming angular amplitude is same]
$\frac{K_1}{K_2}=\frac{\frac{1}{2} m g l \theta^2}{\frac{1}{2} m g(2 L) \theta^2}=\frac{1}{2}$
$K_{2}=2K_{1} \, $