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 the 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 $=\frac{1}{2}mω^{2}A^{2}$
$\omega =\sqrt{\frac{g}{L}}$
$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
$K_{2}=\frac{1}{2}mg\left(2 L\right)\left(\theta \right)^{2}$ [Here we are assuming angular amplitude is same]
$\frac{K_{1}}{K_{2}}=\frac{\frac{1}{2} \left(mgl\theta \right)^{2}}{\frac{1}{2} mg \left(2 L\right) \left(\theta \right)^{2}}=\frac{1}{2}$
$K_{2}=2K_{1} \, $