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 $=\frac{1}{2}m{\omega }^2{A}^2$
$\omega =\sqrt{\frac{g}{L}}$
$A=L\theta$
$KE=\frac{1}{2}m\frac{g}{L}\times L^2{\theta }^2$
$KE=\frac{1}{2}mgL{\theta }^2$
If length is doubled
$K_2=\frac{1}{2}mg\left(2L\right){\left(\theta \right)}^2$ [Here we are assuming angular amplitude is same]
$\frac{K_1}{K_2}=\frac{\frac{1}{2}{\left(mgl\theta \right)}^2}{\frac{1}{2}mg\left(2L\right){\left(\theta \right)}^2}=\frac{1}{2}$
$K_2=2K_1$