Question

If the mass of a pendulum is doubled, the time period

• A. becomes double
• B. becomes half
• C. becomes 4 times
• D. remains the same

Answer 1

The correct option is D remains the same

In pendulum, $mglsin\left(\theta \right)$ is restoring torque.

$\Rightarrow Torque=-mglsin\left(\theta \right)$

$\Rightarrow I\alpha =-mglsin\left(\theta \right)$, Where $I=m{l}^2$ and $\alpha$ is angular acceleration.

$\because \theta$ is very small so $\sin \left(\theta \right)\approx \theta$,

$\Rightarrow I\alpha =-mgl\theta$ $\Rightarrow m{l}^2\alpha =-mgl\theta$ $\Rightarrow \alpha =-\frac{g}{l}\theta$

$\Rightarrow \alpha =-{\omega }^2\theta$,

$\Rightarrow \omega =\sqrt{\frac{g}{l}}$

$\therefore \text{Time period T}=\frac{2\pi }{\omega }$ $\Rightarrow T=2\pi \sqrt{\frac{l}{g}}$

$\therefore$ Time period is independent of mass.