If the mass of the bob in a simple pendulum is increased to twice its original mass and its length i

Question

If the mass of the bob in a simple pendulum is increased to twice its original mass and its length is made one-fifth of its original length, then the new time period of oscillation is $\frac{x}{5}$ times its original time period. Then the value of $x$ is

Professor PC Thomas Classes · Answer

(a) : Time period of a simple pendulum is independent of mass, $T=2 \pi \sqrt{\frac{l}{g}}$. New time period $T^{\prime}=2 \pi \sqrt{\frac{l/5}{g}} = \frac{1}{\sqrt{5}} T$. We are given $T^{\prime}=\frac{x}{5} T$. Therefore, $\frac{1}{\sqrt{5}} T = \frac{x}{5} T$, which implies $x = \frac{5}{\sqrt{5}} = \sqrt{5}$.