Question

The length of a simple pendulum is made one-ninth. Its time period becomes____its initial value.

• A. nine times
• B. half
• C. one-third
• D. twice

Answer 1

The correct option is C

one-third

We have time period of a pendulum $T\propto \sqrt{l}$

$\therefore \frac{T_1}{T_2}=\sqrt{\frac{l_1}{l_2}}$ where $T_1$ and $T_2$ are the initial and final time periods of the simple pendulum and $l_1$ and $l_2$ are the initial and final lengths of the pendulum. If the length of the simple pendulum is made one-ninth i.e.,$l_2=\frac{l_1}{9}$

$\Rightarrow \frac{T_1}{T_2}=\sqrt{\frac{l_1}{\frac{l_1}{9}}}$ $\Rightarrow \frac{T_1}{T_2}=\sqrt{9}$ $\Rightarrow \frac{T_1}{T_2}=3$ $\Rightarrow T_2=\frac{1}{3}{T}_1$

i.e., the time period of the pendulum becomes one third its initial value.