Question

If the length of a simple pendulum at a given place is increased by 6 times, the ratio of the final to the initial time periods of the pendulum is .

• A. $36:1$
• B. $1:\sqrt{6}$
• C. $\sqrt{6}:1$
• D. $1:36$

Answer 1

The correct option is C $\sqrt{6}:1$

Given: the length of the pendulum ($l$) is increased by 6 times at a given place.
If $l_1$ and $l_2$ are the initial and final lengths of the pendulum, then $\frac{l_1}{l_2}=\frac{l}{6l}=\frac{1}{6}$ ----- (1)
We know that the time period of a simple pendulum is directly proportional to the square root of the length of the pendulum at a given place.
$⟹T\propto \sqrt{l}$
If $T_1$ and $T_2$ are the initial and final time periods of the pendulum, then $\frac{T_1}{T_2}=\sqrt{\frac{l_1}{l_2}}=\sqrt{\frac{1}{6}}$
$⟹\frac{T_2}{T_1}=\sqrt{\frac{6}{1}}=\frac{\sqrt{6}}{1}=\sqrt{6}:1$.