Question

The lengths of two simple pendulums at a place are in ratio 9:1. Find the ratio of their time periods

Answer 1

Given :

The lengths of two simple pendulums at a place are in ratio 9:1

To Find :

The ratio of their time periods

Theory :

The Time period of a simple pendulum is

$\sf\blue{T=2\pi\sqrt{\dfrac{l}{g}}}$

where L is the length of the string

and g is the acceleration due to gravity.

Solution :

Let the time period of one pendulum be $\sf\:T_1$ and its length be $\sf\:l_1$ and other pendulum time period be $\sf\:T_2$ and it's length be $\sf\:l_2$

Time period of one pendulum

$\sf\:T_1=2\pi\sqrt{\dfrac{l_1}{g}}...(1)$

Time period of other pendulum

$\sf\:T_2=2\pi\sqrt{\dfrac{l_2}{g}}...(2)$

Now , Divide equation (1) & (2)

$\sf\dfrac{T_1}{T_2}=\dfrac{2\pi\sqrt{\frac{l_1}{g}}}{2\pi\sqrt{\frac{l_2}{g}}}$

$\sf\implies\dfrac{T_1}{T_2}=\sqrt{\dfrac{l_1}{l_2}}$

$\sf\implies\dfrac{T_1}{T_2}=\sqrt{\dfrac{9}{1}}$

$\sf\implies\dfrac{T_1}{T_2}=\dfrac{3}{1}$

Therefore, The ratio of their time periods is 3:1