Question

A simple pendulum has a period of 9 secs. When the length of the pendulum is squared, the period is 4 sec. Calculate the original length. 

Answer 1

The period of the simple pendulum with original length can be written as follows:

$$T_1=2\pi\sqrt{\dfrac{L}{g}}=9\ s.$$

Then,  the length of the pendulum is squared. The new period of the pendulum can be written as follows:

$$T_2=2\pi\sqrt{\dfrac{L^2}{g}}=2\pi\dfrac{L}{\sqrt{g}}=4\ s.$$

Let's divide $T_1$ by $T_2$:

$$\dfrac{T_1}{T_2}=\dfrac{1}{\sqrt{L}}=\dfrac{9\ s}{4\ s}=2.25,$$ $$L=0.197\ m.$$