Question

What happens to the time period of a simple pendulum when its length is increased to four times its value?

Answer 1

1. The time period of a simple pendulum is defined as the time taken by the bob of the pendulum to complete one oscillation.
2. It is represented by T.
3. The mathematical formula for the time period of a simple pendulum is:
4. $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}},\mathrm{where}\mathrm{l}\mathrm{is}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{pendulum}\mathrm{and}\mathrm{g}\mathrm{is}\mathrm{the}\mathrm{gravity}.$
5. So, T is directly proportional to the square root of $\mathrm{l}$.
6. If $\mathrm{l}$ is increased to four times its value i.e. ${\mathrm{l}}^{\text{'}}=4\mathrm{l}$, the resulting time period will be ${\mathrm{T}}^{\text{'}}=2\mathrm{\pi }\sqrt{\frac{4\mathrm{l}}{\mathrm{g}}}=2\times 2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}=2\mathrm{T}$.
7. So, ${\mathrm{T}}^{\text{'}}=2\mathrm{T}$.

Therefore, the time period of a simple pendulum is doubled when its length is increased to four times its value.