Question

In the equation \( T=2 \pi \sqrt{\frac{\ell}{g}} \), If \( \ell \) increases by \( 4 \% \) and \( g \) also increases by \( 2 \% \) then \( \% \) change in T is :-

Answer 1

We know time period of simple pendulum

\(T = 2 \pi \sqrt {\frac {l}{g}} = 2 \pi l ^{\frac {1}{2}}g ^{-{\frac {1}{2}}}\)

\(\Rightarrow \frac {\Delta T}{T} = \frac {1}{2} \frac {\Delta l}{l} - \frac {1}{2} \frac {\Delta g}{g}\)

\(=\frac {1}{2} \times \frac {4}{100} - \frac {1}{2} \times \frac {2}{100}\)

\(=\frac {2-1}{100} = \frac {1}{100} = +1 \% \,\,\text {(increase)}\)

Hence time period will increase by 1%.