Question

Time peirod \(T\) of a simple pendulum of length \(I\) is given by \(T=2 \pi \sqrt{\frac{1}{g}}\). If the length is increased by \(2 \%\), then an approximate change in the time period is
(A) \(2 \%\)
(B) \(1 \%\)
(C) \(\frac{1}{2} \%\)
(D) None of these

Answer 1

ans(B)

\(\begin{aligned} \text { Hint } &: \frac{\mathrm{dT}}{\mathrm{d} \ell}=\frac{2 \pi}{\sqrt{\mathrm{g}}} \cdot \frac{1}{2 \sqrt{\ell}} \\ \therefore & \Delta \mathrm{T}=\frac{\mathrm{d} \mathrm{T}}{\mathrm{d} \ell} \cdot \Delta \ell=\frac{\pi}{\sqrt{\mathrm{g} \ell}} \cdot\left(\frac{2 \ell}{100}\right) \\=& 2 \pi \sqrt{\frac{\ell}{\mathrm{g}} \cdot \frac{1}{100}=\frac{\mathrm{T}}{100}} \\ \therefore & \frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{1}{100} \\ \therefore & \therefore 1 \% \end{aligned}\)