Question

The angle made by the string of a simple pendulum with the vertical depends on time as $\mathrm{\theta }=\frac{\mathrm{\pi }}{90}\sin \left[\left(\mathrm{\pi }{s}^{-1}\right)t\right]$. Find the length of the pendulum if g = π² m⁻².

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}: \\ \mathrm{Angle}\mathrm{made}\mathrm{by}\mathrm{the}\mathrm{simple}\mathrm{pendulum}\mathrm{with}\mathrm{the}\mathrm{vertical},\mathrm{\theta }=\left(\frac{\mathrm{\pi }}{90}\right)\sin \left[\mathrm{\pi }\left({\mathrm{s}}^{-1}\right)t\right] \\ \mathrm{On}\mathrm{comparing}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{with}\mathrm{the}\mathrm{eq}\mathrm{uation}\mathrm{of}\mathrm{S}.\mathrm{H}.\mathrm{M}.,\mathrm{we}\mathrm{get}: \\ \mathit{}\omega =\mathrm{\pi }{\mathrm{s}}^{-1} \\ \Rightarrow \frac{2\mathrm{\pi }}{T}=\mathrm{\pi } \\ \therefore T=2\mathrm{s} \\ \mathrm{Time}\mathrm{period}\mathrm{is}\mathrm{given}\mathrm{by}\mathrm{the}\mathrm{relation}, \\ T=2\mathrm{\pi }\sqrt{\left(\frac{l}{g}\right)} \\ \Rightarrow 2=2\mathrm{\pi }\sqrt{\left(\frac{l}{{\mathrm{\pi }}^2}\right)} \\ \Rightarrow 1=\mathrm{\pi }\frac{1}{\mathrm{\pi }}\sqrt{\left(l\right)} \\ \Rightarrow l=1\mathrm{m} \\ \mathrm{Hence},\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{pendulum}\mathrm{is}1\mathrm{m}. \end{gathered}$$