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Question
The angle made by the string of a simple pendulum with the vertical depends on time as \[\theta = \frac{\pi}{90} \sin \left[ \left( \pi s^{- 1} \right)t \right]\] .Find the length of the pendulum if g = π2 m2.
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Solution
It is given that:
Angle made by the simple pendulum with the vertical, \[ \theta = \left( \frac{\pi}{90} \right)\sin \left[ \pi\left( s^{- 1} \right)t \right]\] On comparing the above equation with the equation of S . H . M . , we get:
\[\omega = \pi s^{- 1} \]
\[ \Rightarrow \frac{2\pi}{T} = \pi\]
\[ \therefore T = 2 s\]
\[\text { Time period is given by the relation, }\]
\[T = 2\pi\sqrt{\left( \frac{l}{g} \right)}\]
\[ \Rightarrow 2 = 2\pi\sqrt{\left( \frac{l}{\pi^2} \right)}\]
\[ \Rightarrow 1 = \pi\frac{1}{\pi}\sqrt{\left( l \right)}\]
\[ \Rightarrow l = 1 m\]
\[\text { Hence, length of the pendulum is 1 m .}\]
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