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.

Answer 1

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: 

\[\omega =   \pi   s^{- 1} \] 

\[ \Rightarrow \frac{2\pi}{T} = \pi\] 

\[ \therefore   T = 2  s\] 

\[\text { Time  period  is  given  by  the  relation, }\] 

\[T = 2\pi\sqrt{\left( \frac{l}{g} \right)}\] 

\[ \Rightarrow 2 = 2\pi\sqrt{\left( \frac{l}{\pi^2} \right)}\] 

\[ \Rightarrow 1 = \pi\frac{1}{\pi}\sqrt{\left( l \right)}\] 

\[ \Rightarrow l = 1  m\] 

\[\text { Hence,   length  of  the  pendulum  is  1  m .}\]