Question

The angle made by the string of a simple pendulum with the vertical depends upon time $\theta =\frac{\pi }{90}sin\pi t$. Find the length of the pendulum if g=${\pi }^{2}m/s$.

Answer 1

We have,

$\Theta =\frac{\pi }{90}sin\pi t$...(1)

and we know that

$\Theta ={\Theta }_{0}sin\omega t$...(ii)

comparing (i) & (ii)

Angular frequency, ${}^{\prime }{\omega }^{\prime }=\pi$

$\Rightarrow \frac{2\pi }{T}=\pi$

$\Rightarrow T=2sec,$

where, $T\to timeperiod$

we know that, in simple pendulum

$T=2\pi \sqrt{\frac{l}{g}}$

$T=2\pi \sqrt{\frac{l}{g}}$

$\Rightarrow 2=2\pi \sqrt{\frac{l}{g}}$

$\Rightarrow \frac{1}{\pi }=\sqrt{\frac{l}{g}}$ $\because g={\pi }^{2}m{s}^{-2}$ given

$\Rightarrow \frac{l}{{\pi }^2}=\frac{1}{{\pi }^2}$

$\Rightarrow l=1m$