Question
The angle made by the string of a simple pendulum with the vertical depends on time as $\theta=\frac{\pi}{90}\sin[(\pi\text{s}^{-1})\text{t}].$ Find the length of the pendulum if $\text{g}=\pi^2\text{m/s}^2.$
✓
Answer
From the equation,$\theta=\pi\sin[\pi\sec^{-1}\text{t}]$
$\therefore\omega=\pi\sec^{-1}$ (comparing with the equation of SHM)
$\Rightarrow\frac{2\pi}{\text{T}}=\pi$
$\Rightarrow\text{T}=2\sec.$
We know that $\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}}}$$\Rightarrow2=2\sqrt{\frac{\ell}{\text{g}}}$
$\Rightarrow1=\sqrt{\frac{\ell}{\text{g}}}$
$\Rightarrow\ell=1\text{m}.$
$\therefore$ Length of the pendulum is 1m.
$\therefore\omega=\pi\sec^{-1}$ (comparing with the equation of SHM)
$\Rightarrow\frac{2\pi}{\text{T}}=\pi$
$\Rightarrow\text{T}=2\sec.$
We know that $\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}}}$$\Rightarrow2=2\sqrt{\frac{\ell}{\text{g}}}$
$\Rightarrow1=\sqrt{\frac{\ell}{\text{g}}}$
$\Rightarrow\ell=1\text{m}.$
$\therefore$ Length of the pendulum is 1m.
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