Question

A bob is attached to a massless string. It has a density $\sigma =12gc{m}^{-3}$ and length of the string $l=1m$ when it is immersed in a liquid of density $\rho =2gc{m}^{-3}$, its time period of oscillation is _______$Hz$. (Take $\sqrt{3}=1.7$ )

Answer 1

Answer: 2.135

$F_{\text{restoring }}=-(mg-V\rho g)\sin \theta$
$=-(mg-V\rho g)\frac{x}{l}$
$a=-\left(g-\frac{V\rho g}{m}\right)\left(\frac{x}{l}\right)$
$\Rightarrow \omega =\sqrt{\frac{\left(g-\frac{V\rho g}{m}\right)}{l}}=\sqrt{\frac{g}{l}\left(1-\frac{\rho }{\sigma }\right)}$
$T=2\pi \sqrt{\frac{l}{g\left(1-\frac{\rho }{\sigma }\right)}}$
$\Rightarrow T=2\pi \sqrt{\frac{1}{10\left(1-\frac{2}{12}\right)}}$
$=\frac{2\pi }{10}\sqrt{12}$
$=\frac{2\times 3.14\times 2\times 1.7}{10}$
$\therefore T=2.135s$