Question

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

Answer 1

$F _{\text {restoring }}=-( mg - V \rho g) \sin \theta$
$=-( 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 =2.135\, s$