A cylindrical plastic bottle of negligible mass of filled with $310\, ml$ of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega $. If the radius of the bottle is $2.5\, cm$ then $\omega $ is close to ..... $rad\, s^{-1}$ (density of water $= 10^3\, kg/m^3$)
JEE MAIN 2019, Diffcult
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$A\rho gx = {F_{restoring}}$
$\pi {r^2}\rho gx = n{\omega ^2}x$
$\therefore \omega \sqrt {\frac{{\pi {r^2}\rho g}}{{\rho v}}} $
$ = r\sqrt {\frac{{\pi g}}{v}} = 2.5 \times {10^{ - 2}}\sqrt {\frac{{3.14 \times 10}}{{310 \times {{10}^{ - 6}}}}} $
$ = 2.5 \times {10^{ - 2}} \times {10^2}\sqrt {10} $
$\omega = 2.5 \times \sqrt {10} $
$\therefore f = \frac{{2.5 \times \sqrt {10} }}{{2\pi }} = 1.25$
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