A) \[2\pi \sqrt{\frac{\sigma \,a}{\rho \,g}}\]
B) \[2\pi \sqrt{\frac{\rho \,a}{\sigma \,g}}\]
C) \[2\pi \sqrt{\frac{\rho \,g}{\sigma \,a}}\]
D) \[2\pi \sqrt{\frac{\sigma \,g}{\rho \,a}}\]
Correct Answer: A
Solution :
As a is the side of cube s is its density. Mass of cube is \[{{a}^{2}}\sigma ,\] its weight \[={{a}^{3}}\sigma g\] Let h be the height of cube immersed in liquid of density r in equilibrium then, \[F={{a}^{2}}h\,\rho \,g=Mg={{a}^{3}}\sigma \,g\] If it is pushed down by y then the buoyant force \[{F}'={{a}^{2}}(h+y)\rho \,g\] Restoring force is \[\Delta F={F}'-F={{a}^{2}}(h+y)\sigma \,g-{{a}^{2}}h\,\sigma \,g\] \[={{a}^{2}}y\,\rho \,g\] Restoring acceleration \[=\frac{\Delta F}{M}=-\frac{{{a}^{2}}y\rho \,g}{M}=-\frac{{{a}^{2}}\rho \,g}{{{a}^{2}}\sigma }y\] Motion is S.H.M. Þ \[T=2\pi \sqrt{\frac{{{a}^{3}}\sigma }{{{a}^{2}}\rho \,g}}=2\pi \sqrt{\frac{a\sigma }{\rho \,g}}\]
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