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Q. 14.18 A cylindrical piece of cork of density of base area A and height h floats in a liquid of density \rho _{\imath }. The cork is depressed slightly and then released

Show that the cork oscillates up and down simple harmonically with a period   T=2\pi \sqrt{\frac{h\rho }{\rho _{ 1}g}} where \rho is the density of cork. (Ignore damping due to viscosity of the liquid).

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Let the cork be displaced by a small distance x in downwards direction from its equilibrium position where it is floating.

The extra volume of fluid displaced by the cork is Ax

Taking the downwards direction as positive we have

\\ma=-\rho _{1}gAx\\ \rho Aha=-\rho _{1}gAx\\ \frac{\mathrm{d}^{2} x}{\mathrm{d} t^{2}}=-\frac{\rho _{1}g}{\rho h}x

Comparing with a=-kx we have

\\k=\frac{\rho _{1}g}{\rho h}\\ T=\frac{2\pi }{\sqrt{k}}\\ T=2\pi \sqrt{\frac{\rho h}{\rho_{1}g }}

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