.14.18 A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ${\mathit{\rho }}_{\mathit{l}}$ .The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

T = 2 $\mathit{\pi }\sqrt{\frac{\mathit{h}\mathit{\rho }}{{\mathit{\rho }}_{\mathit{l}}\mathit{g}}}$

where $\mathit{\rho }$ is the density of cork. (Ignore damping due to viscosity of the liquid).

<p>Base area of the cork = A</p><p>Height of the cork = h</p><p>Density of the liquid =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub></math></span></p><p>Density of the cork =&nbsp;<span title="Click to copy mathml"><math><mi>ρ</mi></math></span></p><p>In equilibrium, Weight of the cork = Weight of the liquid displaced by the floating cork</p><p>Let the cork be depressed slightly by an amount x, as a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides restoring force to the cork.</p><p>Up-thrust (Restoring force) = weight of the extra water displaced</p><p>F = mg =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub><mi>V</mi><mi>g</mi></math></span></p><p>Volume = Area&nbsp;<span title="Click to copy mathml"><math><mo>×</mo></math></span>&nbsp;distance through which the cork is depressed</p><p>V = Ax</p><p>F = A&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub><mi>g</mi><mi>x</mi></math></span>&nbsp;….(i)</p><p>According to force law, F= kx, where k is constant</p><p>k =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>F</mi></mrow></mrow><mrow><mrow><mi>x</mi></mrow></mrow></mfrac></math></span>&nbsp;= A&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub><mi>g</mi></math></span>&nbsp;…(ii)</p><p>Time period of the oscillation of the cork, T = 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;….(iii)</p><p>Where m = mass of the cork = Base area of the cork&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal"></mi><mo>×</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">y</mi></math></span></p><p>m = Ah&nbsp;<span title="Click to copy mathml"><math><mi></mi><mi>ρ</mi></math></span></p><p>So T = 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">h</mi><mi>ρ</mi></mrow></mrow><mrow><mrow><mi mathvariant="normal">A</mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub><mi>g</mi><mi mathvariant="normal"></mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;= 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi mathvariant="normal">h</mi><mi>ρ</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></msub><mi>g</mi></mrow></mrow></mfrac></mrow></msqrt></math></span></p>

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