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A cylindrical piece of cork of density of base area $\mathbf{A}$ and height $h$ floats in a liquid of density $\rho_l$. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
$$
T=2 \pi \sqrt{\frac{h \rho}{\rho_l g}}
$$
where $\rho$ is the density of cork. (Ignore damping due to viscosity of the liquid).
$$
T=2 \pi \sqrt{\frac{h \rho}{\rho_l g}}
$$
where $\rho$ is the density of cork. (Ignore damping due to viscosity of the liquid).
Solution:
1639 Upvotes
Verified Answer
Let initially in equilibrium $y$ height of cylinder is inside the liquid. Then,
Weight of the cylinder $=$ upthrust due to liquid displaced $\therefore \quad \mathrm{A} h \mathrm{gg}=\mathrm{A} y \rho_l \mathrm{~g}$
When the cork cylinder is depressed slightly by $\Delta y$ and released, a restoring force, equal to additional upthrust, acts on it. The restoring force
$$
F=A(y+\Delta y) \rho_l \mathrm{~g}-\mathrm{Ay} \rho_l \mathrm{~g}=A \mathrm{~g} \Delta y
$$
$\therefore$ Acceleration, $a=\frac{F}{m}=\frac{A \rho_l \mathrm{~g} \Delta y}{A h \rho}=\frac{\rho_l \mathrm{~g}}{h \rho}, \Delta y$ and the acceleration is directed in a direction opposite to $\Delta y$. Obviously, as $a \alpha-\Delta y$, the motion of cork cylinder is SHM, whose time period is given by
$$
\begin{aligned}
T &=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}} \\
&=2 \pi \sqrt{\frac{\Delta y}{a}}=2 \pi \sqrt{\frac{h \rho}{\rho_l \mathrm{~g}}}
\end{aligned}
$$
Weight of the cylinder $=$ upthrust due to liquid displaced $\therefore \quad \mathrm{A} h \mathrm{gg}=\mathrm{A} y \rho_l \mathrm{~g}$
When the cork cylinder is depressed slightly by $\Delta y$ and released, a restoring force, equal to additional upthrust, acts on it. The restoring force
$$
F=A(y+\Delta y) \rho_l \mathrm{~g}-\mathrm{Ay} \rho_l \mathrm{~g}=A \mathrm{~g} \Delta y
$$
$\therefore$ Acceleration, $a=\frac{F}{m}=\frac{A \rho_l \mathrm{~g} \Delta y}{A h \rho}=\frac{\rho_l \mathrm{~g}}{h \rho}, \Delta y$ and the acceleration is directed in a direction opposite to $\Delta y$. Obviously, as $a \alpha-\Delta y$, the motion of cork cylinder is SHM, whose time period is given by
$$
\begin{aligned}
T &=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}} \\
&=2 \pi \sqrt{\frac{\Delta y}{a}}=2 \pi \sqrt{\frac{h \rho}{\rho_l \mathrm{~g}}}
\end{aligned}
$$
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