A rectangular block of mass $m$ and area of cross-section $A$ floats in a liquid of density $\rho$. If it is g

Question

A rectangular block of mass $m$ and area of cross-section $A$ floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium, it undergoes oscillation with a time period $T$. Then:

Accepted Answer

1. **Identify the Restoring Force:** When the block is depressed by a small distance $x$, an additional volume of liquid $V = Ax$ is displaced. According to Archimedes' Principle, this creates an additional upthrust (buoyant force) which acts as the restoring force.
   $$ F_{restoring} = -(	ext{Weight of displaced liquid}) = -(Axho)g $$
2. **Determine Spring Constant ($k$):** Comparing this to the standard SHM equation $F = -kx$, we identify the effective force constant $k = Aho g$.
3. **Apply Time Period Formula:** The time period of a particle/body in SHM is given by $T = 2\pi \sqrt{\frac{m}{k}}$ .
   Substituting $k = Aho g$:
   $$ T = 2\pi \sqrt{\frac{m}{Aho g}} $$
4. **Analyze Proportionality:** From the derived formula, the relationship between time period $T$ and cross-sectional area $A$ is:
   $$ T \propto \frac{1}{\sqrt{A}} $$
   (Note: It is also proportional to $\sqrt{m}$ and inversely proportional to $\sqrt{ho}$).

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