Question

A ball of density $\rho$ is released from the surface of a liquid whose density varies with depth $h$ as ${\rho }_l=\alpha h$. Here $\alpha$ is a positive constant. Match the following two columns. (liquid is ideal)

Answer 1

Ball will oscillate simple harmonically. The mean position in at depth
$\rho =\alpha h$
or $h=\frac{\rho }{\alpha }=AB$
$h_{max}=\frac{2\rho }{\alpha }=AC$
Amplitude$=\frac{\rho }{\alpha }=AB$ or $BC$
From A to B: $\rho >{\rho }_l$, weight> upthrust
At B, $\rho ={\rho }_l$ weight=upthrust
From B to C, ${\rho }_e>\rho$ upthrust> weight.
From A to C$\to$upthrust will increase and gravitational potential energy will decrease.
From C to A, upthrust will decrease and gravitational potential energy will increase.
From A to B, speed will increase.
From B to C, speed will decrease.