A body of density $\rho$ is dropped from (at rest) height ' $h$ ' into a lake of density ' $\delta$ '…

A body of density $\rho$ is dropped from (at rest) height ' $h$ ' into a lake of density ' $\delta$ ' $(\delta>\rho)$. The maximum depth to which the body sinks before returning to float on the surface is [Neglect all dissipative forces]
  1. $\frac{(\delta-\rho)}{2 h \rho}$
  2. $\frac{2 h \rho}{(\delta-\rho)}$
  3. $\frac{h \rho}{2(\delta-\rho)}$
  4. $\frac{h \rho}{(\delta-\rho)}$

Solution

The velocity of the body when it reaches the surface of the lake is $\begin{aligned} & \frac{\mathrm{a}}{\mathrm{g}}=\frac{\delta-\rho}{\rho} \\ & \therefore \mathrm{a}=\left(\frac{\delta-\rho}{\rho}\right) \mathrm{g} \end{aligned}$ If $a$ is the acceleration and retardation in the liquid then $\mathrm{v}^2=2 \mathrm{ad}$ by eq. (1) and (4) $\begin{aligned} & 2 \mathrm{ad}=2 \mathrm{gh} \\ & \therefore \mathrm{d}=\frac{\mathrm{g}}{\mathrm{a}} \mathrm{h} \quad \therefore \mathrm{d}=\frac{\rho}{\delta-\rho} \mathrm{h} \end{aligned}$

Asked in: MHT CET 2021 (20 Sep Shift 1)

Practice more Mechanical Properties of Fluids questions on Aicharya