A body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\sigma$, where $\sigma \rho$. Neglecting all dissipative forces, calculate the maximum depth to which the body sinks before returning to float on the surface

Question

A body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\sigma$, where $\sigma > \rho$. Neglecting all dissipative forces, calculate the maximum depth to which the body sinks before returning to float on the surface

Dr. Abubacker · Accepted Answer

(c) The speed of the body just before entering the liquid is$v = \sqrt{2\ gh}$. The buoyant force $B$ of the lake $(ie,$ upward thrust of liquid on the body) is greater than the weight of the body$w$, since$\sigma > \rho$. If $V$ is the volume of the body and $a$ is the acceleration of the body inside the liquid, then
$B - w = ma$
Or $\sigma Vg - \rho Vg = \rho V\ a$
Or $(\sigma - \rho)g = \rho a$ or $a = \frac{(\sigma - \rho)g}{\rho}$
Using the relation, $v^{2} = u^{2} + 2as,$ we have
$0 = \left( \sqrt{2\ gh} \right)^{2} - 2g\ \frac{(\sigma - \rho)}{\rho}s$ or $s = \frac{h\rho}{\sigma - \rho}$

Q. A body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\sigma$, where $\sigma > \rho$. Neglecting all dissipative forces, calculate the maximum depth to which the body sinks before returning to float on the surface

Explanation

Reviewed by Dr. Abubacker

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