Question

A small body of density ${\rho }^{\prime }$ is dropped from rest at a height $h$ into a lake of density $\rho$, where $\rho >{\rho }^{\prime }$. Which of the following statement(s) is/are correct, if all dissipative effects are neglected (neglect viscosity)?

• A. The speed of the body just entering the lake is $\sqrt{2gh}$.
• B. The body in the lake experiences upward acceleration equal to $\{\left(\frac{\rho }{{\rho }^{\prime }}\right)-1\}g$
• C. The maximum depth to which the body sinks in the lake is $\frac{h{\rho }^{\prime }}{(\rho -{\rho }^{\prime })}$.
• D. The body does not come back to the surface of the lake.

Answer 1

Answer: (A), (B), (C)

The correct options are
A The speed of the body just entering the lake is $\sqrt{2gh}$.
B The body in the lake experiences upward acceleration equal to $\{\left(\frac{\rho }{{\rho }^{\prime }}\right)-1\}g$
C The maximum depth to which the body sinks in the lake is $\frac{h{\rho }^{\prime }}{(\rho -{\rho }^{\prime })}$.

By using conservation of energy,
Initially at the top, the body has only potential energy.
When the body is just about to enter the lake, the total potential energy is converted into kinetic energy. So,
$mgh=\frac{1}{2}m{v}^2$
$v=\sqrt{2gh}$
Option 1 is correct
When body is in water it will experience an upward bouyancy force and weight of the body is acting downwards. The equation of force will be,
$F_b-w=ma$
$V\times \rho \times g-V\times {\rho }^{\prime }\times g=V\times {\rho }^{\prime }\times a$
$\frac{\rho -{\rho }^{\prime }}{{\rho }^{\prime }}g=a$
$(\frac{\rho }{{\rho }^{\prime }}-1)g=a$
option B is correct
The density of the body is less than that of the fluid so it will retart in the fluid. As we know the velocity of body before entering the fluid and its retardation in fluid so, by using newton third law of motion
$v^2=u^2-2ad$
$v=0$ ($\because$ body is retarding)
$⟹2gh=2d(\frac{\rho }{{\rho }^{\prime }}-1)g$
$⟹d=\frac{{\rho }^{\prime }h}{\rho -{\rho }^{\prime }}$
$\therefore$option C is correct

As the density of the body is less than the density of fluid so it will eventually come back to the surface.
Option D is not correct