Question

A plank is floating in a non-viscous liquid as shown, Choose the correct options.

• A. Equilibrium of plank is stable in vertical direction
• B. For small oscillations of plank in vertical direction motion is simple harmonic
• C. Even if oscillations are large, motion is simple harmonic till it is not fully immersed
• D. On vertical displacement motion is periodic but not simple harmonic

Answer 1

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

The correct options are
A Equilibrium of plank is stable in vertical direction
B For small oscillations of plank in vertical direction motion is simple harmonic
C Even if oscillations are large, motion is simple harmonic till it is not fully immersed

Let $x$ be the vertical displacement of the plank in water from equilibrium position.

Restoring force acting upwards=$fV\rho g-mg=fV\rho g-Vdg$

where f is the fraction of volume immersed in liquid.

When the plank was in equilibrium,

$mg=f_{0}V\rho g=Vdg$

$Ax=fV-f_{0}V$

$⟹(fV-Ax)\rho g=Vdg$

Thus restoring force=$Ax\rho g\propto x$

So this is a Simple Harmonic Motion.

If the plank is fully immersed, the SHM won't last since the restoring force would be$=Vd\rho g-mg=V(\rho -d)g$ and not proportional to vertical displacement from equilibrium position.