Question

A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ${{\rho }_{1}}$ . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
$T=2\pi \sqrt{\dfrac{h\rho }{{{\rho }_{_{1}}}g}}$
Where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Answer 1

Hint: Archimedes principle is the basic for solving this question. The equation of force including volume, density and acceleration due to gravity will also help in doing this. According to Archimedes’ principle, in equilibrium, the weight of the cork will be the same as the weight of the fluid displaced by the cork.

Complete step by step answer:
Archimedes' principle states that ,’the upward buoyant force exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.’
According to Archimedes’ principle, in equilibrium, the weight of the cork will be the same as the weight of the fluid displaced by the cork.

Let the base area of the cork = A.
Height of the cork =h
Density of liquid displaced=${{\rho }_{1}}$
Density of the floating cork=ρ

Cork has been dipped in the liquid more by a distance x and thus some extra volume of liquid got displaced from it. Hence, some restoring force is gained by the cork due to the upthrust force acting upwards.

Restoring force
$F=-\left( \rho vg \right)$
Where volume v= area× distance
Therefore force
 $F=-Ax\times {{\rho }_{1}}g$
We know that according to force law,
$k=\dfrac{F}{X}$
k is the spring constant.
Now substituting this in above equation,

$k=\dfrac{F}{x}=-A{{\rho }_{1}}g$……………. (1)
Time period of oscillation of a cork
$T=2\pi \sqrt{\dfrac{m}{k}}$…………… (2)
m= Mass of the cork
     = Volume of the cork × Density
     = Base area × Height × Density of the cork
 $=Ah\rho $…………….. (3)
Substituting (2) and (3) in equation (1)
$T=2\pi \sqrt{\dfrac{Ah\rho }{A{{\rho }_{1}}g}}=2\pi \sqrt{\dfrac{h\rho }{{{\rho }_{1}}g}}$
Hence proved.

Note:
Archimedes principle should be known and the equation of time period of oscillation should also be taken care of.
The equation for solving this question is
$T=2\pi \sqrt{\dfrac{Ah\rho }{A{{\rho }_{1}}g}}=2\pi \sqrt{\dfrac{h\rho }{{{\rho }_{1}}g}}$
According to Archimedes’ principle, in equilibrium, the weight of the cork will be the same as the weight of the fluid displaced by the cork.