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Question
The cylindrical piece of the 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 = 2pi sqrt((hrho)/(rho_1g)`
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
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Solution 1
Base area of the cork = A
Height of the cork = h
Density of the liquid = `rho_1`
Density of the cork = ρ
In equilibrium:
Weight of the cork = Weight of the liquid displaced by the floating cork
Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides the restoring force to the cork.
Up-thrust = Restoring force, F = Weight of the extra water displaced
F = –(Volume × Density × g)
Volume = Area × Distance through which the cork is depressed
Volume = Ax
∴ F = – A x `rho_1` g … (i)
According to the force law:
F = kx
`k = F/x`
Where, k is a constant
`k = F/x = -Arho_1g` ...(ii)
The time period of the oscillations of the cork:
`T = 2pi sqrt(m/k)` ...(iii)
Where
m = Mass of the cork
= Volume of the cork × Density
= Base area of the cork × Height of the cork × Density of the cork
= Ahρ
Hence, the expression for the time period becomes:
`T = 2pi sqrt((Ahrho)/(Arho_1g)) = 2 pi sqrt((hrho)/(rho_1g)`
Solution 2
Say, initially in equilibrium, y height of cylinder is inside the liquid. Then,
Weight of the cylinder = upthrust due to liquid displaced
`:. Ahrhog = Ayrho_1g`
When the cork cylinder is depressed slightly by `triangle y` and released, a restoring force, equal to additional upthrust, act on it. The restoring force is
`F = A(y + triangle y) rho_1 g - Ayrho_1g = Arho_1g triangle y`
:. Acceleration, `a = F/m = (Arho_1g triangle y)/(Ahrho) = (rho_1g)/(hrho). triangle y ` and the acceleration is directed in a direction opposite to `triangle y`. Obviously, as `a prop - triangle y`, the motion of cork cylinder is SHM, whose time period is given by
`T = 2pi sqrt("displacement"/"accelertion")`
`=2pi sqrt((triangle y)/a)`
`= 2pi sqrt((hrho)/(rho_1g))`
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