A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ1. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T = $$\dfrac{ 2π \sqrt{hρ}}{ρ_1g}$$ where ρ is the density of cork. (Ignore damping due to viscosity of the liquid)

Asked by Abhisek | 1 year ago |  114

##### Solution :-

Given

Base area of the cork = A

Height of the cork = h

Density of the liquid = ρ1

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 excess water of a certain volume is displaced.

Thus, 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 x Density x g)

Volume = Area x Distance through which the cork is depressed

Volume = Ax

Therefore,

F = – Ax x ρ1g …. (1)

According to the force law:

F = kx

k = $$\dfrac{F}{X}$$

where,

k is constant

k = $$\dfrac{F}{X}$$ = – Aρ1g ……. (2)

The time period of the oscillations of the cork:

T = 2π ($$\sqrt{\dfrac{m}{k} }$$) ……. (3)

Where,

m = Mass of the cork

= Volume of the cork x Density

= Base area of the cork x Height of the cork x Density of the cork

= Ahρ

Therefore, the expression for the time period becomes:

T = 2π $$\sqrt{\dfrac{Ahρ}{Aρ_1g}}$$

T = 2π$$\sqrt{\dfrac{hρ}{ρ_1g}}$$

Answered by Pragya Singh | 1 year ago

### Related Questions

#### A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction

A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

#### A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s.

A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is

(a) 5 cm

(b) 3 cm

(c) 0 cm.

#### A circular disc of mass 10 kg is suspended by a wire attached to its centre.

A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant α is defined by the relation J = –α θ, where J is the restoring couple and θ the angle of twist).

#### Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals.

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.