A cylindrical piece of cork of density ρ of base area A and height h floats in a liquid of density ρ_{l}. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

Where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Now when a cork is kept on Liquid surface there are two forces acting on it equilibrium position (when forces are balanced and it is at rest), its weight acting in the downward direction and up thrust due to liquid which is equal to weight of liquid displaced by the block, now for equilibrium condition suppose A cylindrical piece of cork of density ρ of base area A and height h floats in a liquid of density ρ_{l} and its Y height is inside the liquid

**As shown in the figure**

Now volume of liquid displaced equal to the volume of cork inside liquid, now we know

Volume = Area × length

Here surface area of cork is A, and suppose length inside liquid is Y so the volume of cork inside liquid is

V_{1} = AY

Now total height of cork is h so total volume of the cork is

V = Ah

Now we know mass is given as

m = V × ρ

where m is the mass of the body, V is its volume and ρ is the density

so mass of the cork is

m = Ahρ

where A is the area of cross section of cork, ρ is its density and h is its height

we know weight of a body is given as

W = mg

Here m is the mass of the body and g is acceleration due to gravity, so weight of the cork is

W = Ahρg

now mass of water displaced by the cork is

m_{1} = AYρ_{l}

where A is the area of cross section of cork, ρ_{l} is its density and Y is its height inside liquid

we know weight of a body is given as

W = mg

Here m is the mass of the body and g is acceleration due to gravity, so weight of liquid displaced by the cork is

W_{1} = AYρ_{l}g

Where ρ_{l} is the density of the liquid, Y is the height if cork inside liquid

Now force equal to weight of liquid displaced by cork will act in upward direction and weight of cork will be in down ward direction both must be equal in magnitude and opposite in direction for equilibrium i.e.

W = W_{1}

Now when the cork is pushed inside slightly the amount of liquid displaced by it will increase hence up thrust due to liquid will increase and but weight will remain same hence there will be a net force in upward direction, which will push it towards original equilibrium position, hence after releasing it will start oscillating about the original equilibrium or mean position

**The situation has been shown in the figure**

Now when the cork is at a distance X below mean position total height of cork inside liquid will be (X + Y) so volume of liquid displaced will be

V_{2} = A(X+Y)

Here V_{2} is the volume of liquid displaced and surface area of cork is A

So mass of liquid displaced by cork is

m_{2} = ρ_{l} V_{2}

= ρ_{l} A(X+Y)

Where ρ_{l} is the density of liquid displacement

So weight of liquid displaced will be

W_{2} = ρ_{l} A(X+Y)g

Where g is acceleration due to gravity

Now weight of liquid displaced will be the new force acting upward direction and, weight of the cylindrical cork is acting in downward direction so net force in upward direction is

F = W_{2} – W

W is the weight of cork

We know W = W_{1}

Where W_{1} is the weight liquid displaced by cork in equilibrium position

So we have

F = W_{2} – W_{1}

Or

F = ρ_{l} A(X+Y)g - ρ_{l}AYg

= ρ_{l}AXg

We know force on a Body is

F = ma

Where m is the mass of particle and a is the acceleration of the particle, so acceleration of the body can be written as

a = F/m

here mass of the cork is

m = Ahρ

where A is the area of cross section of cork, ρ is its density and h is its height

so net acceleration of cork is

a = (ρ_{l}AXg)/(Ahρ)

= (ρ_{l}g/hρ)X

we know condition for simple harmonic motion is that acceleration of particle is proportional to distance from mean position and is directed toward mean position

and we have relation between acceleration and displacement as

a = -ω^{2}X

where a is the acceleration of particle undergoing Simple Harmonic Motion with angular frequency ω

here acceleration of the cork is

a = -(ρ_{l}g/hρ)X

(-ve sign because acceleration is opposite to direction of displacement, acceleration is upward and displacement in vertically downward direction)

where the density of liquid ρ_{l}, the density of cork ρ, height of cork h, acceleration due to gravity g al are constant quantities so acceleration of cork is proportional to displacement from mean position or equilibrium position X i.e. Cork is undergoing simple harmonic motion, so comparing with the equation for acceleration of simple harmonic motion we get

ω^{2} = ρ_{l}g/hρ

i.e. the angular frequency is

We know relation between time period T and angular frequency ω T = 2π/ω

So we have the time period of the oscillation as

So time period of the Simple harmonic motion of cork as

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