There are 2 liquids of density d₁ and d₂ in a vessel and a cylinder of density d, cross secion area A and length L. A height h of the cylinder is floating in the liquid of density d₁.

Find h in terms of L, d₁, d₂ and d.

EXPLAIN IT PLEASE.

One of the question of BITSAT 2015

To solve the problem of finding the height \( h \) of a floating cylinder in a liquid, we need to apply the principles of buoyancy and equilibrium. The cylinder is floating in a liquid of density \( d_1 \) while the other liquid of density \( d_2 \) is below it. The key concept here is that the weight of the liquid displaced by the submerged part of the cylinder must equal the weight of the cylinder itself for it to float. Let’s break this down step by step.

Understanding the Forces at Play

When the cylinder floats, it displaces a volume of liquid equal to the submerged part of the cylinder. The weight of the displaced liquid can be expressed as:

• Weight of displaced liquid = Volume of submerged part × Density of liquid
• Weight of the cylinder = Volume of the cylinder × Density of the cylinder

Calculating the Volume of the Cylinder

The total volume \( V \) of the cylinder can be calculated using its cross-sectional area \( A \) and its length \( L \):

V = A × L

Now, if \( h \) is the height of the cylinder that is submerged in the liquid of density \( d_1 \), the volume of the submerged part is:

Volume of submerged part = A × h

Setting Up the Equilibrium Condition

According to Archimedes' principle, for the cylinder to float, the weight of the displaced liquid must equal the weight of the cylinder:

Weight of displaced liquid = Weight of the cylinder

Substituting the expressions we derived:

(A × h) × d_1 = (A × L) × d

Solving for Height \( h \)

We can simplify the equation by canceling out the cross-sectional area \( A \) (assuming it is non-zero):

h × d_1 = L × d

Now, to find \( h \), we rearrange the equation:

h = (L × d) / d_1

Considering the Two Liquids

However, we must also consider the scenario where the cylinder is floating between two liquids. If the density of the liquid below the cylinder is \( d_2 \), we need to ensure that the buoyant force from both liquids is accounted for. The total weight of the cylinder must equal the weight of the liquid displaced by both densities:

h × d_1 + (L - h) × d_2 = L × d

Here, \( (L - h) \) represents the height of the cylinder submerged in the liquid of density \( d_2 \). Rearranging this equation gives us:

h × d_1 + L × d_2 - h × d_2 = L × d

Combining like terms leads to:

h × (d_1 - d_2) = L × d - L × d_2

Finally, solving for \( h \) yields:

h = (L × (d - d_2)) / (d_1 - d_2)

Final Expression

Thus, the height \( h \) of the cylinder submerged in the liquid of density \( d_1 \) can be expressed in terms of the given parameters as:

h = (L × (d - d_2)) / (d_1 - d_2)

This equation shows how the height of the submerged part of the cylinder depends on the densities of the liquids and the density of the cylinder itself. Understanding these relationships is crucial in problems involving buoyancy and floating objects.