Question

A body P floats in water with half its volume immersed. Another body Q floats in a liquid of density $\frac{3}{4}$ of the density of water with two-third of the volume immersed. The ratio of density of P to that of Q is

• A. 1:2
• B. 1:1
• C. 2:1
• D. 2:3
• E. 3:4

Hint:

The fraction of volume immersed is directly proportional to the ratio of the body's density to the liquid's density.

Answer 1

The Correct Option is B

Concept:
For a floating body, the weight of the body equals the buoyant force (weight of the displaced liquid): \[ \rho_{body} V_{body} g = \rho_{liquid} V_{immersed} g \implies \rho_{body} = \rho_{liquid} \left( \frac{V_{immersed}}{V_{body}} \right) \]

Step 1: Calculate the density of body P.
Body P floats in water ($\rho_w$) with half volume immersed ($V_i/V = 1/2$): \[ \rho_P = \rho_w \times \left( \frac{1}{2} \right) = \frac{1}{2} \rho_w \]

Step 2: Calculate the density of body Q.
Body Q floats in a liquid of density $\rho_L = \frac{3}{4} \rho_w$ with two-thirds volume immersed ($V_i/V = 2/3$): \[ \rho_Q = \left( \frac{3}{4} \rho_w \right) \times \left( \frac{2}{3} \right) = \frac{2}{4} \rho_w = \frac{1}{2} \rho_w \]

Step 3: Find the ratio.
\[ \frac{\rho_P}{\rho_Q} = \frac{\frac{1}{2} \rho_w}{\frac{1}{2} \rho_w} = 1 \] The ratio is 1:1.