Question

A body is floating in oil. The density of the oil is \( \rho_{\text{oil}} \) and the density of the body is \( \rho_{\text{body}} \). If the body is partially submerged, what is the fraction of the body's volume submerged in the oil?
Given:
\( \rho_{\text{oil}} \) = density of the oil,
\( \rho_{\text{body}} \) = density of the body.

• A. \( \frac{\rho_{\text{body}}}{\rho_{\text{oil}}} \)
• B. \( \frac{\rho_{\text{oil}}}{\rho_{\text{body}}} \)
• C. \( 1 - \frac{\rho_{\text{oil}}}{\rho_{\text{body}}} \)
• D. \( 1 - \frac{\rho_{\text{body}}}{\rho_{\text{oil}}} \)

Hint:

In fluid mechanics, the fraction of a floating body submerged in a liquid is the ratio of the density of the liquid to the density of the body.

Answer 1

The Correct Option is B

When a body floats in a fluid, the weight of the body is equal to the buoyant force exerted by the fluid. The buoyant force is given by Archimedes' principle and depends on the volume of the displaced fluid. The fraction of the body's volume submerged in the fluid is given by the ratio of the densities of the fluid and the body: \[ \text{Fraction submerged} = \frac{\rho_{\text{oil}}}{\rho_{\text{body}}} \] This fraction represents the proportion of the body that is submerged in the oil. Thus, the correct answer is Option (B): \( \frac{\rho_{\text{oil}}}{\rho_{\text{body}}} \).