Q A cylinder of mass M and density d1 hanging from a string is lowered into a vessel of cross sectional area A containing a liquid of density d2(d2

A)Md2g/d1A

B)Mg/A

C) Md1g/d2A

D) Zero

To tackle this problem, we need to analyze the situation involving the cylinder, the liquid, and the forces at play. The key here is to understand how the pressure changes when the cylinder is fully immersed in the liquid. Let's break it down step by step.

Understanding the Pressure Change

When the cylinder is lowered into the liquid, it displaces some of the liquid. The pressure at a certain depth in a fluid is given by the formula:

P = P₀ + ρgh

Where:

• P is the pressure at depth.
• P₀ is the atmospheric pressure above the liquid.
• ρ is the density of the liquid.
• g is the acceleration due to gravity.
• h is the height of the liquid column above the point where pressure is being measured.

Analyzing the Forces

When the cylinder is fully immersed, it experiences a buoyant force equal to the weight of the liquid displaced. The buoyant force can be calculated using Archimedes' principle:

F_b = V_displaced * d2 * g

Where:

• F_b is the buoyant force.
• V_displaced is the volume of the cylinder submerged in the liquid.
• d2 is the density of the liquid.

The weight of the cylinder is:

W = M * g

Calculating the Pressure Increase

When the cylinder is fully immersed, the pressure at the bottom of the vessel increases due to the weight of the liquid column above it and the weight of the cylinder itself. The total pressure at the bottom can be expressed as:

P_total = P₀ + d2 * g * h + (M * g) / A

Here, h is the height of the liquid column above the bottom of the vessel. However, since we are interested in the increase in pressure due to the cylinder being immersed, we can focus on the additional pressure contributed by the weight of the cylinder:

ΔP = (M * g) / A

Final Considerations

Given that the density of the liquid (d2) is less than the density of the cylinder (d1), the cylinder will not float but will instead sink, contributing to the pressure at the bottom of the vessel. Therefore, the increase in pressure at the bottom of the vessel due to the cylinder being immersed is:

ΔP = (M * g) / A

Thus, the correct answer to the question is:

B) Mg/A

This result shows that the increase in pressure at the bottom of the vessel is directly proportional to the weight of the cylinder and inversely proportional to the cross-sectional area of the vessel. This relationship is crucial in understanding how objects interact with fluids and the resulting pressure changes.