Problem 19 - Master Resource Book in JEE Main Physics

Question

A cylinder of mass \(m\) and density \(\rho\) hanging from a string is lowered into a vessel of cross-sectional area \(A\) containing a liquid of density \(\sigma(<\rho)\) until it is fully immersed. The increase in pressure at the bottom of the vessel is (a) Zcro (b) \(\frac{m g}{A}\) (c) \(\frac{m g \rho}{a A}\) (d) \(\frac{\operatorname{mog}}{\rho \mathrm{A}}\)

Step-by-Step Solution

Answer

The computation for pressure increase does not match the options given.

Step 1: Understand the Scenario

We have a cylinder being immersed in a liquid, leading to an increase in pressure at the bottom of the vessel due to its displacement of the liquid.

Step 2: Examine the Buoyancy and Displacement

When the cylinder is fully immersed, it displaces a volume of liquid equivalent to its own volume. The buoyant force is equal to the weight of the displaced liquid.

Step 3: Calculate the Displaced Volume

The volume of the cylinder, denoted as \( V \), can be calculated using its mass \( m \) and density \( \rho \): \( V = \frac{m}{\rho} \).

Step 4: Determine the Weight of the Displaced Liquid

The weight of the displaced liquid \( W_l \) can be found using its density \( \sigma \): \( W_l = \sigma V g \), which becomes \( W_l = \sigma \left(\frac{m}{\rho}\right) g \).

Step 5: Relate to Pressure Increase

The increase in pressure \( \Delta P \) at the bottom of the vessel is caused by this extra force spread over the area of the vessel. Thus, \( \Delta P = \frac{W_l}{A} = \frac{\sigma \left(\frac{m}{\rho}\right) g}{A} \).

Step 6: Substitute Given Values

Substitute back the expressions we derived to evaluate \( \Delta P = \frac{m g \sigma}{\rho A} \), recognizing it as different from any of the provided options. It looks like the relevant computations have not correctly matched the answer choices.

Key Concepts

Pressure Change in Fluids

When we immerse an object in a fluid, it causes a change in pressure at various points within the fluid. This change is usually due to the object's effect on the fluid's equilibrium.
When a cylinder is immersed in a fluid, it displaces part of the fluid, causing an increase in pressure. Depending on where the object is within the fluid, this change can be analyzed to understand how forces exert within the fluid.
The pressure increase at the bottom of the vessel can be determined by dividing the weight of the displaced fluid by the area of the vessel. Mathematically, this is expressed as \( \Delta P = \frac{W_l}{A} \). This relationship highlights how displaced fluid affects pressure distribution within the fluid's body.

• \( \Delta P \) – Pressure change at the bottom.
• \( W_l \) – Weight of the displaced liquid.
• \( A \) – Cross-sectional area of the vessel.

Fluid Density and Mass Relationship

Density is a crucial property of fluids influencing how they interact with immersed objects. The density of an object or fluid is its mass per unit volume, expressed as \( \rho = \frac{m}{V} \).
When objects are submerged into fluids, understanding density assists in calculating the volume of fluid displaced by the object, which is pivotal in determining how buoyancy and pressure changes occur.
The relationship between the mass and density of the cylinder allows us to find its volume using the equation \( V = \frac{m}{\rho} \). This equation enables us to relate the object's attributes directly to its interaction with the surrounding fluid.

• \( m \) – Mass of the object.
• \( \rho \) – Density of the object.
• \( V \) – Volume of the object.

Archimedes' Principle in Physics

Archimedes' Principle is a fundamental principle unifying the concepts of buoyancy and fluid displacement. It states that any object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object.
This principle helps us understand why ships float and balloons rise. For our cylinder, when it is fully immersed, the buoyant force equals the weight of the water displaced.
Mathematically expressed, the buoyant force \( F_b \) is \( F_b = \sigma V g \), where \( \sigma \) is the fluid's density, \( V \) is the volume of displaced fluid, and \( g \) is the acceleration due to gravity. This direct correlation aids in calculating the resulting changes in water pressure and understanding the object's stability in the fluid.

• Buoyant Force (\( F_b \)) – Force exerted by the fluid on a submerged object.
• \( \sigma \) – Density of the fluid.
• \( V \) – Volume of the fluid displaced.
• \( g \) – Acceleration due to gravity.