Problem 4 - Master Resource Book in JEE Main Physics

Question

A U-tube contains water and methylated spirit separated by mereury. The mercury columns in the two arms are in level with \(10.0 \mathrm{~cm}\) of water in one arm and \(12.5 \mathrm{~cm}\) of spirit in the other. The specific gravity of spirit would be. (a) \(0.70\) (b) \(0.80\) (c) \(0.90\) (d) \(0.60\)

Step-by-Step Solution

Answer

The specific gravity of the spirit is 0.80 (option b).

Step 1: Understanding the Problem

The U-tube is balanced, meaning the pressure due to the water on one arm is equal to the pressure due to the spirit on the other arm. We need to find the specific gravity of the spirit.

Step 2: Calculate Water Pressure

The pressure exerted by the column of water can be calculated using the formula: \( P = h \cdot \rho \cdot g \), where \( h = 10.0 \mathrm{~cm} = 0.10 \mathrm{~m} \), \( \rho \) is the density of water (\(1000 \mathrm{~kg/m^3}\)), and \( g = 9.8 \mathrm{~m/s^2} \). Thus, \( P_{water} = 0.10 \cdot 1000 \cdot 9.8 = 980 \mathrm{~Pa} \).

Step 3: Calculate Spirit Pressure

Use the same pressure formula for the spirit: \( P = h \cdot \rho_{spirit} \cdot g \). Given \( h = 12.5 \mathrm{~cm} = 0.125 \mathrm{~m} \), the equation becomes \( 0.125 \cdot \rho_{spirit} \cdot 9.8 = 980 \mathrm{~Pa} \).

Step 4: Solve for Spirit Density

Rearrange the equation from the spirit pressure to solve for the density of spirit: \( \rho_{spirit} = \frac{980}{0.125 \cdot 9.8} = 800 \mathrm{~kg/m^3} \).

Step 5: Calculate Specific Gravity of Spirit

Specific gravity is the ratio of the density of the spirit to the density of water. Therefore, \( \text{Specific Gravity} = \frac{\rho_{spirit}}{\rho_{water}} = \frac{800}{1000} = 0.80 \).

Key Concepts

Understanding Hydrostatic Pressure

Hydrostatic pressure is an important concept in fluid dynamics. It refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. When dealing with fluids in a U-tube, like water and methylated spirit separated by mercury, hydrostatic pressure is crucial for understanding fluid behavior.
The hydrostatic pressure can be calculated using the formula:

• \( P = h \cdot \rho \cdot g \)

In this equation:

• \( P \) stands for pressure,
• \( h \) is the height of the fluid column,
• \( \rho \) represents the fluid's density, and
• \( g \) is the acceleration due to gravity, typically \( 9.8 \ \text{m/s}^2 \).

By understanding hydrostatic pressure, you can better analyze the conditions needed for balancing different fluids in a U-tube. This concept helps in calculating equivalent pressures exerted by different fluids, leading to deeper insights into fluid behavior.

Understanding Specific Gravity

Specific gravity is an essential concept in fluid dynamics that allows for comparisons between the densities of different substances. It is a ratio, meaning it does not have any units, which makes it very useful.
Specific gravity is defined as:

• \( \text{Specific Gravity} = \frac{\rho_{\text{object}}}{\rho_{\text{reference}}} \)

Where:

• \( \rho_{\text{object}} \) is the density of the substance in question,
• \( \rho_{\text{reference}} \) is the density of the reference substance (usually water for liquids).

In the context of the U-tube problem, calculating the specific gravity of the spirit helps us understand how heavy or light it is compared to water. With a specific gravity of 0.80, it indicates that the methylated spirit is lighter than water, since a specific gravity less than one means the substance is less dense than the reference.

Basics of Density Calculations

Density plays a key role in understanding how different substances interact in fluid dynamics. Density is defined as mass per unit volume. It is often expressed in units of kg/m³.
To calculate density, use the formula:

• \( \rho = \frac{m}{V} \)

Where:

• \( \rho \) is the density,
• \( m \) is the mass, and
• \( V \) is the volume of the substance.

In fluid problems like the U-tube exercise, knowing the density of each liquid involved helps in determining their behavior, specifically their tendencies to sink or float when in equilibrium with other fluids. By calculating the density of the methylated spirit and comparing it with water's density, we can derive useful insights into the specific gravity and overall behavior of the fluid mixture. This understanding is vital for solving complex problems involving multiple interacting fluids.