Problem 34 - Master Resource Book in JEE Main Physics

Question

The volume of a metal sphere increases by \(0.24 \%\) when its temperature is raised by \(40^{\circ} \mathrm{C}\). The coefficient of linear expansion of the metal is ... \({ }^{\circ} \mathrm{C}\). (a) \(2 \times 10^{-5} \mathrm{per}^{\circ} \mathrm{C}\) (b) \(6 \times 10^{-5}\) per \(^{\circ} \mathrm{C}\) (c) \(2.1 \times 10^{-5} \mathrm{per}^{\circ} \mathrm{C}\) (d) \(1.2 \times 10^{-5} \mathrm{per}^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The coefficient of linear expansion is (a) \(2 \times 10^{-5}\) per \(^{\circ}\mathrm{C}\).

Step 1: Understanding Thermal Expansion

The volume expansion of a sphere due to temperature change is governed by the formula: \( \Delta V = \beta V_0 \Delta T \), where \( \beta \) is the coefficient of volume expansion, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.

Step 2: Relating Volume and Linear Expansion

The coefficient of volume expansion \( \beta \) is three times the coefficient of linear expansion \( \alpha \) (i.e., \( \beta = 3\alpha \)) because volume expansion in three dimensions is considered.

Step 3: Setting Up the Equation

Since the volume increases by \(0.24\%\), we have \( \Delta V/V_0 = 0.0024 \). Using \( \Delta V = \beta V_0 \Delta T \), we get \( \beta \times 40 = 0.0024 \).

Step 4: Solving for \( \beta \) and \( \alpha \)

Solve \( \beta \times 40 = 0.0024 \) to get \( \beta = \frac{0.0024}{40} = 0.00006 \). Since \( \beta = 3\alpha \), solve \( 3\alpha = 0.00006 \) to find \( \alpha = \frac{0.00006}{3} = 0.00002 \).

Step 5: Choosing the Correct Option

Compare the calculated \( \alpha = 0.00002 \) or \( 2 \times 10^{-5} \) with the given options. The correct answer is (a) \(2 \times 10^{-5} \) per \(^{\circ}\mathrm{C}\).

Key Concepts

Thermal Expansion

When materials are heated, they tend to expand. This phenomenon is known as thermal expansion. It occurs because molecules vibrate more intensely and take up more space as the temperature increases.

For solids, thermal expansion is often summarized with the coefficient of linear expansion, which measures how much a unit length of a material expands per degree of temperature increase. This concept applies to various objects, including wires, rods, and even complex structures.

• Linear expansion: Occurs in one dimension, such as the length of a rod.
• Area expansion: Affects two-dimensional surfaces, like sheets or plates.
• Volume expansion: Applies to three-dimensional objects, such as cubes or spheres.

Understanding thermal expansion is crucial in engineering and construction, as materials like metal expand and contract with temperature changes, influencing structures and machine parts.

Volume Expansion

Volume expansion is a specific type of thermal expansion concerning three-dimensional objects. It describes how objects like spheres, cylinders, and cubes increase in volume when heated.

In mathematical terms, volume expansion can be expressed with the formula: \[ \Delta V = \beta V_0 \Delta T \] where:

• \(\Delta V\) is the change in volume.
• \(\beta\) is the coefficient of volume expansion.
• \(V_0\) is the original volume.
• \(\Delta T\) is the temperature change.

Floating on the relationship between physics and math, this formula establishes how much larger an object becomes when heated. Identifying \(\beta\) helps determine how susceptible a material is to volume changes with temperature, informing everything from material selection to design parameters.

Temperature Change Effects

Temperature change is central to understanding thermal and volume expansion. As an object's temperature rises, the kinetic energy of its molecules increases. This movement causes the material to take up more space.

The extent of expansion depends on:

• The initial size of the object.
• The material's expansion coefficient.
• The degree of temperature change.

Materials have unique expansion coefficients, meaning steel, aluminum, and other metals respond differently to temperature fluctuations. These coefficients are crucial in various applications, from crafting thermostats to constructing bridges, ensuring that size changes due to temperature are predictable and manageable.

Metal Sphere Expansion

When a metal sphere undergoes a temperature increase, it experiences volume expansion due to the increase in kinetic energy of its molecules.

For a metal sphere, the volume change becomes significant, especially for precision-required applications. The problem presented involves calculating the linear expansion coefficient from the known volume expansion and temperature increase.

The connection between volume and linear expansion lies in the relationship \(\beta = 3\alpha\).

• Volume expansion reflects the total change in the sphere's size.
• Linear expansion refers to changes per unit length.

This coefficient multiplication by three accounts for the three spatial dimensions affecting volume expansion. Understanding these concepts enables better appreciation of how temperature changes impact real-world materials.