Problem 174 - Master Resource Book in JEE Main Physics

Question

The value of a metal sphere increase by \(0.24 \%\) when its temperature is raised by \(40^{\circ} \mathrm{C}\). The coefficient of linear expansion of the metal is... \({ }^{\circ} \mathrm{C}^{-1}\). [BVP Engg. 2007] (a) \(2 \times 10^{-5}\) (b) \(6 \times 10^{-5}\) (c) \(18 \times 10^{-5}\) (d) \(1.2 \times 10^{-5}\)

Step-by-Step Solution

Answer

The coefficient of linear expansion is \( 2 \times 10^{-5} \ { }^{\circ} \mathrm{C}^{-1} \).

Step 1: Understanding the Problem

The problem provides the percent increase in volume of a metal sphere due to a temperature increase and asks for the coefficient of linear expansion. We use the formula for the volume expansion: \( \Delta V = \beta V_0 \Delta T \). Given the percent increase, the change in volume \( \Delta V = 0.24\% \times V_0 \), and \( \Delta T = 40^{\circ} \mathrm{C} \).

Step 2: Volume Expansion and Linear Expansion Relations

The coefficient of volume expansion \( \beta \) is related to the coefficient of linear expansion \( \alpha \) by the formula \( \beta = 3\alpha \). This relationship arises because volume changes in three dimensions.

Step 3: Calculate Coefficient of Volume Expansion

Using the formula \( \beta = \frac{\Delta V}{V_0 \Delta T} = \frac{0.0024}{40} \), calculate \( \beta \). Substitute \( \Delta V = 0.0024 V_0 \) for the 0.24% increase.

Step 4: Substitute and Solve for \( \alpha \)

With \( \beta = \frac{0.0024}{40} = 0.00006 \), calculate \( \alpha = \frac{\beta}{3} = \frac{0.00006}{3} = 0.00002 = 2 \times 10^{-5} \).

Step 5: Identify the Correct Answer

Compare the obtained value of \( \alpha = 2 \times 10^{-5} \) to the given options. The correct match is option (a) \( 2 \times 10^{-5} \).

Key Concepts

Volume Expansion

When a solid object, like a metal sphere, is heated, its molecules vibrate more vigorously. This increased motion causes the object to expand in size. This concept is known as **volume expansion**. In the case of the metal sphere, the entire volume of the sphere increases as temperature rises.
Volume expansion essentially means the object occupies more space. The extent of this expansion depends on the material and its properties, particularly its coefficient of volume expansion, denoted as \( \beta \). The formula to express volume expansion is:

• \( \Delta V = \beta V_0 \Delta T \)

where \( \Delta V \) is the change in volume, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature. The greater the temperature change, the greater the sphere's expansion, assuming \( \beta \) remains constant. For our metal sphere, the volume increased by 0.24% when the temperature increased by 40°C. Understanding volume expansion is essential in calculating how much the object will enlarge due to temperature changes.

Temperature Increase

Temperature increase impacts the physical state of materials, leading to expansion or contraction. For solids, like metals, as temperature increases, the molecules gain energy and start vibrating more. This leads to an expansion in length, area, or volume, depending on the shape and constraints of the material.
In our context of a metal sphere, the material experiences a uniform expansion throughout its volume because of heating. The change in temperature \( \Delta T \) is crucial in determining the resultant change in the object's size. When dealing with temperature changes:

• A small temperature increase typically causes a subtle expansion.
• A large temperature change leads to a more noticeable size change.

The exercise was centered around a specific temperature increase of 40°C, resulting in a measurable 0.24% increase in the sphere's volume. These changes form core calculations in determining the coefficient of linear expansion.

Metal Sphere

A metal sphere is typically considered for study in expansion problems because it is symmetrical and simplifies calculations. When heated, every part of the metal sphere expands uniformly due to its shape. This uniformity is crucial when calculating both linear and volume expansion.
For a metal sphere, the concept of linear expansion is extended to three dimensions, resulting in the **coefficient of volume expansion**, \( \beta \). It's important to note:

• The shape of the object affects how it expands.
• Metal spheres expand the same way in all directions.

This makes them ideal for deriving straightforward formulas relating linear and volume expansions. For any sphere, including our example, the volume expansion \( \beta \) is related to the linear expansion coefficient \( \alpha \) by \( \beta = 3 \alpha \). This relationship helps bridge the sphere’s increase in volume due to temperature changes with the science of linear expansion, helping to calculate the sphere's change in dimensions.