Question

Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is ' $\alpha^{\prime}$. The metal sheet is heated uniformly, by a small temperature $\Delta T ,$ so that its new temperature is $T +\Delta T$. Calculate the increase in the volume of the metal box.

• A. $3 a ^{3} \alpha \Delta T$
• B. $4 a ^{3} \alpha \Delta T$
• C. $4 \pi a ^{3} \alpha \Delta T$
• D. $\frac{4}{3} \pi a ^{3} \alpha \Delta T$

Answer 1

The Correct Option is A

To understand the problem, we need to calculate the increase in volume of a metal box when it is heated by a small temperature $\Delta T$. The box is in the shape of a cube with side length 'a' at the initial temperature 'T'. The coefficient of linear expansion of the metal sheet is given as $\alpha'$.

First, we need to know how thermal expansion works in a three-dimensional object like a cube:

• The linear expansion in solids is given by the formula: $L = L_0 (1 + \alpha' \Delta T)$, where $L_0$ is the original length, and $\alpha'$ is the coefficient of linear expansion.
• For a cube, each side will expand, and its new side length will be: $a_{\text{new}} = a (1 + \alpha' \Delta T)$

Now, the original volume of the cube is $V_0 = a^3$ . The volume of the cube after heating, $V_{\text{new}}$, will be:

$V_{\text{new}} = (a_{\text{new}})^3 = (a(1 + \alpha' \Delta T))^3$

Expanding this expression using binomial theorem, we get:

• $V_{\text{new}} = a^3 (1 + \alpha' \Delta T)^3$
• Assuming $\alpha' \Delta T$ is small, we approximate using binomial expansion up to the first order:
  $(1 + \alpha' \Delta T)^3 \approx 1 + 3 \alpha' \Delta T$

Next, substituting back to find the new volume:

$V_{\text{new}} = a^3 (1 + 3 \alpha' \Delta T)$

The increase in volume $\Delta V$ is given by:

$\Delta V = V_{\text{new}} - V_0 = a^3(1 + 3 \alpha' \Delta T) - a^3$

This simplifies to:

$\Delta V = 3 a^3 \alpha' \Delta T$

Thus, the correct answer is $3 a^3 \alpha' \Delta T$, which corresponds to the first option.