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 'α'. The metal sheet is heated uniformly, by a small temperature ΔT, so that its new temperature is T + ΔT. Calculate the increase in the volume of the metal box.

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

Hint:

For small expansions: $\beta \approx 2\alpha$ (area) and $\gamma \approx 3\alpha$ (volume).

Answer 1

The Correct Option is D

To solve the problem of finding the increase in the volume of a metal box when it is heated, we need to understand the concept of thermal expansion. The box is initially a cube with side length a and the coefficient of linear expansion for the material is given as \alpha.

When the temperature of the box increases by \Delta T, each dimension of the box expands. For a linear expansion, the change in length \Delta L for each side is given by:

\Delta L = a \alpha \Delta T

Therefore, the new length of each side of the cube becomes:

a_{\text{new}} = a + \Delta L = a(1 + \alpha \Delta T)

To find the change in volume, we calculate the new volume of the cube:

V_{\text{new}} = (a_{\text{new}})^3 = \left[a(1 + \alpha \Delta T)\right]^3

Expanding this expression using the binomial theorem (for small \alpha \Delta T), we get:

V_{\text{new}} = a^3 (1 + 3\alpha \Delta T + 3(\alpha \Delta T)^2 + (\alpha \Delta T)^3)

Since \alpha \Delta T is small, the higher-order terms (\alpha \Delta T)^2 and (\alpha \Delta T)^3 can be neglected. Thus, the new volume becomes approximately:

V_{\text{new}} \approx a^3 (1 + 3\alpha \Delta T)

Therefore, the increase in volume \Delta V is:

\Delta V = V_{\text{new}} - V_{\text{original}}

\Delta V = a^3(1 + 3\alpha \Delta T) - a^3

\Delta V = a^3 + 3a^3 \alpha \Delta T - a^3

\Delta V = 3a^3 \alpha \Delta T

Therefore, the correct answer is 3a^3 \alpha \Delta T.

This formula shows that the increase in volume is directly proportional to the original volume, the coefficient of linear expansion, and the temperature change.