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

Step 1: Initial volume of the cube $V = a^3$.
Step 2: Coefficient of volume expansion $\gamma = 3\alpha$ for isotropic solids.
Step 3: The increase in volume $\Delta V = V \gamma \Delta T$.
Step 4: $\Delta V = (a^3) (3\alpha) \Delta T = 3a^3 \alpha \Delta T$.