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.

Solution

Physics sir · 4 steps

A hollow box grows exactly like a solid cube of the same metal, so use γ = 3α on V = a³.

1. Hollow behaves like solid. When the sheet is heated uniformly, every length on the box scales by the same factor (1 + αΔ T). So the empty space inside expands as if it were filled with the same metal.
2. Each side grows. The new side is a' = a(1 + αΔ T). The new volume is a'³ = a³(1 + αΔ T)³.

   a³(1+αΔ T)³ ≈ a³(1 + 3αΔ T) ⇒ Δ V = 3a³αΔ T

3. Why drop terms. αΔ T is very small, so its square and cube are negligible. That leaves the volume coefficient γ = 3α, one α for each of the three dimensions.
4. So. The increase in volume is Δ V = γ V Δ T = 3a³αΔ T, which is option (a).

The idea to keep. A cavity expands as if it were made of the surrounding material. For small Δ T, use Δ V = 3α V Δ T, whatever the shape.

Answer from the NTA final answer key. Solution by PrepWiser.