A non-isotropic solid metal cube has coefficients of linear expansion as :
5 × 10⁻⁵/ᵒC along the x-axis and 5 × 10⁻⁶/ᵒC along the y and the z-axis. If the coefficient of volume expansion of the solid is C × 10⁻⁶/ᵒC then the value of C is

Solution

Physics sir · 3 steps

Each side of the cube stretches on its own, so just add the three linear coefficients.

1. Why they add. Volume is V = xyz. A small temperature rise Δ T changes each length by its own fraction α Δ T. The fractional change in a product is the sum of the fractional changes, so (Δ V)/V = (αₓ + α_y + α_z)Δ T.

   γ = αₓ + α_y + α_z

2. Match the powers. Write every coefficient with the same power of ten: αₓ = 5 × 10⁻⁵ = 50 × 10⁻⁶ °C⁻¹, while α_y = α_z = 5 × 10⁻⁶ °C⁻¹.
3. So. γ = (50 + 5 + 5) × 10⁻⁶ = 60 × 10⁻⁶ °C⁻¹, so C = 60.

The common slip: 15. Adding 5 + 5 + 5 without converting 5 × 10⁻⁵ to 50 × 10⁻⁶ gives 15, not 60.

The idea to keep. The rule γ = 3α is only the isotropic case. In general γ = αₓ + α_y + α_z, and for area it is the sum of the two in-plane coefficients.

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