Calculate the compressional force required to prevent the metallic rod of length l cm and cross-sectional area A cm 2 when heated through t ° C , from expanding lengthwise. Young's modulus of elasticity of the metal is E and mean coefficient of linear expansion is α per degree celsius.

Step-by-step solution

The change in natural length = Δ l ⁡ t = l ⁡ α t The natural length of rod at temperature t ∘ C = l ⁡ + l ⁡ α t The decrease in natural length due to developed stress = Δ l ⁡ But the length of rod remains constant. ∴ Δ l ⁡ t - Δ l ⁡ = 0 ∴ Δ l ⁡ = Δ l ⁡ t = l ⁡ α t ∴ E = stress strain = F/A - Δ l ⁡ l ⁡ + Δ l ⁡ t ∴ E = - EA Δ l ⁡ l ⁡ + Δ l ⁡ t = - EA Δ l ⁡ t l ⁡ + Δ l ⁡ t = - EA l ⁡ α t l ⁡ + l ⁡ α t = - EA α t 1 + α t