A metallic rod of length `l cm` and cross-sectional area `A cm^(2)` is heated through `t^(@)C`. After expansion if a mechanical force is applied normal to its length on both sides of the rod and restore its original length, what is the value of force? The young's modulus of elasticity of the metal is `E` and mean coefficient of linear expansion is `alpha` per degree Celsius.

Explanation

Change in the length `= Deltal = lalphat`
Length of rod at `t^(@)C` is `l + lalphal`
Decrease in length due to stress `= Deltal`
But length of rod remains constant `:. Deltal_(t) - Deltal = 0`
`:. Deltal = Deltal_(t) = lat rArr E = ("stress")/("strain") = (F)/(A) xx (l + Deltal_(t))/(-Delta_(t))`
`:. F = (EADeltal_(t))/(l + Deltal_(t)) = (-EAlalphat)/(l + lalphat) = -(EAalphat)/((l + alphat))`
Here, negative sign indicates that forces are compressive in nature.