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 `alpha` per degree celsius.

Explanation

The change in natural length = `Delta_(t) = 1 alpha t `
The natural length of rod at temperature `t^(@)` C is 1 + `alpha` t
The decrease in natural length due to developed stress ` = Delta ` l But the length of rod remains constant.
`therefore Delta l_(t) - Delta l = 0 " " therefore Delta l = Delta l_(t) = l alpha t `
`therefore E = ("stess")/("strain") = (((F)/(A))/(-Delta L))/(l + Delta l_(t))`
`therefore F = (EA Delta l )/(l + Delta l_(t)) = (- E A l alpha t )/(l + l alpha t ) = - (E A alpha t )/( (1 + alpha t ) )`
Here, negative sign indicates that the forces is compressive in nature.