A metal rod of Young's modulus and coefficient of thermal expansion `alpha` is held at its two ends such that its length remains invariant. If its temperature is raised by `t^(@)C,` the linear stress developed it its is .....

Options

A. `(alpha t )/(F)`
B. `(Y)/(alpha t)`
C. `Y prop t `
D. `(1)/(Y prop t)`

Answer

C

Explanation

`Delta l = alpha l Delta T`
`therefore (Delta l )/(l) = alpha Delta T " "...(1)`
Young modulus `Y = ("stress" sigma _(l))/("strain" G_(l))`
`therefore epsi _(l ) = (sigma _(l))/(Y)`
`therefore (Delta l )/(l) = (sigma _(l))/( Y) " "...(2)`
From equation (1) and (2)
`(sigma _(1))/(Y) = alpha Delta T`
`therefore sigma _(l) = Y prop Delta T`
`therefore sigma _(l) = Y prop T` (where `Delta T =l ^(@)C)`