A metal rod of length 'L', cross-sectional area 'A', Young's modulus 'Y' and coefficient of linear expansion `'alpha'` is heated to `'t'^(@)C`. The work that can be perfomred by the rod when heated is

Options

A. `(YA alpha Lt^(2))/(2)`
B. `(Y A alpha^(2) Lt^(2))/(2)`
C. `(YA alpha^(2) L^(2)t^(2))/(2)`
D. `(YA alpha L T)/(2)`

Answer

b

Explanation

work performed by the rod
`Delta w` = change in potential energy of the rod
` = (1)/(2) xx` stress `xx` strain `xx` volume
`= (1)/(2) xx Y xx ("strain")^(2) xx AL` `[:' Y = ("Stress")/("strain")]`
`= (1)/(2) xx Y xx [(Delta L)/(L)]^(2) xx AL`
`(1)/(2) xx Y xx [alpha Delta f]^(2) xx AL` `[:. Delta L = L alpha F]`
`= (1)/(2) xx Y xx alpha^(2) Delta t^(2) xx AL = (Y A alpha^(2) Lt^(2))/(2)`
where Y `to` = Young's modulus of the material
`alpha to` = Thermal coefficient of linear expansion
`Delta t to` = change in temperature = t
`A to` = Corss section area