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

Answer

B

Explanation

Work done or (energy stored) per unit volume
or `(W)/(V) = (1)/(2) "stress" xx "strain"`
`:. W = (1)/(2) "stress" xx "strain" xx "volume"` …(1)
In this case, we consider the thermal stress
`+ Y alpha Delta theta = Y alpha t` …(2)
where `t = Delta theta` = temperature difference
Strain `= (dl)/(L)` and `because alpha = (dl)/(L(Delta theta)) = (dl)/(L(t)) = ("strain")/(t)`
`:.` Strain `= alpha t` ...(3)
and volume of the wire `= Delta L` ...(4)
Using (2), (3) and (4) in (1), we get
`W = (1)/(2) xx Y alpha t xx alpha t xx AL`
`W = (1)/(2) YA L alpha^(2) t^(2)`