What is the Work Done by a Heated Metal Rod?

[gracy]

Homework Statement

a metal rod of length "L",cross sectional area "A",young modulus "Y" and coefficient of linear expansion "α " is heated to "t" degree celsius .The work that can be performed by the rod when heated is

Homework Equations

work done by rod when heated=1/2 force exerted by rod(when heated) multiplied by extension
I know force exerted by rod(when heated)=Y α ∆t multiplied by A
(here ∆t is increase in temperature by due to heating)and extension would be final length -initial length let L' be final length.So extension would be L' - L
As L'=L(1 +α ∆t )
So L' - L=Lα ∆t

The Attempt at a Solution

with the help of equations above we can see
work done by rod when heated=1/2 Y α ∆t A multiplied by Lα ∆t
=1/2 YAL(α ∆t )^2
right?but in my book answer is given 1/2 YAL(αt)^2
which one is right?

 

[BvU]

It looks like the reference for the exercise writer is zero degrees celsius. You take a more general approach where the environment temperature can be 0 or 20 or whatever.
Apart from that I see no difference between your answer and the book answer ..

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[gracy]

It looks like the reference for the exercise writer is zero degrees celsius.

if that's the case then

no difference between your answer and the book answer ..

but as we can see nothing such is mentioned in the question