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Question

A metal rod, fixed at both ends, has a length l,area of cross section A and youngs modulus Y and coefficient of linear expansion α is heated through t0 C. The work done in heating the rod is


A

YAl2t2

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B

14YAlα2t2

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C

12YAlα2t2

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D

12YAlαt

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Solution

The correct option is C

12YAlα2t2


The work done is nothing but the potential energy stored.

Also, the thermal stress is given by TA=Yαt. We have,

W=12 (stress) (strain) (volume)

W=12×Yl×x2l×(Al) ..... (1)
Since x=TlYA and T=YAαt, we have on substituting in (1),

W=12YAlα2t2


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