# A compressive force, F is applied at the two ends of a long thin steel rod.  It is heated, simultaneously, such that its temperature increases by T.  The net change in its length is zero.  Let l be the length of the rod, A its area of cross-section, Y its Young’s modulus, and α its coefficient of linear expansion.  Then, F is equal to : Option 1)   Option 2) Option 3)   Option 4)

As we learnt in

Young Modulus -

Ratio of normal stress to longitudnal strain

it denoted by Y

$Y= \frac{Normal \: stress}{longitudnal\: strain}$

- wherein

$Y=\frac{F/A}{\Delta l/L}$

F -  applied force

A -  Area

$\Delta l$ -  Change in lenght

l - original length

Here $l=l_{0}(l+\alpha\Delta T)$

$l=l_{0}+l_{0}\alpha\Delta T$

$l-l_{0}=l_{0}\alpha \Delta T$

$\Rightarrow\ \; \Delta l=l_{0}\alpha \Delta T$

$\Rightarrow\ \; \frac{\Delta l}{l_{0}}=\alpha \Delta T$

$Y=\frac{F}{A} \frac{l_{0}}{\Delta l}$

$\Rightarrow\ \; F=\frac{Y_{A}\Delta l}{l_{0}}$

$F=Y_{A} \alpha.\Delta T$

Correct option is 3.

Option 1)

This is an incorrect option.

Option 2)

This is an incorrect option.

Option 3)

This is the correct option.

Option 4)

This is an incorrect option.

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