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Calculate the force required to prevent a steel wire of 1 sq.mm cross-section from contracting when it cools from $ 60^\circ C $ to $ 15^\circ C $ , if Young's modulus for steel is $ 2{\text{ }}x{\text{ }}{10^{11}}{\text{ }}N/{m^2} $ and its coefficient of linear expansion is $ 0.000011{\text{ /}}^\circ C $ .
(A) 44 N
(B) 66N
(C) 71N
(D) 99 N

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Last updated date: 16th Jun 2024
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Answer
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Hint When the temperature of a solid body drops, it decreases in length due to length contraction. We must provide an amount of force that will compensate for the decrease in length of the rod by providing lateral stress on the rod. So we need to calculate this stress by the formula using the coefficient of linear expansion and the young’s modulus of the wire.
In the solution we will be using the following formula,
 $\Rightarrow \Delta L = L\alpha \Delta T $ where $ \Delta L $ is the change in length of the rod of length $ L $ , coefficient of linear expansion $ \alpha = {10^{ - 5}}/^\circ C $ due to temperature change $ \Delta T $
 $\Rightarrow Y = \dfrac{{TL}}{{A\Delta L}} $ where $ Y $ is the young’s modulus, $ T $ is the tension in the rod, $ A $ is the cross-sectional area of the rod.

Complete step by step answer
We know that when the temperature of the rod drops, its length will decrease according to the equation
 $\Rightarrow \Delta L = L\alpha \Delta T $
Substituting the values of $ \alpha = 0.000011 $ and $ \Delta T = (15 - 60) = - 45 $ , we get
 $\Rightarrow \Delta L = L(0.000011 \times 45) $
 $\Rightarrow 0.000495L $
Now to keep the rod at the same length, we must provide enough force that can increase the length of the rod by the same amount. So, substituting the value of $ \Delta L $ in the equation of young’s modulus $ Y = \dfrac{{TL}}{{A\Delta L}} $ , we get
 $\Rightarrow Y = \dfrac{{TL}}{{A0.000495L}} $
Substituting the value of $ A = 1m{m^2} = {10^{ - 6}}{m^2} $ and $ Y = 2 \times {10^{11}} $ ,we get
 $\Rightarrow 2 \times {10^{11}} = \dfrac{T}{{{{10}^{ - 6}} \times 0.000495}} $
Solving for $ T $ , we get
$ T = 99\,N $ which corresponds to option (D).

Note
Here we have assumed that the decrease in length occurs almost instantaneously however in reality, the temperature of the rod will decrease at a slow rate and so its length will decrease gradually as well. So, we must increase the force applied on the rod gradually too otherwise we might over-extend the rod in length. While we don’t know the length of the rod, it isn’t needed in the final solution since it cancels out and the force that we must apply is independent of the length of the rod.