Question

A ruler is accurate when the temperature is ${25}^{\circ }\text{C}$. When the temperature drops to $-{15}^{\circ }\text{C}$, the ruler shrinks and no longer measures distances accurately.However the ruler can be made to read correctly if a force of magnitude $1.2\times {10}^3\text{N}$ applied to each end so as to stretch it back to its original length. The ruler has a cross sectional area of $1.6\times {10}^{-5}{\text{m}}^2$, and it is made from a material whose coefficient of linear expansion is $2.5\times {10}^{-5}(C^{\circ }{)}^{-1}$. What is Young's modulus (in ${10}^9{\text{N/m}}^2$) for the material from which the ruler is made?

Answer 1

Change in length of rod due to change in temp should be balanced by extension of rod due to application of force.
Change in length of rod due to change in temperature is $l\alpha \Delta T$
Extension of rod due to application of force$\Delta l=\frac{Tl}{YA}$
$\Rightarrow \frac{Tl}{YA}=l\alpha \Delta T$

$\Rightarrow Y=\frac{T}{A\alpha \Delta T}=\frac{1200}{1.6\times {10}^{-5}\times 2.5\times {10}^{-5}\times 40}=\frac{1200\times {10}^9}{16}=75\times {10}^9{\text{N/m}}^2$