A rail track made of steel having length 10 m is clamped on a railway line at its two ends (Fig). On a summer day due to rise in temperature by 20° C , it is deformed as shown in figure. Find x (displacement of the centre) if αsteel = 1.2* 10^-5 /°C.

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Payal Gupta | Contributor-Level 10

8 months ago

This is a long answer type question as classified in NCERT Exemplar

By applying Pythagoras theorem in given figure

^{${(\frac{L + ? L}{2})}^{2} = {(\frac{L}{2})}^{2} + x$ 2}

x= $\sqrt[]{{(\frac{L + ? L}{2})}^{2} - {(\frac{L}{2})}^{2}}$

=1/2 $\sqrt[]{{(L + ? L)}^{2} - L^{2}}$

= ½ $\sqrt[]{L^{2} + {? L}^{2} + 2 L ? L - L^{2}}$

=1/2 $\sqrt[]{({? L}^{2} + 2 L ? L)}$

As Increase in $? L$ is very small so we neglect it

=1/2 $* \sqrt[]{2 ? L}$

$? L = L α ? t$

By using this value in above in equation

x=1/2 $* \sqrt[]{2 L * L α ? T}$ = ½ L $\sqrt[]{2 α ? t}$

= $\frac{10}{2} * \sqrt[]{2 * 1.2 * 10^{- 5} * 20}$

= 5 $* \sqrt[]{4 * 1.2 * 10^{- 4}}$

= 5 $* 2 * 1.1 * 10^{- 2} = 0.11 = 11 c m$

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A rail track made of steel having length 10 m is clamped on a railway line at its two ends (Fig). On a summer day due to rise in temperature by 20° C , it is deformed as shown in figure. Find x (displacement of the centre) if αsteel = 1.2* 10-5 /°C.