A rail track made of steel having length $10\,m$ is clamped on a railway line at its two ends as shown in figure. On a summer day due to rise in temperature by $20\,^oC$ , it is deformed as shown in figure. Find $x$ (displacement of the centre) if $\alpha _{steel} = 1.2 \times 10^{-5} \,^oC^{-1}$
A rail track made of steel having length $10\,m$ is clamped on a railway line at its two ends as shown in figure. On a summer day due to rise in temperature by $20\,^oC$ , it is deformed as shown in figure. Find $x$ (displacement of the centre) if $\alpha _{steel} = 1.2 \times 10^{-5} \,^oC^{-1}$
By using Pythagoras theorem, for given right angle triangle,
$\left(\frac{\mathrm{L}+\Delta \mathrm{L}}{2}\right)^{2}=\left(\frac{\mathrm{L}}{2}\right)^{2}+x^{2}$
$x=\sqrt{\left(\frac{\mathrm{L}+\Delta \mathrm{L}}{2}\right)^{2}-\left(\frac{\mathrm{L}}{2}\right)^{2}}$
$=\frac{1}{2}\left(\sqrt{(\mathrm{L}+\Delta \mathrm{L})^{2}-\mathrm{L}^{2}}\right)$
$=\frac{1}{2}\left[\sqrt{\mathrm{L}^{2}+2 \mathrm{~L} \Delta \mathrm{L}+\Delta \mathrm{L}^{2}-\mathrm{L}^{2}}\right]$
$=\frac{1}{2}\left[\sqrt{2 \mathrm{~L} \Delta \mathrm{L}+\Delta \mathrm{L}^{2}}\right]$
As $\Delta \mathrm{L}$ is very small, $\Delta \mathrm{L}^{2}$ is neglected.
$x=\frac{1}{2} \sqrt{2 \mathrm{~L} \Delta \mathrm{L}}$
But $\Delta \mathrm{L}=\mathrm{L} \alpha \Delta \mathrm{T}$
$\therefore x=\frac{1}{2} \sqrt{2 \mathrm{~L} \times \mathrm{L} \alpha \Delta \mathrm{T}}$
$=\frac{1}{2} \mathrm{~L} \sqrt{20 \Delta \mathrm{T}}$
$\therefore x=\frac{10}{2} \times \sqrt{2 \times 1.2 \times 10^{-5} \times 20}$
$=5 \times \sqrt{4.8 \times 10^{-4}}$
$=5 \times 2 \times 1.1 \times 10^{-2}$
$=11 \times 10^{-2}$
$=0.11 \mathrm{~m}=11 \mathrm{~cm}$
Here, the increase in length of mercury is neglected as it is very small.
Similar Questions
Explain linear expansion. Write a unit.
Explain linear expansion. Write a unit.
Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T _1=300 K$ and $T _2=100 K$, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K _1$ and $K _2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 K$, then $K _1 / K _2=$ . . . . .
Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T _1=300 K$ and $T _2=100 K$, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K _1$ and $K _2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 K$, then $K _1 / K _2=$ . . . . .
- [IIT 2018]
A flask of volume $10^3 cc$ is completely filled with mercury at $0 ^oC$. The coefficient of cubical expansion of mercury is $180 × 10^{-6} / ^oC$ and that of glass is $40 × 10^{-6}/ ^oC$. If the flask is now placed in boiling water at $100\ ^oC$, ........ $cc$ mercury will overflow
A flask of volume $10^3 cc$ is completely filled with mercury at $0 ^oC$. The coefficient of cubical expansion of mercury is $180 × 10^{-6} / ^oC$ and that of glass is $40 × 10^{-6}/ ^oC$. If the flask is now placed in boiling water at $100\ ^oC$, ........ $cc$ mercury will overflow
Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57\,^oC$ is drunk. You can take body (tooth) temperature to be $37\,^oC$ and $\alpha = 1.7 \times 10^{-5}/^oC$, bulk modulus for copper $ = 140 \times 10^9\, N/m^2 $.
Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57\,^oC$ is drunk. You can take body (tooth) temperature to be $37\,^oC$ and $\alpha = 1.7 \times 10^{-5}/^oC$, bulk modulus for copper $ = 140 \times 10^9\, N/m^2 $.
A solid metallic cube having total surface area $24\;m ^{2}$ is uniformly heated. If its temperature is increased by $10\,^{\circ} C$, calculate the increase in volume of the cube $\left(\right.$ Given $\left.: \alpha=5.0 \times 10^{-4}{ }^{\circ} C ^{-1}\right)$
A solid metallic cube having total surface area $24\;m ^{2}$ is uniformly heated. If its temperature is increased by $10\,^{\circ} C$, calculate the increase in volume of the cube $\left(\right.$ Given $\left.: \alpha=5.0 \times 10^{-4}{ }^{\circ} C ^{-1}\right)$
- [JEE MAIN 2022]