A thin rod having length $L_0$ at $0\,^oC$ and coefficient of linear expansion $\alpha $ has its two ends maintained at temperatures $\theta _1$ and $\theta _2$, respectively. Find its new length.
The temperature in rod changed by going linearly from its one end to another end and temperature at midpoint is $\theta$. In thermal steady state heat current $=\frac{d \mathrm{Q}}{d t}=$ constant.
$\therefore \mathrm{KA} \frac{\theta_{1}-\theta}{\left(\mathrm{L}_{0} / 2\right)}=\frac{\mathrm{KA}\left(\theta-\theta_{2}\right)}{\left(\mathrm{L}_{0 / 2}\right)}$
where $\mathrm{K}$ is thermal conductivity,
$\therefore \theta_{1}-\theta=\theta-\theta_{2}$
$\therefore \theta_{1}+\theta_{2}=2 \theta$
$\therefore \theta=\frac{\theta_{1}+\theta_{2}}{2} \text { temperature of midpoint }$
Now, its length increases with increase in temperature,
$\therefore \mathrm{L}=\mathrm{L}_{0}(1+\alpha \theta)$
$\therefore \mathrm{L}=\mathrm{L}_{0}\left[1 \times \alpha\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right] \text { which is new length. }$
When vapour condenses into liquid
A bimetallic strip consists of metals $X$ and $Y$. It is mounted rigidly at the base as shown. The metal $X$ has a higher coefficient of expansion compared to that for metal $Y$. When the bimetallic strip is placed in a cold bath:
If the length of a cylinder on heating increases by $2\%$, the area of its base will increase by ....... $\%$
Coefficient of linear expansion of a vessel completely filled with $Hg$ is $1 \times 10^{-5} /{ }^{\circ} C$. If there is no overflow of $Hg$ on heating the vessel, then coefficient of cubical expansion of $Hg$ is ......
On heating a uniform metallic cylinder length increases by $3 \%$. The area of cross-section of its base will increase by ........... $\%$