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,
$\therefore \mathrm{L}=\mathrm{L}_{0}(1+\alpha \theta)$
$\therefore \mathrm{L}=\mathrm{L}_{0}\left[1 \times \alpha\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right] \text { which is new length. }$
A litre of alcohol weighs
The volume of a gas at $20°C$ is $100 \,cm^3$ at normal pressure. If it is heated to $100°C$, its volume becomes $125\, cm^3$ at the same pressure, then volume coefficient of the gas at normal pressure is
What is areal expansion ? Give definition and unit of coefficient of areal expansion.
Give the value of coefficient of volume expansion at $0\,^oC$ for ideal gas.
An aluminium sphere of $20 \;cm$ diameter is heated from $0^{\circ} C$ to $100^{\circ} C$. Its volume changes by (given that coefficient of linear expansion for aluminium $\alpha_{A l}=23 \times 10^{-6}\;/{^o}C$