A copper rod of $88\; \mathrm{cm}$ and an aluminum rod of unknown length have their increase in length independent of increase in temperature. The length of aluminum rod is….$cm$

$( \alpha_{Cu}=1.7 \times 10^{-5}\; \mathrm{K}^{-1}$ and $\alpha_{Al}=2.2 \times 10^{-5} \;\mathrm{K}^{-1} ) $

Three rods of equal length $l$ are joined to form an equilateral triangle $PQR.$ $O$ is the mid point of $PQ.$ Distance $OR$ remains same for small change in temperature. Coefficient of linear expansion for $PR$ and $RQ$ is same i.e. ${\alpha _2}$ but that for $PQ$ is ${\alpha _1}$. Then relation between ${\alpha _1}$ and ${\alpha _2}$ is

A rod of length $l$ and radius $r$ is held between two rigid walls so that it is not allowed to expand. If its temperature is increased, then the force developed in it is proportional to ………

The coefficient of volume expansion of glycerin is $49 \times 10^{-5} \,K ^{-1}$. The fractional change in the density on a $30^{\circ} C$ rise in temperature is ……….

The coefficient of superficial expansion of a solid is $2 \times 10^{-5} {°C^{-1}}$. It's coefficient of linear expansion is