In a vertical $U-$tube containing a liquid, the two arms are maintained at different temperatures ${t_1}$ and ${t_2}$. The liquid columns in the two arms have heights ${l_1}$ and ${l_2}$ respectively. The coefficient of volume expansion of the liquid is equal to

The bulk modulus of copper is $1.4 × 10^{11}$ $Pa$ and the coefficient of linear expansion is $1.7 × 10^{-5} (C^o )^{-1}$. What hydrostatic pressure is necessary to prevent a copper block from expanding when its temperature is increased from $20^o C$ to $30^o C$

A steel rod of diameter $1\,cm$ is clamped firmly at each end when its temperature is $25\,^oC$ so that it cannot contract on cooling. The tension in the rod at $0\,^oC$ is approximately ……… $N$ $(\alpha = 10^{-5}/\,^oC,\,\,Y = 2 \times 10^{11}\,N/m^2)$

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 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} ) $