The gap between any two rails, each of length $l$ laid on a railway track equal $x$ at $27\,^oC$ . When the temperature rises to $40\,^oC$ , the gap close up. The coefficient of linear expansion of the material of the rail is $\alpha $ . The length $l$ of a rail at $27\,^oC$ will be

A thin rod having a length of $1\; m$ and area of cross-section $3 \times 10^{-6}\,m ^2$ is suspended vertically from one end. The rod is cooled from $210^{\circ}\,C$ to $160^{\circ}\,C$. After cooling, a mass $M$ is attached at the lower end of the rod such that the length of rod again becomes $1\,m$. Young's modulus and coefficient of linear expansion of the rod are $2 \times 10^{11} Nm ^{-2}$ and $2 \times 10^{-5} K ^{-1}$, respectively. The value of $M$ is $…….kg .\left(\right.$ Take $\left.g=10\,m s ^{-2}\right)$

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 $K _1 / K _2=$ . . . . .

The coefficient of volumetric expansion of mercury is $18 × 10^{-5}{°C^{-1}}$. A thermometer bulb has a volume $10^{-6}\, m^3$ and cross section of stem is $ 0.004 \,cm^2$. Assuming that bulb is filled with mercury at $0°C$ then the length of the mercury column at $100°C$ is

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 $30^o C$