The gap between any two rails, each of length $l$ laid on a railway track, is equal to $x$ at $27\,^{\circ}C$. When the temperature rises to $40\,^{\circ}C$, the gap closes up. If the coefficient of linear expansion of the material of the rail is $\alpha$, the length $l$ of a rail at $27\,^{\circ}C$ is:

The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is :

Two rods,one of copper $(Cu)$ and the other of iron $(Fe)$,having initial lengths $L_1$ and $L_2$ respectively,are connected together to form a single rod of length $L_1+L_2$. The coefficients of linear expansion of $Cu$ and $Fe$ are $\alpha_c$ and $\alpha_i$ respectively. If the length of each rod increases by the same amount when their temperatures are raised by $t^{\circ}C$,then the ratio $\frac{L_1-L_2}{L_1+L_2}$ will be:

Two rods,one of aluminum and the other made of steel,having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1 + l_2$. The coefficients of linear expansion for aluminum and steel are $\alpha_a$ and $\alpha_s$ respectively. If the length of each rod increases by the same amount when their temperature is raised by $t ^\circ C$,then find the ratio $\frac{l_1}{l_1 + l_2}$.

Coefficient of linear expansion of brass and steel rods are $\alpha_1$ and $\alpha_2$. Lengths of brass and steel rods are $l_1$ and $l_2$ respectively. If $(l_2 - l_1)$ is maintained same at all temperatures,which one of the following relations holds good?

$A$ clock which keeps correct time at $20^\circ C$ is subjected to $40^\circ C$. If the coefficient of linear expansion of the pendulum is $12 \times 10^{-6} /^\circ C$,how much time will it gain or lose per day?