If a cylinder of diameter $1.0 \, cm$ at $30^{\circ}C$ is to be slid into a hole of diameter $0.9997 \, cm$ in a steel plate at the same temperature,then the minimum required rise in the temperature of the plate is ........ $^{\circ}C$ (Coefficient of linear expansion of steel $\alpha = 12 \times 10^{-6} \, ^{\circ}C^{-1}$).

When a meter rod made of silver at $0^{\circ} C$ is heated to $100^{\circ} C$,its length increases by $0.19 \ cm$. Find the coefficient of volume expansion of the silver.

Two rods of different materials have lengths $\ell_1$ and $\ell_2$ whose coefficients of linear expansion are $\alpha_1$ and $\alpha_2$ respectively. If the difference between the two lengths is independent of temperature,then:

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 $\frac{l_1}{l_1 + l_2}$.

Two rods,one of aluminium and the other 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 of aluminium and steel are $\alpha_1$ and $\alpha_2$ respectively. If the length of each rod increases by the same amount,when their temperatures are raised by $t^{\circ}C$,then the ratio $\frac{L_1}{L_1+L_2}$ will be

The ratio of linear expansivity to the coefficient of superficial expansion of a rectangular sheet of a solid is