A liquid with coefficient of volume expansion $\gamma$ is filled in a container of a material having the coefficient of linear expansion $\alpha$. If the liquid overflows on heating, then

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

Suppose there is a hole in a copper plate. On heating the plate, diameter of hole, would

A steel tape is calibrated at $20^{\circ} C$. On a cold day when the temperature is $-15^{\circ} C$, percentage error in the tape will be ........... $\%$ $\left[\alpha_{\text {steel }}=1.2 \times 10^{-5}{ }^{\circ} C ^{-1}\right]$

Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be

( ${\alpha _1},\,{\alpha _2},$ = linear expansion coefficient

$A_1, A_2$ = Area of rods

$Y_1, Y_2$ = Young modulus)

Thermal expansion of a solid is due to the