$A$ circular metallic disc of radius $R$ has a small circular cavity of radius $r$ as shown in the figure. On heating the system:

Two straight metallic strips each of thickness $t$ and length $\ell$ are riveted together. Their coefficients of linear expansion are $\alpha_1$ and $\alpha_2$. If they are heated through a temperature change $\Delta T$,the bimetallic strip will bend to form an arc of radius:

The coefficients of linear expansion of brass and steel are ${\alpha _1}$ and ${\alpha _2}$ respectively. If we take a brass rod of length ${l_1}$ and a steel rod of length ${l_2}$ at $0^{\circ}C$,their difference in length $({l_2} - {l_1})$ will remain the same at any temperature if:

$A$ brass disc fits snugly in a hole of a steel plate. To loosen the disc from the hole, the system should be:

$A$ rod of length $2 \ m$ at $0^\circ C$ has a linear expansion coefficient $\alpha = (3x + 2) \times 10^{-6} \ ^\circ C^{-1}$,where $x$ is the distance (in $cm$) from one end of the rod. Find the length of the rod at $20^\circ C$ in meters.

An equilateral triangle $ABC$ is formed by two copper rods $AB$ and $BC$ and one aluminium rod $AC$. The system is heated such that the temperature of each rod increases by $\Delta T$. Find the change in the angle $\angle ABC$. (Coefficient of linear expansion for copper is $\alpha_1$ and for aluminium is $\alpha_2$).