On what value does the coefficient of linear expansion $\alpha_l$ depend? Write its unit.

$A$ thin rod having length $L_0$ at $0\,^{\circ}C$ and coefficient of linear expansion $\alpha$ has its two ends maintained at temperatures $\theta_1$ and $\theta_2$,respectively. Find its new length.

Two rods,one of aluminum and another of steel,have initial lengths $ℓ_1$ and $ℓ_2$ respectively. They are joined 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 increase in length of both rods is the same when the temperature is increased by $t \ ^\circ C$,find the ratio $\frac{ℓ_1}{ℓ_1 + ℓ_2}$.

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 $\alpha_2$).

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

$A$ sheet of steel is $40 \ cm$ long and $5 \ cm$ broad at $0^{\circ} C$. The surface area of the sheet increases by $1.4 \ cm^2$ at $100^{\circ} C$. The coefficient of linear expansion of steel is: