At some temperature $T$,a bronze pin is slightly too large to fit into a hole drilled in a steel block. The change in temperature required for an exact fit is minimum when:

The coefficient of linear expansion of a crystal in one direction is ${\alpha _1}$ and that in every direction perpendicular to it is ${\alpha _2}$. The coefficient of cubical expansion is

$A$ 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.

What is areal expansion? Give the definition and unit of the coefficient of areal expansion.

$A$ blacksmith fixes a circular iron frame on the wooden wheel of a bullock cart. The diameters of the wooden wheel and the circular iron frame are $5.012 \ m$ and $5 \ m$ respectively at $27^{\circ} C$. The temperature (in $^{\circ} C$) to which the iron ring must be heated so as to fit the wooden wheel is (Coefficient of linear expansion of iron $= 1.2 \times 10^{-5} \ ^{\circ} C^{-1}$).

$A$ copper rod of length $l_1$ and an iron rod of length $l_2$ are always maintained at the same common temperature $T$. If the difference $(l_2 - l_1)$ is $15\,cm$ and is independent of the value of $T$,then $l_1$ and $l_2$ have the values (given the linear coefficients of expansion for copper and iron are $\alpha_c = 2.0 \times 10^{-6}\,^{\circ}C^{-1}$ and $\alpha_i = 1.0 \times 10^{-6}\,^{\circ}C^{-1}$ respectively).