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 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:

$A$ meter scale made of steel reads accurately at $25^{\circ} C$. Suppose in an experiment an accuracy of $0.06 \,mm$ in $1 \,m$ is required, the range of temperature in which the experiment can be performed with this meter scale is (Coefficient of linear expansion of steel is $11 \times 10^{-6} /{ }^{\circ} C$)

$A$ hole is drilled in a metal sheet. At $27^{\circ}C$,the diameter of the hole is $5\,cm$. When the sheet is heated to $177^{\circ}C$,the change in the diameter of the hole is $d \times 10^{-3}\,cm$. The value of $d$ will be $...........$ if the coefficient of linear expansion of the metal is $1.6 \times 10^{-5} /^{\circ}C$.

$A$ circular metallic ring of radius $R$ has a small gap of width $d$. The coefficient of thermal expansion of the metal is $\alpha$. If we increase the temperature of the ring by an amount $\Delta T$,then the width of the gap:

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