$A$ cylinder has a piston at a temperature of $30^{\circ} C$. There is an all-round clearance of $0.08 \ mm$ between the piston and the cylinder wall. If the internal diameter of the cylinder is $15 \ cm$,what is the temperature at which the piston will fit into the cylinder exactly (in $^{\circ} C$)? $(\alpha_p = 1.6 \times 10^{-5} /^{\circ} C$ and $\alpha_c = 1.2 \times 10^{-5} /^{\circ} C)$

$A$ solid expands upon heating because

Show that the coefficient of area expansion,$(\Delta A / A) / \Delta T,$ of a rectangular sheet of a solid is twice its linear expansivity,$\alpha_{l}$.

The focal length of a spherical mirror made of steel is $150 \,cm$. If the temperature of the mirror increases by $200 \,K$, its focal length becomes (coefficient of linear expansion of steel $\alpha = 12 \times 10^{-6} \,^{\circ}C^{-1}$). (in $\,cm$)

$A$ large steel wheel is to be fitted onto a shaft of the same material. At $27^{\circ}C$,the outer diameter of the shaft is $8.70\;cm$ and the diameter of the central hole in the wheel is $8.69\;cm$. The shaft is cooled using 'dry ice'. At what temperature (in $^{\circ}C$) of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range: $\alpha_{steel} = 1.20 \times 10^{-5}\;K^{-1}$.

$A$ metal rod of length $20 \, cm$ expands by $0.075 \, cm$ when its temperature is raised from $0 \, ^\circ C$ to $100 \, ^\circ C$. Another rod $B$ of a different metal of the same length expands by $0.045 \, cm$ for the same rise in temperature. $A$ third rod of the same length is composed of two metals $A$ and $B$ joined together. This composite rod expands by $0.060 \, cm$ for the same temperature difference. The length of the portion of metal $A$ is ..... $cm$.