In a thermostat, two metal strips are used, which have different ............

$A$ hole of diameter $5 \ cm$ is drilled in a metal sheet at $30^{\circ} C$. The coefficient of linear expansion of the metal is $2 \times 10^{-5} K^{-1}$. The diameter of the hole when the temperature is raised to $230^{\circ} C$ is equal to: (in $cm$)

The gap between any two rails, each of length $l$ laid on a railway track, is equal to $x$ at $27\,^{\circ}C$. When the temperature rises to $40\,^{\circ}C$, the gap closes up. If the coefficient of linear expansion of the material of the rail is $\alpha$, the length $l$ of a rail at $27\,^{\circ}C$ is:

$A$ steel rod at $25^{\circ} C$ is observed to be $1 \ m$ long when measured by another metal scale which is correct at $0^{\circ} C$. The exact length of the steel rod at $0^{\circ} C$ is $(\alpha_{\text{steel}} = 12 \times 10^{-6} \ ^{\circ} C^{-1}, \alpha_{\text{metal}} = 20 \times 10^{-6} \ ^{\circ} C^{-1})$. (in $m$)

Obtain the relation between the coefficient of volume expansion $(\alpha_V)$ and the coefficient of linear expansion $(\alpha_l)$.

$A$ steel tape $1 \; m$ long is correctly calibrated for a temperature of $27.0 \; ^{\circ}C$. The length of a steel rod measured by this tape is found to be $63.0 \; cm$ on a hot day when the temperature is $45.0 \; ^{\circ}C$. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is $27.0 \; ^{\circ}C$? (Coefficient of linear expansion of steel $\alpha = 1.20 \times 10^{-5} \; K^{-1}$)