The volume of a metal sphere increases by $0.33 \%$ when its temperature is raised by $50^{\circ} C$. The coefficient of linear expansion of the metal is

We would like to make a vessel whose volume does not change with temperature. We can use brass and iron $\left( {{\gamma _{{\text{brass}}}} = 6 \times {{10}^{ - 5}}/K} \right.$ and $\left. {{\gamma _{{\text{iron}}}} = 3.55 \times {{10}^{ - 5}}/K} \right)$ to create a volume of $100 \, cc$. How can you achieve this?

$A$ circular copper ring at $30^{\circ} C$ has a hole with an area of $9.98 \ cm^2$. It is made to slip onto a steel rod of cross-sectional area of $10 \ cm^2$,by raising the temperature of both the ring and the rod simultaneously by an amount $\Delta T$. If the coefficients of linear expansion of copper and steel are $17 \times 10^{-6} /{ }^{\circ} C$ and $11 \times 10^{-6} /{ }^{\circ} C$ respectively,then the minimum value of $\Delta T$ should be: (in $^{\circ} C$)

If two rods of length $L$ and $2L$ having coefficients of linear expansion $\alpha$ and $2\alpha$ respectively are connected so that the total length becomes $3L$,the average coefficient of linear expansion of the composite rod equals:

The temperature of a metal coin is increased by $100^{\circ} C$ and its diameter increases by $0.15 \%$. Its area increases by nearly (in $\%$)

$A$ metal rod having a coefficient of linear expansion $2 \times 10^{-5} /{ }^{\circ} C$ is $0.75 \ m$ long at $45^{\circ} C$. When the temperature rises to $65^{\circ} C$,the increase in length of the rod will be: (in $mm$)