A pendulum clock loses $12 \mathrm{s}$ a day if the temperature is $40°\mathrm{C}$ and goes fast by $4 \mathrm{s}$ a day if the temperature is $20°\mathrm{C}$. Find the temperature at which the clock will show correct time and the coefficient of linear expansion of the metal of the pendulum shaft.

A brass wire $1.8 \mathrm{m}$ long at $27 °C$ is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of $-39 °C$, what is the tension developed in the wire, if its diameter is $2.0 \mathrm{mm}$? Co-efficient of linear expansion of brass is $2.0\times {10}^{-5} {\mathrm{K}}^{-1}$; Young’s modulus of brass is $0.91\times 10 \mathrm{Pa}$.

A rod has length $2 \mathrm{m}$ at a temperature of $20°\mathrm{C}$. Find the free expansion of the rod, if the temperature is increased to $50°\mathrm{C}$, then find stress produced when the rod is fully prevented to expand. $Y=2\times {10}^{11} {\mathrm{Nm}}^{-2}; \alpha =15\times {10}^{-6}/{}^{\circ }\mathrm{C}$

A rod has length $2 \mathrm{m}$ at a temperature of $20°\mathrm{C}$. Find the free expansion of the rod, if the temperature is increased to $50°\mathrm{C}$, then find stress produced when the rod is permitted to expand by $0.4 \mathrm{mm}$. $Y=2\times {10}^{11} {\mathrm{Nm}}^{-2}; \alpha =15\times {10}^{-6}/{}^{\circ }\mathrm{C}$

Two rods of different metals having the same area of cross section $A$ are placed between the two massive walls as shown in the given figure. The first rod has a length $l_1$, coefficient of linear expansion ${\alpha }_1$ and Young's modulus $Y_1$. The corresponding quantities for second rod are $l_2,{\alpha }_2$ and $Y_2$. The temperature of both the rods is now raised by $t°\mathrm{C}$.

(a) Find the force with which the rods act on each other (at higher temperature) in terms of given quantities.

(b) Also find the length of the rods at higher temperature.

A steel ball initially at a pressure $1.0\times {10}^5 \mathrm{Pa}$ is heated from $20°\mathrm{C}$ to $120°\mathrm{C}$ keeping its volume constant. Find the pressure inside the ball.

The coefficient of linear expansion of steel is $\left(12\times {10}^{-6}\right)°{\mathrm{C}}^{-1}$.

Bulk modulus of steel is $1.6\times {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$.

A composite rod is made by joining a copper rod, end to end, with a second rod of different material but of the same area of cross section. At $25°\mathrm{C}$, the composite rod is $1 \mathrm{m}$ long and the copper rod is $30 \mathrm{cm}$ long. At $125°\mathrm{C}$ the length of the composite rod increases by $1.91 \mathrm{mm}$. When the composite rod is prevented from expanding by holding it between two rigid walls, it is found that the constituent rods have remained unchanged in length in spite of rise of temperature. Find Young's modulus and the coefficient of linear expansion of the second rod ($Y$ of copper $=1.3\times {10}^{10} \mathrm{N} {\mathrm{m}}^{-2}$ and $\alpha$ of copper $=17\times {10}^{-6} {\mathrm{K}}^{-1}$ ).

At room temperature $\left(25°\mathrm{C}\right)$ the length of a steel rod is measured using a brass centimetre scale. The measured length is $20 \mathrm{cm}$. If the scale is calibrated to read accurately at temperature $0°\mathrm{C}$, find the actual length of steel rod at room temperature.

A steel tape $1 m$ long is correctly calibrated for a temperature of $27.0{ }^{o}C$. 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 $45.0{ }^{o}C$. 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 $27.0{ }^{o}C$? Coefficient of linear expansion of steel $=1.20\times {10}^{-5} K^{-1}$.