When the temperature of a wire of length $L_0$ is increased by $T$,what is the energy density? The volume expansion coefficient of the wire is $\gamma$ and the Young's modulus is $Y$.

$A$ steel wire and a copper wire are joined end to end having equal cross-sections. The elongation of the two wires is found to be equal under tension. What is the ratio of the length of the steel wire to the length of the copper wire? (Young's modulus of steel $= 2.0 \times 10^{11} \ N \ m^{-2}$ and Young's modulus of copper $= 1.1 \times 10^{11} \ N \ m^{-2}$)

$A$ sphere of mass $2 \,kg$ and diameter $4.5 \,cm$ is attached to the lower end of a steel wire of $2 \,m$ length and area of cross-section $0.24 \times 10^{-6} \,m^2$. The wire is suspended from a $205 \,cm$ high ceiling of a room. When the system is made to oscillate as a simple pendulum, the sphere just grazes the floor at its lowest position. Find the velocity of the sphere at the lowest position. (Young's modulus of steel $= 2 \times 10^{11} \,Nm^{-2}$ and acceleration due to gravity $= 10 \,ms^{-2}$) (in $\,ms^{-1}$)

$A$ wire of length $2 \, m$ is made from copper having a volume of $10 \, cm^3$. When a force $F$ is applied,the extension in its length is $2 \, mm$. If a wire of length $8 \, m$ is made from the same volume of copper,what will be the extension in its length in $cm$ when the same force $F$ is applied?

$A$ copper wire of length $1.0\, m$ and a steel wire of length $0.5\, m$ having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by $1\, mm$. If the Young's moduli of copper and steel are respectively $1.0 \times 10^{11}\, N/m^2$ and $2.0 \times 10^{11}\, N/m^2$,the total extension of the composite wire is ........ $mm$.

$A$ copper wire $(Y = 1 \times 10^{11} \, N/m^2)$ of length $6 \, m$ and a steel wire $(Y = 2 \times 10^{11} \, N/m^2)$ of length $4 \, m$,each of cross-sectional area $10^{-5} \, m^2$,are fastened end to end and stretched by a tension of $100 \, N$. The elongation produced in the copper wire is ......... $mm$.