What is the velocity of sound in a perfectly rigid rod? Why?

$A$ rod of length $1000\, mm$ and coefficient of linear expansion $\alpha = 10^{-4} / ^\circ C$ is placed symmetrically between fixed walls separated by $1001\, mm$. The Young's modulus of the rod is $10^{11} N/m^2$. If the temperature is increased by $20^\circ C$,then the stress developed in the rod is ........... $MPa$.

Young's modulus of the material of a wire of length $L$ and cross-sectional area $A$ is $Y$. If the length of the wire is doubled and the cross-sectional area is halved,then the Young's modulus will be:

$A$ metal rod of cross-sectional area $3 \times 10^{-6} \,m^{2}$ is suspended vertically from one end and has a length of $0.4 \,m$ at $100^{\circ} C$. The rod is cooled to $0^{\circ} C$, but prevented from contracting by attaching a mass '$m$' at the lower end. Find the value of '$m$'. (Given: $Y = 10^{11} \,N/m^{2}$, coefficient of linear expansion $\alpha = 10^{-5} /K$, $g = 10 \,m/s^{2}$) (in $\,kg$)

$A$ block of weight $100 \ N$ is suspended by copper and steel wires of same cross-sectional area $0.5 \ cm^2$ and lengths $\sqrt{3} \ m$ and $1 \ m$,respectively. Their other ends are fixed on a ceiling as shown in the figure. The angles subtended by the copper and steel wires with the ceiling are $30^{\circ}$ and $60^{\circ}$,respectively. If the elongation in the copper wire is $\Delta \ell_C$ and the elongation in the steel wire is $\Delta \ell_S$,then the ratio $\frac{\Delta \ell_C}{\Delta \ell_S}$ is. . . . . .
[Young's modulus for copper and steel are $1 \times 10^{11} \ N/m^2$ and $2 \times 10^{11} \ N/m^2$ respectively]

$(a)$ $A$ steel wire of mass $\mu$ per unit length with a circular cross-section has a radius of $0.1\,cm$. The wire is of length $10\,m$ when measured lying horizontal and hangs from a hook on the wall. $A$ mass of $25\,kg$ is hung from the free end of the wire. Assuming the wire to be uniform and lateral strains $\ll$ longitudinal strains,find the extension in the length of the wire. The density of steel is $7860\,kg/m^3$ and Young's modulus $Y = 2 \times 10^{11}\,N/m^2$.
$(b)$ If the yield strength of steel is $2.5 \times 10^8\,N/m^2$,what is the maximum weight that can be hung at the lower end of the wire?