Two uniform rods $AB$ and $BC$ have Young's moduli $1.2 \times 10^{11} \, N/m^2$ and $1.5 \times 10^{11} \, N/m^2$ respectively. If the coefficient of linear expansion of $AB$ is $1.5 \times 10^{-5} /^{\circ}C$ and both have equal area of cross-section,then the coefficient of linear expansion of $BC$,for which there is no shift of the junction at all temperatures,is ............. $\times 10^{-5} /^{\circ}C$.

Two wires are made of the same material and have the same volume. The first wire has cross-sectional area $A$ and the second wire has cross-sectional area $3A$. If the length of the first wire is increased by $\Delta l$ on applying a force $F$,how much force is needed to stretch the second wire by the same amount?

$A$ metallic ring of radius $r$ and cross-sectional area $A$ is fitted into a wooden circular disc of radius $R$ $(R > r)$. If the Young's modulus of the material of the ring is $Y$, the force with which the metal ring expands is

$A$ 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]

The length of a metal wire is $L_1$ when the tension is $T_1$ and $L_2$ when the tension is $T_2$. The unstretched length of the wire is:

$A$ steel wire of length $L$ at $40^{\circ} C$ is suspended from the ceiling and then a mass $m$ is hung from its free end. The wire is cooled down from $40^{\circ} C$ to $30^{\circ} C$ to regain its original length $L$. The coefficient of linear thermal expansion of the steel is $\alpha = 10^{-5} /^{\circ} C$,Young's modulus of steel is $Y = 10^{11} N/m^2$,and the radius of the wire is $r = 1 \ mm$. Assume that $L \gg$ diameter of the wire. Then the value of $m$ in $kg$ is nearly: