The ratio of Young's modulus of the material of two wires is $2 : 3.$ If the same stress is applied on both,then the ratio of elastic energy per unit volume will be

If the temperature of a wire of length $2 \,m$ and area of cross-section $1 \,cm^2$ is increased from $0^{\circ}C$ to $80^{\circ}C$ and is not allowed to increase in length,then the force required for it is ............$N$ $\{Y=10^{10} \,N/m^2, \alpha=10^{-6}/^{\circ}C\}$

$A$ steel ring of radius $r$ and cross-sectional area $A$ is fitted onto a wooden disc of radius $R$ $(R > r)$. If Young's modulus is $Y$,then the force with which the steel ring is expanded is

$A$ rigid massless rod of length $6L$ is suspended horizontally by means of two elastic rods $PQ$ and $RS$ as shown in the figure. Their area of cross-section,Young's modulus,and lengths are mentioned in the figure. Find the deflection of end $S$ in the equilibrium state. The free end of the rigid rod is pushed down by a constant force $F$. $A$ is the area of cross-section,$Y$ is Young's modulus of elasticity.

If the ratio of lengths,radii,and Young's moduli of steel and brass wires in the figure are $a, b,$ and $c$ respectively,then the corresponding ratio of increase in their lengths is

Two metallic wires $P$ and $Q$ have the same volume and are made up of the same material. If their areas of cross-section are in the ratio $4:1$ and a force $F_1$ is applied to $P$,an extension of $\Delta l$ is produced. The force required to produce the same extension in $Q$ is $F_2$. The value of $\frac{F_1}{F_2}$ is . . . . . . .