One end of a metal wire is fixed to a ceiling and a load of $2 \ kg$ hangs from the other end. $A$ similar wire is attached to the bottom of the load and another load of $1 \ kg$ hangs from this lower wire. Then the ratio of longitudinal strain of the upper wire to that of the lower wire will be . . . . . . .
[Area of cross section of wire $= 0.005 \ cm^2$,$Y = 2 \times 10^{11} \ Nm^{-2}$ and $g = 10 \ ms^{-2}$]

$A$ mild steel wire of length $2l$ meter and cross-sectional area $A \; m^2$ is fixed horizontally between two pillars. $A$ small mass $m \; kg$ is suspended from the midpoint of the wire. If the extension in the wire is within the elastic limit,then the depression $x$ at the midpoint of the wire will be:

$A$ steel ring of radius '$r$' is to be fitted over a wooden disc of radius '$R$' $(R > r)$. The force required to expand the ring so that it fits over the disc is ($Y =$ Young's modulus of steel,$A =$ area of cross-section of the wire).

$A$ metal rod is fixed rigidly at two ends so as to prevent its thermal expansion. If $L$,$\alpha$,and $Y$ denote the length of the rod,the coefficient of linear thermal expansion,and the Young's modulus of its material respectively,then for an increase in temperature of the rod by $\Delta T$,the longitudinal stress developed in the rod is

Two metal wires $A$ and $B$ have lengths $L$ and $3L$ respectively. The radii of the cross-sectional circular areas of wires $A$ and $B$ are $R$ and $2R$,respectively. These wires are joined end-to-end along their axis. When one end of the combined system is fixed and the other end is pulled with a constant force $F$,the elongation in both wires is equal. If $Y_A$ and $Y_B$ are the Young's moduli of wires $A$ and $B$,respectively,then the ratio $Y_B / Y_A$ is:

Which of the following graphs correctly depicts the spring constant $k$ versus the length $l$ of the spring?