One end of a steel wire of radius $r$ is fixed to a ceiling and a load of $3 \ kg$ is attached to the free end of the wire. Another wire made of copper of radius $2r$ is attached to the bottom of the $3 \ kg$ load and a $2 \ kg$ load is attached to the free end of the copper wire. The ratio of longitudinal strains produced in copper and steel wires is (Young modulus of steel $= 20 \times 10^{10} \ Nm^{-2}$,Young modulus of copper $= 12 \times 10^{10} \ Nm^{-2}$)

Two wires $A$ and $B$ made of different materials of length $6.0 \ cm$ and $5.4 \ cm$, respectively, and area of cross-sections $3.0 \times 10^{-5} \ m^2$ and $4.5 \times 10^{-5} \ m^2$, respectively, are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is . . . . . . . . . . .

In the determination of Young's modulus $\left(Y=\frac{4 MLg}{\pi \ell d^2}\right)$ by using Searle's method,a wire of length $L=2 \ m$ and diameter $d=0.5 \ mm$ is used. For a load $M=2.5 \ kg$,an extension $\ell=0.25 \ mm$ in the length of the wire is observed. Quantities $d$ and $\ell$ are measured using a screw gauge and a micrometer,respectively. They have the same pitch of $0.5 \ mm$. The number of divisions on their circular scale is $100$. The contributions to the maximum probable error of the $Y$ measurement:

In Young's experiment,if the length of the wire and the radius are both doubled,then the value of $Y$ will become:

$A$ wire of length $L$ and radius $r$ is rigidly fixed at one end. On stretching the other end of the wire with a force $F$,the increase in its length is $l$. If another wire of same material but of length $2L$ and radius $2r$ is stretched with a force of $2F$,the increase in its length will be

$A$ wire is loaded by $6 \ kg$ at its one end,the increase in length is $12 \ mm$. If the radius of the wire is doubled and all other magnitudes are unchanged,then the increase in length will be ......... $mm$.