$A$ wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$,its length increases by $l$. Another wire of the same material of length $2L$ and radius $2r$ is pulled by a force $2f$. Then the increase in its length will be

$A$ rigid bar of mass $15 \, kg$ is supported symmetrically by three wires,each $2 \, m$ long. The wires at each end are made of copper,and the middle one is made of steel. The Young's modulus of elasticity for copper and steel are $110 \times 10^9 \, N/m^2$ and $190 \times 10^9 \, N/m^2$ respectively. If each wire is to have the same tension,the ratio of their diameters (diameter of copper wire to diameter of steel wire) will be ............

$A$ structural steel rod has a radius of $10 \;mm$ and a length of $1.0 \;m$. $A$ $100 \;kN$ force stretches it along its length. Calculate $(a)$ stress,$(b)$ elongation,and $(c)$ strain on the rod. Young's modulus of structural steel is $2.0 \times 10^{11} \;N \;m^{-2}$.

$A$ wire elongates by $l \ mm$ when a load $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends,the elongation of the wire will be (in $mm$)

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:

$A$ 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 ......... $mm$.