Two wires $A$ and $B$ are stretched by the same force. If,for $A$ and $B$,$Y_A: Y_B = 1: 2$,$r_A: r_B = 3: 1$,and $L_A: L_B = 4: 1$,then the ratio of their extension $\left(\frac{\Delta L_A}{\Delta L_B}\right)$ will be .............

$A$ metallic rod $1 \, cm$ long with a square cross-section is heated through $1^{\circ} C$. If Young's modulus of elasticity of the metal is $E$ and the mean coefficient of linear expansion is $\alpha$ per degree Celsius,then the compressional force required to prevent the rod from expanding along its length is: (Neglect the change of cross-sectional area)

$A$ steel wire of length $3.2 \, m$ $(Y_{S} = 2.0 \times 10^{11} \, N/m^{2})$ and a copper wire of length $4.4 \, m$ $(Y_{C} = 1.1 \times 10^{11} \, N/m^{2})$,both of radius $1.4 \, mm$,are connected end to end. When stretched by a load,the net elongation is found to be $1.4 \, mm$. The load applied,in Newtons,is. (Given $\pi = \frac{22}{7}$)

One end of a uniform wire of length $L$ and of weight $W$ is attached rigidly to a point in the roof and a weight $W_1$ is suspended from its lower end. If $A$ is the area of cross-section of the wire,the stress in the wire at a height $3L/4$ from its lower end is

$A$ load $W$ produces an extension of $1 \ mm$ in a thread of radius $r$. If the load is increased to $4W$ and the radius is increased to $2r$,while all other factors remain the same,what will be the new extension in $mm$?

$A$ wire of cross-sectional area $10^{-6} \, m^2$ is stretched such that its length increases by $0.1\%$. If the tension produced in the wire is $1000 \, N$, calculate the Young's modulus of the wire.