The length of a wire becomes $l_1$ and $l_2$ when $100\,N$ and $120\,N$ tensions are applied respectively. If $10l_2 = 11l_1$,the natural length of the wire will be $\frac{1}{x} l_1$. Here,the value of $x$ is ........

$A$ sphere of mass $2 \,kg$ and diameter $4.5 \,cm$ is attached to the lower end of a steel wire of $2 \,m$ length and area of cross-section $0.24 \times 10^{-6} \,m^2$. The wire is suspended from a $205 \,cm$ high ceiling of a room. When the system is made to oscillate as a simple pendulum, the sphere just grazes the floor at its lowest position. Find the velocity of the sphere at the lowest position. (Young's modulus of steel $= 2 \times 10^{11} \,Nm^{-2}$ and acceleration due to gravity $= 10 \,ms^{-2}$) (in $\,ms^{-1}$)

$A$ wire of length $2 \, m$ is made from copper having a volume of $10 \, cm^3$. When a force $F$ is applied,the extension in its length is $2 \, mm$. If a wire of length $8 \, m$ is made from the same volume of copper,what will be the extension in its length in $cm$ when the same force $F$ is applied?

$A$ wire of cross-sectional area $A$,modulus of elasticity $2 \times 10^{11} \text{ N m}^{-2}$,and length $2L = 2 \text{ m}$ is stretched between two vertical rigid supports. When a mass of $2 \text{ kg}$ is suspended at the middle,it sags from its original position,making an angle $\theta = \frac{1}{100} \text{ radian}$ with the horizontal at the points of support. The value of $A$ is . . . . . . $\times 10^{-4} \text{ m}^2$. (Given: $g = 10 \text{ m/s}^2$)

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:

Two rods of same material and volume having circular cross-section are subjected to tension $T$. Within the elastic limit,the same force is applied to both the rods. If the diameter of the first rod is half of the second rod,then the ratio of the extension of the first rod to the second rod will be: (in $: 1$)