A wire is stretched by 0.01 m by a certain force F. Another wire of same material whose diameter and length are double to the original wire is stretched by the same force. Then its elongation will be

The correct option is 0.005 m.

To find the elongation of the second wire, we can use the relation between force, elongation, and the properties of the wire defined by Hooke's Law and Young's Modulus.
Young's Modulus ($Y$) of a wire is given by the formula:
$Y=\frac{\text{Stress}}{\text{Strain}}=\frac{F/A}{\Delta L/L}=\frac{F\cdot L}{A\cdot \Delta L}$
where:
- $F$ is the applied force,
- $L$ is the original length of the wire,
- $A$ is the cross-sectional area of the wire, and
- $\Delta L$ is the elongation (stretch) of the wire.

Since the wire has a circular cross-section of diameter $d$, its area is:
$A=\frac{\pi \cdot d^2}{4}$
Substituting this area into the Young's Modulus formula gives:
$Y=\frac{4\cdot F\cdot L}{\pi \cdot d^2\cdot \Delta L}$

Rearranging this formula to solve for the elongation $\Delta L$:
$\Delta L=\frac{4\cdot F\cdot L}{\pi \cdot d^2\cdot Y}$

Let the parameters of the original wire be $L_1=L$, $d_1=d$, and its elongation $\Delta L_1=0.01\text{m}$.
For the second wire, we are given:
- It is made of the same material, so Young's Modulus remains the same: $Y_2=Y$.
- It is stretched by the same force: $F_2=F$.
- Its length is doubled: $L_2=2L$.
- Its diameter is doubled: $d_2=2d$.

Now, write the expression for the elongation of the second wire, $\Delta L_2$:
$\Delta L_2=\frac{4\cdot F\cdot L_2}{\pi \cdot d_2^2\cdot Y}$
Substitute the values of $L_2$ and $d_2$ into the equation:
$\Delta L_2=\frac{4\cdot F\cdot \left(2L\right)}{\pi \cdot {\left(2d\right)}^2\cdot Y}$
$\Delta L_2=\frac{4\cdot F\cdot 2L}{\pi \cdot 4d^2\cdot Y}$
$\Delta L_2=\frac{2}{4}\cdot \left(\frac{4\cdot F\cdot L}{\pi \cdot d^2\cdot Y}\right)$
$\Delta L_2=\frac{1}{2}\cdot \Delta L_1$

Substitute the initial elongation $\Delta L_1=0.01\text{m}$:
$\Delta L_2=\frac{0.01\text{m}}{2}=0.005\text{m}$

Thus, the elongation of the second wire is 0.005 m.