Two wire mode of the same material are subjected to force in the ratio of `1:2`. Their lengths are in the ratio `8:1` and diameters in the raio `2:1`. Find the ratio of their extension.

Explanation

To find the ratio of the extensions of two wires made of the same material, we can use the formula for Young's modulus and the relationships given in the problem. Let's break it down step by step.

### Step 1: Understand the given ratios
- The ratio of forces \( F_1 : F_2 = 1 : 2 \)
- The ratio of lengths \( L_1 : L_2 = 8 : 1 \)
- The ratio of diameters \( d_1 : d_2 = 2 : 1 \)

### Step 2: Calculate the radii
Since the diameters are in the ratio \( 2 : 1 \), the radii will also be in the same ratio:
- \( R_1 : R_2 = 2 : 1 \)

### Step 3: Calculate the areas of cross-section
The area of cross-section \( A \) for a wire is given by the formula:
\[ A = \pi R^2 \]
Thus, the areas for the two wires will be:
- \( A_1 = \pi R_1^2 = \pi (2R)^2 = 4\pi R^2 \)
- \( A_2 = \pi R_2^2 = \pi (R)^2 = \pi R^2 \)

Now, the ratio of the areas is:
\[ A_1 : A_2 = 4\pi R^2 : \pi R^2 = 4 : 1 \]

### Step 4: Use Young's modulus relation
Young's modulus \( Y \) is defined as:
\[ Y = \frac{F}{A} \cdot \frac{L}{\Delta L} \]
For both wires, since they are made of the same material, we can equate their Young's modulus:
\[ \frac{F_1}{A_1} \cdot \frac{L_1}{\Delta L_1} = \frac{F_2}{A_2} \cdot \frac{L_2}{\Delta L_2} \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{F_1}{F_2} \cdot \frac{A_2}{A_1} \cdot \frac{L_1}{L_2} \]

### Step 6: Substitute the known ratios
Substituting the known ratios:
- \( \frac{F_1}{F_2} = \frac{1}{2} \)
- \( \frac{A_2}{A_1} = \frac{1}{4} \)
- \( \frac{L_1}{L_2} = \frac{8}{1} \)

Now substituting these values into the equation:
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{1}{2} \cdot \frac{1}{4} \cdot 8 \]

### Step 7: Simplify the expression
Calculating the right-hand side:
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{1}{2} \cdot \frac{1}{4} \cdot 8 = \frac{1 \cdot 8}{2 \cdot 4} = \frac{8}{8} = 1 \]

### Conclusion
Thus, the ratio of their extensions is:
\[ \Delta L_1 : \Delta L_2 = 1 : 1 \]