Two wires A and B are of the same maeterial. Their lengths are in the ratio 1 : 2 and the diameters are in the ratio 2 : 1. IF they are pulled by the same force, their increases in length will be in the ratio

Options

A. `2 : 1`
B. ` 1: 4`
C. ` 1 : 8`
D. `8 : 1`

Explanation

To solve the problem of finding the ratio of increases in length of two wires A and B, we will follow these steps:

### Step 1: Understand the given ratios
We are given:
- The lengths of wires A and B are in the ratio \( L_1 : L_2 = 1 : 2 \).
- The diameters of wires A and B are in the ratio \( D_1 : D_2 = 2 : 1 \).

### Step 2: Convert diameter ratio to radius ratio
Since the radius is half of the diameter, the radius ratio will be:
\[ R_1 : R_2 = \frac{D_1}{2} : \frac{D_2}{2} = 2 : 1 \quad \Rightarrow \quad R_1 : R_2 = 2 : 1 \]
Thus, we can write:
\[ \frac{R_1}{R_2} = 2 : 1 \]

### Step 3: Use Young's modulus formula
The increase in length (\( \Delta L \)) of a wire can be expressed using the formula:
\[ \Delta L = \frac{F \cdot L}{Y \cdot A} \]
where:
- \( F \) is the force applied,
- \( L \) is the original length of the wire,
- \( Y \) is Young's modulus (constant for the same material),
- \( A \) is the cross-sectional area of the wire.

For a circular cross-section, the area \( A \) can be expressed as:
\[ A = \pi R^2 \]

### Step 4: Write the equations for both wires
For wire A:
\[ \Delta L_1 = \frac{F \cdot L_1}{Y \cdot A_1} = \frac{F \cdot L_1}{Y \cdot \pi R_1^2} \]
For wire B:
\[ \Delta L_2 = \frac{F \cdot L_2}{Y \cdot A_2} = \frac{F \cdot L_2}{Y \cdot \pi R_2^2} \]

### Step 5: Find the ratio of increases in length
Now we find the ratio \( \frac{\Delta L_1}{\Delta L_2} \):
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{\frac{F \cdot L_1}{Y \cdot \pi R_1^2}}{\frac{F \cdot L_2}{Y \cdot \pi R_2^2}} = \frac{L_1 \cdot R_2^2}{L_2 \cdot R_1^2} \]
Since \( F \), \( Y \), and \( \pi \) cancel out.

### Step 6: Substitute the known ratios
Substituting \( L_1 : L_2 = 1 : 2 \) and \( R_1 : R_2 = 2 : 1 \):
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{1 \cdot (1)^2}{2 \cdot (2)^2} = \frac{1}{2 \cdot 4} = \frac{1}{8} \]

### Conclusion
Thus, the ratio of increases in length of wires A and B is:
\[ \Delta L_1 : \Delta L_2 = 1 : 8 \]

### Final Answer
The increases in length will be in the ratio \( 1 : 8 \).

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