Force constant of two wires `A` and `B` of the same material are `K` and `2K` respectively. If the two wires are stretched equally, then the ratio of work done in stretching `((W_(A))/(W_(B)))` is

Options

A. `(1)/(3)`
B. `(1)/(3)`
C. `(1)/(2)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to find the ratio of work done in stretching two wires A and B, given that their force constants are K and 2K respectively, and they are stretched equally.

### Step-by-Step Solution:

1. **Understand the Work Done Formula**:
The work done (W) in stretching a wire can be expressed as:
\[ W = \frac{1}{2} K x^2 \]
where \( K \) is the force constant (spring constant) and \( x \) is the extension (stretch) of the wire.

2. **Calculate Work Done for Wire A**:
For wire A, the force constant is \( K \). Thus, the work done in stretching wire A (denoted as \( W_A \)) is:
\[ W_A = \frac{1}{2} K x^2 \]

3. **Calculate Work Done for Wire B**:
For wire B, the force constant is \( 2K \). Thus, the work done in stretching wire B (denoted as \( W_B \)) is:
\[ W_B = \frac{1}{2} (2K) x^2 = K x^2 \]

4. **Find the Ratio of Work Done**:
Now, we need to find the ratio of the work done in stretching wire A to the work done in stretching wire B:
\[ \frac{W_A}{W_B} = \frac{\frac{1}{2} K x^2}{K x^2} \]

5. **Simplify the Ratio**:
When we simplify this expression, we can cancel out \( K x^2 \) from the numerator and denominator:
\[ \frac{W_A}{W_B} = \frac{1/2}{1} = \frac{1}{2} \]

6. **Final Ratio**:
Therefore, the ratio of work done in stretching wire A to wire B is:
\[ \frac{W_A}{W_B} = \frac{1}{2} \]

### Conclusion:
The ratio of work done in stretching wire A to wire B is \( 1:2 \).

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