Two wires of same diameter of the same material having the length `l` and `2l` If the force `F` is applied on each, the ratio of the work done in two wires will be

Options

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

Explanation

To solve the problem, we need to find the ratio of the work done in two wires of different lengths when the same force is applied to each. Let's denote the lengths of the wires as follows:

- Length of the first wire, \( L_1 = l \)
- Length of the second wire, \( L_2 = 2l \)

### Step 1: Understand the Work Done Formula
The work done \( W \) in stretching a wire can be expressed as:
\[ W = \frac{1}{2} \times F \times \text{(stretched length)} \]
where \( F \) is the force applied.

### Step 2: Determine the Stretched Length
For a wire under tension, the stretched length is proportional to the original length of the wire when the same force is applied. Therefore, if the force \( F \) is constant, the work done will be directly proportional to the length of the wire.

### Step 3: Write the Work Done for Each Wire
- For the first wire (length \( L_1 = l \)):
\[ W_1 = \frac{1}{2} \times F \times l \]

- For the second wire (length \( L_2 = 2l \)):
\[ W_2 = \frac{1}{2} \times F \times (2l) = \frac{1}{2} \times F \times 2l = F \times l \]

### Step 4: Find the Ratio of Work Done
Now, we can find the ratio of the work done in the two wires:
\[ \frac{W_1}{W_2} = \frac{\frac{1}{2} \times F \times l}{F \times l} \]

### Step 5: Simplify the Ratio
The \( F \) and \( l \) terms cancel out:
\[ \frac{W_1}{W_2} = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the work done in the two wires is:
\[ \frac{W_1}{W_2} = 1 : 2 \]

### Final Answer
The ratio of the work done in the two wires is \( 1 : 2 \).

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