A particle is moved from (0, 0) to (a, a) under a force a `F=(3hat(i)+4hat(j))` from two paths. Path 1 is OP and path 2 is OPQ. Let `W_(1)` and `W_(2)` be the work done by this force in these two paths. Then,

Options

A. `W_(1)=W_(2)`
B. `W_(1)=2W_(2)`
C. `W_(2)=2W_(1)`
D. `W_(2)=4W_(1)`

Answer

To solve the problem, we need to calculate the work done by the force \( \mathbf{F} = 3\hat{i} + 4\hat{j} \) along two different paths from the point \( O(0, 0) \) to the point \( A(a, a) \).

### Step-by-Step Solution:

**Step 1: Define the paths.**
- **Path 1 (OP)**: Directly from \( O(0, 0) \) to \( A(a, a) \).
- **Path 2 (OPQ)**: From \( O(0, 0) \) to \( P(a, 0) \), then to \( Q(a, a) \).

**Step 2: Calculate the work done along Path 1 (OP).**
- The displacement vector for Path 1 is:
\[ \mathbf{d_1} = A - O = (a, a) - (0, 0) = a\hat{i} + a\hat{j} \]
- The work done \( W_1 \) is given by the dot product of the force and the displacement:
\[ W_1 = \mathbf{F} \cdot \mathbf{d_1} = (3\hat{i} + 4\hat{j}) \cdot (a\hat{i} + a\hat{j}) \]
- Calculating the dot product:
\[ W_1 = 3a + 4a = 7a \]

**Step 3: Calculate the work done along Path 2 (OPQ).**
- For Path 2, we break it into two segments: \( OP \) and \( PQ \).

1. **Work done along OP**:
- Displacement vector for OP is the same as before:
\[ \mathbf{d_{OP}} = a\hat{i} + 0\hat{j} \]
- Work done \( W_{OP} \):
\[ W_{OP} = \mathbf{F} \cdot \mathbf{d_{OP}} = (3\hat{i} + 4\hat{j}) \cdot (a\hat{i} + 0\hat{j}) = 3a \]

2. **Work done along PQ**:
- Displacement vector for PQ is:
\[ \mathbf{d_{PQ}} = (a, a) - (a, 0) = 0\hat{i} + a\hat{j} \]
- Work done \( W_{PQ} \):
\[ W_{PQ} = \mathbf{F} \cdot \mathbf{d_{PQ}} = (3\hat{i} + 4\hat{j}) \cdot (0\hat{i} + a\hat{j}) = 4a \]

- Total work done along Path 2 \( W_2 \):
\[ W_2 = W_{OP} + W_{PQ} = 3a + 4a = 7a \]

**Step 4: Compare the work done along both paths.**
- We found that:
\[ W_1 = 7a \quad \text{and} \quad W_2 = 7a \]
- Therefore, we conclude that:
\[ W_1 = W_2 \]

### Final Answer:
The work done along both paths is equal:
\[ W_1 = W_2 \]