A particle is acted upon by only a conservative force `F=(7hat(i)-6hat(j))` N (no other force is acting on the particle). Under the influence of this force particle moves from (0, 0) to (-3m, 4m) then

Options

A. work done by the force is 3 J
B. work done by the force is `-45 J`
C. at (0, 0) speed of the particle must be zero
D. at (0, 0) speed of the particle must not be zero

Answer

To solve the problem step by step, we need to calculate the work done by the conservative force as the particle moves from the initial position (0, 0) to the final position (-3 m, 4 m).

### Step 1: Identify the Initial and Final Positions
- Initial Position \( \mathbf{r_i} = (0, 0) \)
- Final Position \( \mathbf{r_f} = (-3, 4) \)

### Step 2: Calculate the Displacement Vector
The displacement vector \( \mathbf{x} \) can be calculated using the formula:
\[ \mathbf{x} = \mathbf{r_f} - \mathbf{r_i} \]
Substituting the values:
\[ \mathbf{x} = (-3, 4) - (0, 0) = (-3, 4) \]
In vector notation, this can be expressed as:
\[ \mathbf{x} = -3 \hat{i} + 4 \hat{j} \]

### Step 3: Write the Force Vector
The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 7 \hat{i} - 6 \hat{j} \]

### Step 4: Calculate the Work Done
The work done \( W \) by the force when moving through the displacement can be calculated using the dot product:
\[ W = \mathbf{F} \cdot \mathbf{x} \]
Substituting the vectors:
\[ W = (7 \hat{i} - 6 \hat{j}) \cdot (-3 \hat{i} + 4 \hat{j}) \]
Calculating the dot product:
\[ W = (7 \cdot -3) + (-6 \cdot 4) \]
\[ W = -21 - 24 \]
\[ W = -45 \text{ Joules} \]

### Final Answer
The work done by the force is:
\[ W = -45 \text{ Joules} \]