To find the work done on a particle by a force as it moves from one point to another, we can use the formula for work done, which is given by the dot product of the force vector and the displacement vector. Here’s a step-by-step solution to the problem:
### Step 1: Identify the Force Vector
The force acting on the particle is given as:
\[ \mathbf{F} = 3x^2 \hat{i} + 4 \hat{j} \]
where \( x \) is in meters.
### Step 2: Determine the Initial and Final Positions
The particle moves from the coordinates:
- Initial position \( (x_1, y_1) = (2 \, \text{m}, 3 \, \text{m}) \)
- Final position \( (x_2, y_2) = (3 \, \text{m}, 0 \, \text{m}) \)
### Step 3: Calculate the Displacement Vector
The displacement vector \( \mathbf{dr} \) can be expressed as:
\[ \mathbf{dr} = dx \hat{i} + dy \hat{j} \]
where \( dx = x_2 - x_1 = 3 - 2 = 1 \, \text{m} \) and \( dy = y_2 - y_1 = 0 - 3 = -3 \, \text{m} \).
### Step 4: Substitute the Values into the Work Done Formula
The work done \( W \) is given by the integral of the force along the path of displacement:
\[ W = \int \mathbf{F} \cdot \mathbf{dr} \]
### Step 5: Express the Dot Product
Substituting \( \mathbf{F} \) and \( \mathbf{dr} \):
\[ W = \int (3x^2 \hat{i} + 4 \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) \]
Calculating the dot product:
\[ W = \int (3x^2 dx + 4 dy) \]
### Step 6: Set Up the Integrals
We will integrate each term separately. The limits for \( x \) will be from 2 to 3 and for \( y \) from 3 to 0.
1. For the \( 3x^2 \) term:
\[ W_1 = \int_{2}^{3} 3x^2 \, dx \]
2. For the \( 4 \) term:
\[ W_2 = \int_{3}^{0} 4 \, dy \]
### Step 7: Calculate the Integrals
1. Calculate \( W_1 \):
\[
W_1 = 3 \left[ \frac{x^3}{3} \right]_{2}^{3} = [27 - 8] = 19 \, \text{J}
\]
2. Calculate \( W_2 \):
\[
W_2 = 4 \left[ y \right]_{3}^{0} = 4(0 - 3) = -12 \, \text{J}
\]
### Step 8: Combine the Results
Now, combine the results of both integrals:
\[ W = W_1 + W_2 = 19 - 12 = 7 \, \text{J} \]
### Final Answer
The total work done on the particle as it moves from (2m, 3m) to (3m, 0m) is:
\[ \boxed{7 \, \text{J}} \]
