The gravitational field in a region is given by `vec(g)=(2hat(i)+3hat(j))` N/kg. The work done in moving a particle of mass 1 kg from (1, 1) to `(2, (1)/(3))` along the line 3y + 2x = 5 is

Options

A. Zero
B. 20 J
C. `-15 J`
D. 18 J

Answer

To solve the problem of finding the work done in moving a particle of mass 1 kg in a gravitational field, we will follow these steps:

### Step 1: Understand the gravitational field
The gravitational field is given by:
\[ \vec{g} = 2\hat{i} + 3\hat{j} \text{ N/kg} \]
This means that the gravitational force acting on a mass \( m \) can be calculated using:
\[ \vec{F} = m \vec{g} \]
For a mass of 1 kg, the force is:
\[ \vec{F} = 1 \cdot (2\hat{i} + 3\hat{j}) = 2\hat{i} + 3\hat{j} \text{ N} \]

### Step 2: Identify the path of movement
The particle is moved from the point (1, 1) to the point \((2, \frac{1}{3})\) along the line defined by the equation:
\[ 3y + 2x = 5 \]
We can check if both points lie on this line:
- For point (1, 1):
\[ 3(1) + 2(1) = 3 + 2 = 5 \quad \text{(True)} \]
- For point \((2, \frac{1}{3})\):
\[ 3\left(\frac{1}{3}\right) + 2(2) = 1 + 4 = 5 \quad \text{(True)} \]

### Step 3: Calculate the displacement vector
The displacement vector \(\vec{d}\) from point (1, 1) to point (2, \(\frac{1}{3}\)) is:
\[ \vec{d} = (2 - 1)\hat{i} + \left(\frac{1}{3} - 1\right)\hat{j} = 1\hat{i} - \frac{2}{3}\hat{j} \]

### Step 4: Calculate the work done
The work done \(W\) in moving the particle in a gravitational field is given by the dot product of the force and the displacement:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the values:
\[ W = (2\hat{i} + 3\hat{j}) \cdot (1\hat{i} - \frac{2}{3}\hat{j}) \]
Calculating the dot product:
\[ W = 2 \cdot 1 + 3 \cdot \left(-\frac{2}{3}\right) = 2 - 2 = 0 \text{ J} \]

### Conclusion
The work done in moving the particle from (1, 1) to \((2, \frac{1}{3})\) along the line \(3y + 2x = 5\) is:
\[ \boxed{0 \text{ J}} \]