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 calculating the work done in moving a particle of mass 1 kg in a given gravitational field, we will follow these steps:

### Step 1: Identify the gravitational field and the force acting on the particle
The gravitational field is given by:
\[ \vec{g} = 2\hat{i} + 3\hat{j} \text{ N/kg} \]
The force acting on a particle of mass \( m = 1 \text{ kg} \) can be calculated using the formula:
\[ \vec{F} = m \vec{g} \]
Substituting the values:
\[ \vec{F} = 1 \cdot (2\hat{i} + 3\hat{j}) = 2\hat{i} + 3\hat{j} \text{ N} \]

### Step 2: Determine the displacement vector \( d\vec{r} \)
The particle is moved from the point \( (1, 1) \) to the point \( (2, \frac{1}{3}) \). The displacement vector \( d\vec{r} \) can be calculated as:
\[ d\vec{r} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} \]
Where:
- \( x_1 = 1, y_1 = 1 \)
- \( x_2 = 2, y_2 = \frac{1}{3} \)

Calculating the components:
\[ d\vec{r} = (2 - 1)\hat{i} + \left(\frac{1}{3} - 1\right)\hat{j} = 1\hat{i} - \frac{2}{3}\hat{j} \]

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

### Conclusion
The work done in moving the particle is:
\[ W = 0 \text{ J} \]