The gravitational field in a region is given by `vecg=(5hati+12hatj)"N kg"^(-1)`. The change in the gravitational potential energy of a particle of mass 2kg when it is taken from the origin to a point `(7m, -3m)` is

Options

A. 71 J
B. `13sqrt(58)J`
C. 2 J
D. 1 J

Answer

To solve the problem, we need to find the change in gravitational potential energy of a particle of mass 2 kg when it is moved from the origin (0, 0) to the point (7 m, -3 m) in a gravitational field given by \(\vec{g} = (5 \hat{i} + 12 \hat{j}) \, \text{N kg}^{-1}\).

### Step-by-Step Solution:

1. **Identify the Gravitational Field and Mass:**
The gravitational field is given as:
\[ \vec{g} = 5 \hat{i} + 12 \hat{j} \, \text{N kg}^{-1} \]
The mass of the particle is:
\[ m = 2 \, \text{kg} \]

2. **Determine the Displacement Vector:**
The displacement vector \(\Delta \vec{r}\) from the origin (0, 0) to the point (7 m, -3 m) is:
\[ \Delta \vec{r} = (7 \hat{i} - 3 \hat{j}) \, \text{m} \]

3. **Calculate the Change in Gravitational Potential (\(\Delta V\)):**
The change in gravitational potential is given by the dot product of the gravitational field and the displacement vector:
\[ \Delta V = \vec{g} \cdot \Delta \vec{r} \]
Substituting the values:
\[ \Delta V = (5 \hat{i} + 12 \hat{j}) \cdot (7 \hat{i} - 3 \hat{j}) \]
Calculating the dot product:
\[ \Delta V = (5 \cdot 7) + (12 \cdot -3) = 35 - 36 = -1 \, \text{J} \]

4. **Calculate the Change in Potential Energy (\(\Delta U\)):**
The change in gravitational potential energy is given by:
\[ \Delta U = m \Delta V \]
Substituting the values:
\[ \Delta U = 2 \, \text{kg} \cdot (-1 \, \text{J}) = -2 \, \text{J} \]

### Final Answer:
The change in gravitational potential energy of the particle when moved from the origin to the point (7 m, -3 m) is:
\[ \Delta U = -2 \, \text{J} \]