The gravitational potential in a region is given by, `V = 200(X + Y) J//kg`. Find the magnitude of the gravitational force on a particle of mass `0.5 kg` placed at the orgin.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Gravitational Potential
The gravitational potential \( V \) is given by:
\[ V = 200(X + Y) \text{ J/kg} \]
This potential is a function of the coordinates \( X \) and \( Y \).

### Step 2: Find the Gravitational Field
The gravitational field \( \mathbf{E} \) can be derived from the gravitational potential using the formula:
\[ \mathbf{E} = -\nabla V \]
where \( \nabla V \) is the gradient of the potential. In Cartesian coordinates, this is given by:
\[ \mathbf{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} \right) \]

### Step 3: Calculate the Partial Derivatives
Now we will calculate the partial derivatives of \( V \):
1. For \( \frac{\partial V}{\partial x} \):
\[ \frac{\partial V}{\partial x} = 200 \]
2. For \( \frac{\partial V}{\partial y} \):
\[ \frac{\partial V}{\partial y} = 200 \]

### Step 4: Substitute into the Gravitational Field Equation
Now substituting these values into the equation for the gravitational field:
\[ \mathbf{E} = -\left(200 \hat{i} + 200 \hat{j}\right) = -200 \hat{i} - 200 \hat{j} \]

### Step 5: Calculate the Gravitational Force
The gravitational force \( \mathbf{F} \) on a mass \( m \) in a gravitational field \( \mathbf{E} \) is given by:
\[ \mathbf{F} = m \mathbf{E} \]
Given that the mass \( m = 0.5 \text{ kg} \):
\[ \mathbf{F} = 0.5 \left(-200 \hat{i} - 200 \hat{j}\right) = -100 \hat{i} - 100 \hat{j} \]

### Step 6: Find the Magnitude of the Gravitational Force
To find the magnitude of the force \( |\mathbf{F}| \):
\[ |\mathbf{F}| = \sqrt{(-100)^2 + (-100)^2} = \sqrt{10000 + 10000} = \sqrt{20000} = 100\sqrt{2} \]

### Final Answer
The magnitude of the gravitational force on the particle of mass \( 0.5 \text{ kg} \) placed at the origin is:
\[ |\mathbf{F}| = 100\sqrt{2} \text{ N} \]