The gravitational potential in a region by V = (20x + 40y)j/kg. Find out the gravitational field (in newton /kg) at a point co-ordinates (2,4). Also find out the magnitude of the gravitational force on a particle of 0.250 kg placed at the point (2,4) .

Answer

To solve the given problem step by step, we will first find the gravitational field from the gravitational potential and then calculate the gravitational force on the particle.

### Step 1: Identify the Gravitational Potential
The gravitational potential \( V \) is given by:
\[ V = (20x + 40y) \, \text{J/kg} \]

### Step 2: Calculate the Gravitational Field
The gravitational field \( \mathbf{g} \) can be determined using the formula:
\[ \mathbf{g} = -\nabla V \]
Where \( \nabla V \) is the gradient of the potential \( V \).

The gradient in Cartesian coordinates is given by:
\[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]

### Step 3: Compute the Partial Derivatives
1. Calculate \( \frac{\partial V}{\partial x} \):
\[ \frac{\partial V}{\partial x} = 20 \, \text{J/kg} \]

2. Calculate \( \frac{\partial V}{\partial y} \):
\[ \frac{\partial V}{\partial y} = 40 \, \text{J/kg} \]

3. Since there is no \( z \) term in \( V \), we have:
\[ \frac{\partial V}{\partial z} = 0 \, \text{J/kg} \]

### Step 4: Formulate the Gravitational Field
Now substituting the partial derivatives into the gradient:
\[ \nabla V = (20, 40, 0) \, \text{J/kg} \]
Thus, the gravitational field is:
\[ \mathbf{g} = -\nabla V = (-20, -40, 0) \, \text{N/kg} \]

### Step 5: Evaluate the Gravitational Field at Point (2, 4)
The gravitational field at the point (2, 4) is:
\[ \mathbf{g} = -20 \hat{i} - 40 \hat{j} \, \text{N/kg} \]

### Step 6: Calculate the Gravitational Force on the Particle
The gravitational force \( \mathbf{F} \) on a particle of mass \( m = 0.250 \, \text{kg} \) is given by:
\[ \mathbf{F} = m \mathbf{g} \]
Substituting the values:
\[ \mathbf{F} = 0.250 \, \text{kg} \cdot (-20 \hat{i} - 40 \hat{j}) \, \text{N/kg} \]
\[ \mathbf{F} = (-5 \hat{i} - 10 \hat{j}) \, \text{N} \]

### Step 7: Calculate the Magnitude of the Gravitational Force
The magnitude of the force \( |\mathbf{F}| \) is given by:
\[ |\mathbf{F}| = \sqrt{(-5)^2 + (-10)^2} \]
\[ |\mathbf{F}| = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \, \text{N} \]

### Final Answers
- The gravitational field at point (2, 4) is \( \mathbf{g} = -20 \hat{i} - 40 \hat{j} \, \text{N/kg} \).
- The magnitude of the gravitational force on the particle is \( 5\sqrt{5} \, \text{N} \).