Gravitational potential in a region is given by `V = -(x+y+z)` J/kg . Find the gravitational intensity at (2,2,2)

Options

A. `(hat(i)+hat(j)+hat(k))`N/kg
B. `2(hat(i)+hat(j)+hat(k))` N/kg
C. `3(hat(i)+hat(j)+hat(k))`N/kg
D. `4(hat(i)+hat(j)+hat(k))`N/kg

Answer

To find the gravitational intensity at the point (2, 2, 2) given the gravitational potential \( V = -(x + y + z) \) J/kg, we can follow these steps:

### Step 1: Understand the relationship between gravitational potential and gravitational intensity
The gravitational intensity \( \vec{g} \) (or gravitational field) is related to the gravitational potential \( V \) by the equation:
\[ \vec{g} = -\nabla V \]
where \( \nabla V \) is the gradient of the potential.

### Step 2: Calculate the gradient of the potential
The gradient \( \nabla V \) in three dimensions is given by:
\[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]

### Step 3: Find the partial derivatives of \( V \)
Given \( V = -(x + y + z) \):
- The partial derivative with respect to \( x \) is:
\[ \frac{\partial V}{\partial x} = -1 \]
- The partial derivative with respect to \( y \) is:
\[ \frac{\partial V}{\partial y} = -1 \]
- The partial derivative with respect to \( z \) is:
\[ \frac{\partial V}{\partial z} = -1 \]

### Step 4: Write the gradient vector
Now, substituting these values into the gradient:
\[ \nabla V = \left( -1, -1, -1 \right) \]

### Step 5: Calculate the gravitational intensity
Using the relationship \( \vec{g} = -\nabla V \):
\[ \vec{g} = -(-1, -1, -1) = (1, 1, 1) \]

### Step 6: Write the final answer
Thus, the gravitational intensity at the point (2, 2, 2) is:
\[ \vec{g} = \hat{i} + \hat{j} + \hat{k} \text{ N/kg} \]

### Summary of the solution
The gravitational intensity at the point (2, 2, 2) is \( \hat{i} + \hat{j} + \hat{k} \) N/kg.

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