Two particle of masses `4kg` and `8kg` are kept at `x=-2m` and `x=4m` respectivley. Then, the gravitational field intensity at the origin is

Options

A. `G`
B. `2G`
C. `G//2`
D. `G//4`

Explanation

To find the gravitational field intensity at the origin due to two particles of masses 4 kg and 8 kg located at x = -2 m and x = 4 m respectively, we will follow these steps:

### Step 1: Identify the positions of the masses
- Mass \( m_1 = 4 \, \text{kg} \) is located at \( x = -2 \, \text{m} \).
- Mass \( m_2 = 8 \, \text{kg} \) is located at \( x = 4 \, \text{m} \).

### Step 2: Calculate the distances from the origin
- The distance from the origin to \( m_1 \) is \( r_1 = 2 \, \text{m} \) (since it's at -2 m).
- The distance from the origin to \( m_2 \) is \( r_2 = 4 \, \text{m} \) (since it's at 4 m).

### Step 3: Calculate the gravitational field intensity due to each mass
The gravitational field intensity \( E \) due to a mass \( m \) at a distance \( r \) is given by the formula:
\[ E = \frac{G \cdot m}{r^2} \]
where \( G \) is the gravitational constant.

#### For mass \( m_1 \):
\[ E_1 = \frac{G \cdot 4}{2^2} = \frac{G \cdot 4}{4} = G \]
The direction of \( E_1 \) is towards the mass \( m_1 \), which is to the left (negative x-direction), so:
\[ E_1 = -G \hat{i} \]

#### For mass \( m_2 \):
\[ E_2 = \frac{G \cdot 8}{4^2} = \frac{G \cdot 8}{16} = \frac{G}{2} \]
The direction of \( E_2 \) is towards the mass \( m_2 \), which is to the left (negative x-direction), so:
\[ E_2 = -\frac{G}{2} \hat{i} \]

### Step 4: Calculate the net gravitational field intensity at the origin
The net gravitational field intensity \( E_{\text{net}} \) at the origin is the vector sum of \( E_1 \) and \( E_2 \):
\[ E_{\text{net}} = E_1 + E_2 = -G \hat{i} - \frac{G}{2} \hat{i} \]
\[ E_{\text{net}} = -\left(G + \frac{G}{2}\right) \hat{i} = -\frac{3G}{2} \hat{i} \]

### Final Result
Thus, the gravitational field intensity at the origin is:
\[ E_{\text{net}} = -\frac{3G}{2} \hat{i} \]