Two particles of masses 4kg and 8kg are kept at x=-2m and x=4m respectively. Then the net gravitational force acting on a third particle of mass 1kg kept at the origin is

Options

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

Answer

To solve the problem, we need to calculate the gravitational force acting on a third particle of mass 1 kg located at the origin due to two other particles of masses 4 kg and 8 kg located at x = -2 m and x = 4 m, respectively.

### Step-by-Step Solution:

1. **Identify the Positions and Masses**:
- Mass \( m_1 = 4 \, \text{kg} \) is located at \( x_1 = -2 \, \text{m} \).
- Mass \( m_2 = 8 \, \text{kg} \) is located at \( x_2 = 4 \, \text{m} \).
- The third mass \( m_3 = 1 \, \text{kg} \) is at the origin \( x = 0 \, \text{m} \).

2. **Calculate the Gravitational Force from \( m_1 \)**:
- The distance between \( m_1 \) and \( m_3 \) is \( d_1 = |0 - (-2)| = 2 \, \text{m} \).
- The gravitational force \( F_1 \) exerted by \( m_1 \) on \( m_3 \) is given by Newton's law of gravitation:
\[ F_1 = \frac{G \cdot m_1 \cdot m_3}{d_1^2} = \frac{G \cdot 4 \cdot 1}{2^2} = \frac{G \cdot 4}{4} = G \]
- The direction of this force is towards \( m_1 \) (to the left).

3. **Calculate the Gravitational Force from \( m_2 \)**:
- The distance between \( m_2 \) and \( m_3 \) is \( d_2 = |0 - 4| = 4 \, \text{m} \).
- The gravitational force \( F_2 \) exerted by \( m_2 \) on \( m_3 \) is:
\[ F_2 = \frac{G \cdot m_2 \cdot m_3}{d_2^2} = \frac{G \cdot 8 \cdot 1}{4^2} = \frac{G \cdot 8}{16} = \frac{G}{2} \]
- The direction of this force is towards \( m_2 \) (to the right).

4. **Determine the Net Gravitational Force**:
- The net gravitational force \( F_{\text{net}} \) acting on the mass at the origin is the vector sum of \( F_1 \) and \( F_2 \):
\[ F_{\text{net}} = F_1 - F_2 = G - \frac{G}{2} = \frac{G}{2} \]
- The net force is directed towards the left (towards the 4 kg mass).

5. **Final Answer**:
- The net gravitational force acting on the 1 kg mass at the origin is \( \frac{G}{2} \).