Mass particles of 1 kg each are placed along x-axus at `x=1, 2, 4, 8, .... oo`. Then gravitational force o a mass of 3 kg placed at origin is (G= universal gravitation constant) :-

Options

A. `4G`
B. `(4G)/3`
C. `2G`
D. `oo`

Answer

To solve the problem, we need to calculate the gravitational force exerted on a mass of 3 kg placed at the origin (x = 0) by an infinite series of 1 kg masses located at positions along the x-axis at x = 1, 2, 4, 8, ..., and so on.

### Step-by-Step Solution:

1. **Identify the Positions of the Masses:**
The masses are located at x = 1, 2, 4, 8, ..., which can be represented as \( x_n = 2^{n-1} \) for \( n = 1, 2, 3, ... \).

2. **Calculate the Gravitational Force by One Mass:**
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
Here, \( m_1 = 1 \, \text{kg} \) (mass at position \( x_n \)), \( m_2 = 3 \, \text{kg} \) (mass at the origin), and \( r = x_n \).

3. **Express the Force for Each Mass:**
For the mass at position \( x_n \):
\[ F_n = \frac{G \cdot 1 \cdot 3}{(2^{n-1})^2} = \frac{3G}{4^{n-1}} \]

4. **Sum the Forces from All Masses:**
Since the masses are located along the positive x-axis, the forces will be attractive towards the origin. The total gravitational force \( F \) on the 3 kg mass at the origin is the sum of the forces from all the 1 kg masses:
\[ F = \sum_{n=1}^{\infty} F_n = \sum_{n=1}^{\infty} \frac{3G}{4^{n-1}} \]

5. **Recognize the Series:**
The series \( \sum_{n=1}^{\infty} \frac{3G}{4^{n-1}} \) is a geometric series with the first term \( a = 3G \) and the common ratio \( r = \frac{1}{4} \).

6. **Calculate the Sum of the Geometric Series:**
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Applying this to our series:
\[ S = \frac{3G}{1 - \frac{1}{4}} = \frac{3G}{\frac{3}{4}} = 4G \]

7. **Final Result:**
Thus, the total gravitational force on the 3 kg mass at the origin is:
\[ F = 4G \]

### Final Answer:
The gravitational force on a mass of 3 kg placed at the origin is \( 4G \).