An infinite number of particles each of mass `m` are placed on the positive X-axis of `1m, 2m, 4m, 8m,...` from the origin. Find the magnitude of the resultant gravitational force on mass `m` kept at the origin.

Options

A. `-8Gm`
B. `-3Gm`
C. `-4Gm`
D. `-2Gm`

Answer

To find the magnitude of the resultant gravitational force on a mass \( m \) kept at the origin due to an infinite number of particles each of mass \( m \) placed at positions \( 1m, 2m, 4m, 8m, \ldots \) on the positive X-axis, we will follow these steps:

### Step 1: Understand the Gravitational Force Formula
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
In our case, both masses are \( m \), so the force exerted by each mass on the mass at the origin will be:
\[ F = \frac{G m^2}{r^2} \]

### Step 2: Identify the Positions and Distances
The particles are located at distances \( r_1 = 1m, r_2 = 2m, r_3 = 4m, r_4 = 8m, \ldots \). The general position of the \( n \)-th particle can be expressed as:
\[ r_n = 2^{n-1} \text{ for } n = 1, 2, 3, \ldots \]

### Step 3: Calculate the Gravitational Force from Each Particle
The gravitational force exerted by the \( n \)-th particle on the mass at the origin is:
\[ F_n = \frac{G m^2}{(2^{n-1})^2} = \frac{G m^2}{4^{n-1}} \]

### Step 4: Sum the Forces from All Particles
The total gravitational force \( F_0 \) on the mass at the origin is the sum of the forces from all particles:
\[ F_0 = \sum_{n=1}^{\infty} F_n = \sum_{n=1}^{\infty} \frac{G m^2}{4^{n-1}} \]

### Step 5: Recognize the Series as a Geometric Series
The series can be rewritten as:
\[ F_0 = G m^2 \sum_{n=0}^{\infty} \frac{1}{4^n} \]
This is a geometric series with first term \( a = 1 \) and common ratio \( r = \frac{1}{4} \).

### Step 6: Calculate the Sum of the Geometric Series
The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

### Step 7: Substitute Back to Find \( F_0 \)
Now substituting back into the expression for \( F_0 \):
\[ F_0 = G m^2 \cdot \frac{4}{3} \]

### Final Result
Thus, the magnitude of the resultant gravitational force on the mass \( m \) kept at the origin is:
\[ F_0 = \frac{4}{3} G m^2 \]