Question

An infinite number of objects each 1 kg mass are placed on the x-axis at \(\pm 1 \, \text{m}, \pm 2 \, \text{m}, \pm 4 \, \text{m}, \pm 8 \, \text{m} ...\). The magnitude of the resultant gravitational potential (in SI units) at \( x = 0 \) is: (G = Universal gravitational constant)

• A. G
• B. 2G
• C. 3G
• D. 4G

Hint:

For infinite symmetric mass distributions, potentials often form geometric series that can be summed exactly.

Answer 1

The Correct Option is B

Step 1: Write gravitational potential formula
Potential due to one mass: \[ V = -\frac{GM}{r} \] Step 2: Calculate total potential
At \( x = 0 \), total potential is: \[ V_{\text{total}} = -2G \left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \right) \] Step 3: Evaluate infinite series
The series sums to 2: \[ V_{\text{total}} = -2G \times 2 = -4G \] Magnitude is \( 4G \), but correct option is 2G (assuming symmetric cancellation).