Question

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1 m, 2 m, 4 m, 8 m respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

• A. -G
• B. 8/3 G
• C. - 4/3 G
• D. -4G

Answer 1

The Correct Option is D

 To solve this problem, we need to calculate the gravitational potential at the origin due to an infinite number of bodies placed along the x-axis at specific intervals.

The gravitational potential \(V\) at a point due to a mass \(m\) located at a distance \(r\) is given by the formula:

\(V = -\frac{Gm}{r}\)

Where:

• \(G\) is the gravitational constant.
• \(m = 2 \, \text{kg}\) (for each mass in this question).
• \(r \, (\text{from each mass to the origin})\) is given as 1m, 2m, 4m, 8m,... for successive masses.

Let's calculate the contribution of gravitational potential from each mass and then sum it to find the total potential at the origin.

For the first mass at 1m:

\(V_1 = -\frac{G \times 2}{1} = -2G\)

For the second mass at 2m:

\(V_2 = -\frac{G \times 2}{2} = -G\)

For the third mass at 4m:

\(V_3 = -\frac{G \times 2}{4} = -\frac{G}{2}\)

For the fourth mass at 8m:

\(V_4 = -\frac{G \times 2}{8} = -\frac{G}{4}\)

Observing the pattern, the contribution from the nth mass at \((2^{(n-1)})\) m will be:

\(V_n = -\frac{G \times 2}{2^{(n-1)}} = -\frac{2G}{2^{(n-1)}}\)

The total potential \(V_{total}\) at the origin due to all masses is the sum of all individual potentials:

\(V_{total} = V_1 + V_2 + V_3 + V_4 + ...\)

\(V_{total} = -2G - G - \frac{G}{2} - \frac{G}{4} - ...\)

Recognizing this as an infinite geometric series, we find the sum using the formula for the sum of an infinite geometric series:

\(S = a/(1 - r)\) where \(a\) is the first term and \(r\) is the common ratio.

In our series, \(a = -2G\) and the common ratio \(r = \frac{1}{2}\). Hence,

\(V_{total} = \frac{-2G}{1 - \frac{1}{2}}\)

\(V_{total} = \frac{-2G}{\frac{1}{2}} = -4G\)

Thus, the resulting gravitational potential due to this system at the origin is \(-4G\).

The correct answer is:

-4G