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An infinite number of particles each of mass 1kg are placed on the positive x-axis at 1m, 2m, 4m, 8m…. from the origin. The magnitude of the resultant gravitational force on 1kg mass kept at the origin is

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\frac{4 G}{3}

\frac{3 G}{4}

answer is D.

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Detailed Solution

We need to calculate the resultant gravitational force exerted by an infinite number of particles on a 1 kg mass placed at the origin.

Mass of each particle = 1 kg
Distance of particles from the origin: 1m, 2m, 4m, 8m, ...

The gravitational force between two masses is given by:

F = G × (m₁ × m₂) / r²

Where:

G = Gravitational constant
m₁ = m₂ = 1 kg
r = Distance from the origin

Force exerted by the particle at distance 1m:

F₁ = G

Force exerted by the particle at distance 2m:

F₂ = G / 4

Force exerted by the particle at distance 4m:

F₃ = G / 16

The total force is:

F = G ( 1 + 1/4 + 1/16 + 1/64 + ... )

This is an infinite geometric series with:

First term (a) = 1
Common ratio (r) = 1/4

Using the sum of an infinite geometric series formula:

S = a / (1 - r) = 1 / (1 - 1/4) = 1 / (3/4) = 4/3

F = G × 4/3 = 4G / 3

Final Answer:

(d) 4G / 3

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