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An infinite number of masses, each of one kg are placed on the +ve X axis at 1 m, 2 m, 4 m,
_______ from the origin . The magnitude of the gravitational field at origin due to this distribution of masses is :

A
2G
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B
4G3
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C
G4
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D
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Solution

The correct option is B 4G3
Each mass is of m=1 kg placed on the +ve X axis.

1st mass is at (211=20=1 m), 2nd mass is at (221=21=2 m), 3rd mass is at (231=22=4 m) from the origin,
and so on.
Thus, we can see that the nth mass is placed at a distance of rn=2n1m from the origin on the +ve X axis.

Thus, the gravitational field intensity at origin for the nth particle will be given as

gn=Gmr2n

gn=G(2n1)2=G4n1

Thus, the net magnitude of the gravitational field at the origin due to this distribution will be a sum of this individual gn

gnet=n=1G4n1

gnet=Gn=114n1

n=114n1 is the sum of an infinite Geometrical Progression with ratio = 1/4,

n=114n1=11(1/4)=13/4=43

substituting

gnet=4G3

Magnitude at origin for the given distribution of masses is 4G3.

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