Question

# 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 4G3Each mass is of m=1 kg placed on the +ve X axis.∵1st mass is at (21−1=20=1 m), 2nd mass is at (22−1=21=2 m), 3rd mass is at (23−1=22=4 m) from the origin, and so on.Thus, we can see that the nth mass is placed at a distance of rn=2n−1m from the origin on the +ve X axis.Thus, the gravitational field intensity at origin for the nth particle will be given asgn=Gmr2n∴gn=G(2n−1)2=G4n−1Thus, the net magnitude of the gravitational field at the origin due to this distribution will be a sum of this individual gn∴gnet=∞∑n=1G4n−1∴gnet=G∞∑n=114n−1∵∞∑n=114n−1 is the sum of an infinite Geometrical Progression with ratio = 1/4,∴∞∑n=114n−1=11−(1/4)=13/4=43substituting∴gnet=4G3Magnitude at origin for the given distribution of masses is 4G3.

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