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Question

# The mass density of a planet of radius R varies with the distance r from its centre as ρ(r)=ρ0(1−r2R2). Then the gravitational field is maximum at:

A
r=34R
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B
r=R
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C
r=13R
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D
r=59R
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Solution

## The correct option is D r=√59R Mass of small element of planet of radius x and thickness dx is given by, dm=ρ×43πx2dx dm=ρ0(1−x2R2)×43πx2dx Mass of the planet M=∫dm=43πρ0∫r0(x2−x4R2)dx ⇒M=43πρ0(r33−r55R2) Gravitational field, E=GMr2=Gr2×43πρ0(r33−r55R2) ⇒E=43πGρ0(r3−r35R2) E is maximum, when dEdr=0 ⇒dEdr=43πGρ0(13−3r25R2)=0 ⇒r=√59R Hence, option (D) is correct.

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