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Question

# If the gravitational force had varied as r−5/2 instead of r−2; the potential energy of a particle at a distance 'r' from the centre of the earth would be proportional to

A
r1
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B
r2
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C
r3/2
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D
r5/2
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Solution

## The correct option is C r−3/2The potential energy for a conservative force is defined asF=−dUdr or U=∫r∞¯F.d¯r .......(i)or Ur=∫r∞GM1M2r2dr=−GM1M2r .......(ii)(∵U∞=0)If we bring the mass from the infinity to the centre of earth, then we obtain to work, 'so it has negative (gravitational force do work on the object) sign & potential energy decreases. But if we bring the mass from the surface of earth to infinite, then we must do work against gravitational force & potential energy of the mass increases.Now in equation (i) if F=GM1M2r5/2 instead of F=GM1M2r2 thenUr=∫r∞GM1M2r5/2dr=−23GM1M2r3/2⇒Ur∝1r3/2

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