If the gravitational force had varied as r to the power -5/2 instead of r to the power -2; the - Physics - Gravitation

Question

If the gravitational force had varied as r to the power -5/2
instead of r to the power -2; the potential energy of a particle at
distance r from the centre of the earth would be proportional to r
to the power n What is the value of n

Vara · Accepted Answer

Dear
Student,
Please find below
the solution to the asked query:
Potential energy:U=-∫F dr   =-∫k r-52 dr   =-kr-52+1-52+1   =-kr-32-32But, U is proportional to rnHence, n = -32
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If the gravitational force had varied as r to the power -5/2 instead of r to the power -2; the potential energy of a particle at distance r from the centre of the earth would be proportional to r to the power n. What is the value of n.