From universal law of gravitation, gravitational force exerted on a body of mass m by another body of mass M is F=GMmx2 where x is the distance between the centres of both the objects.
So, work done by gravitational force in bringing the object of mass m from infinity to a distance r from the centre of body of mass M is W=∫r∞→F(x).→dx =∫r∞GMmx2ˆx.→dx (where ˆx is the unit vector in the direction in which the body of mass M is attracting the body of mass m, i.e. the direction of →dx which results the angle between both vectors 0) =∫r∞GMmx2dx cos0
=−GMm(1r−1∞) =−GMmr
Now, we know that W=-(∆U) -\frac{GMm}{r} = -(U_r - U_\infty) -\frac{GMm}{r} = (U_\infty - U_r) Since, Zero of potential energy is at infinity by convention, so U_\infty = 0 -\frac{GMm}{r} = -U_r \frac{GMm}{r} = U_r
I get potential energy at a distance r as positive, then why is it that gravitational potential energy is -\frac{GMm}{r}
What is wrong in my derivation?
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