From universal law of gravitation, gravitational force exerted on a body of mass m by another body of mass M is $$ \mathbf F = \frac{GMm}{x^2} $$ 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 $$ W = \int_\infty^r \vec{F(x)}.\vec{dx}$$ $$=\int_\infty^r \frac{GMm}{x^2}\hat x.\vec{dx}$$ (where $\hat x$ 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 $\vec{dx}$ which results the angle between both vectors $0$) $$ =\int_\infty^r \frac{GMm}{x^2} {dx}\ cos0$$
$$ = - GMm\left(\frac{1}{r}-\frac{1}{\infty}\right) $$ $$= -\frac{GMm}{r}$$
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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