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My derivation:

Let mass M be at the origin O. Let another mass m be at an arbitrary position r from the origin. The grav. force on m due to M is directed towards the origin. Let the mass m move a distance dr towards the origin due to this force.

The infinitisemal work done by the grav. force is, GMm/r

^{2}.dr. I have expanded the dot product here, since the force and displacement dr both are in the same direction, and theta is 0 so cos(theta)=1.

Now we can integrate this infinitisemal work dW from the point a to b, and we get,

-GMm[(1/b) - (1/a)]

This is equal to the negative of the change in pot. energy. The change in pot. energy is U(b) - U(a).

So ,

-GMm[(1/b) - (1/a)] = -[U(b) - U(a)]

and thus,

GMm[(1/b) - (1/a)] = U(b) - U(a)

at a=infinity, we can chose U(a) to be 0.

We then get,

GMm/b = U(b)

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as you see, my answer differs by a minus sign. In the book derivation, the only difference is that they have conventionally taken the infinitisemal displacement dr, to be in the outward direction.

That is, the unit vector r[tex]\widehat{}[/tex], is taken to be positive in the outward direction from the origin.

And then, according to this convention, the grav. force would be GMm/r

^{2}(-r[tex]\widehat{}[/tex])

and the answer we would get is U(r) = -GMm/r

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Though in a way, I have understood why the paradox arises, (due to the convention of taking r[tex]\widehat{}[/tex] positive in the outward dir.), I still want to know why we can't use my method?

Is it just a matter of convention?

One problem that I figured out with my method is this,

My method gives teh answer U(r)=GMm/r

This means, as the particle, travels from infinity towards the origin, under the attarctive force, r decreases, and hence acc. to my result, the pot. energy "increases" along with an increasing kinetic energy.

Please explain, what exactly is wrong with my approach and why it gives a physically non-meaningful result like the above?

Thanks