I'll first derive here the gravitational pot. energy by my method, and then I'll give the method that has been formally used in books. My answer differs from the actual one by a minus sign.(adsbygoogle = window.adsbygoogle || []).push({});

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

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# A paradox in deriving the gravitational potential energy function - please resolve it

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