Question

Newton's first law of gravitation by integration.

Answer 1

My question is: can you get the same result by integrating the force on all points of an object (planet) by, say, the sun, as if you treated just its centre of mass?

Answer 2

Firstly, consider the sun. If we draw a circle around the it, we can say that the same force will act on 1kg anywhere in the circle.

Answer 3

Now, consider a planet orbiting the sun

Answer 4

All points on the arc will feel the same force. Now...

Answer 5

Okay, the area of the arc will be:
\[\pi2q*\theta/360\]
z and r are constants.
q=sqrt{y^2+p^2}
x=sqrt{z^2-y^2}
y=sqrt{z^2-x^2}
p=r-x
t=q-p
theta=arctan(y/p)

What would you do now to integrate the arc with respect to d(x-t) (as the force is the same as on the arc at x-t, not x)?