Gravitational Potential Energy formula

[Fizziks_Fan]

Hello all.
I'm studying for sat 2 physics but I still don't understand the formula for gravitational potential in outer space. U = -Gm1m2/r
Can anyone explain this to me? Particularly why G is negative and r is used instead of r-squared.
Newton was such a genius.

 

[J Hann]

The gravitational potential is negative by definition. The definition is that
the gravitational potential at a point is the work done in bringing a unit
test mass from infinity to that point. Since the force involved is an attractive force negative work is done in bringing the unit test mass from
infinity to a distance r from the mass being considered.
Also, remember that Work = Force * Distance and that the gravitational
force between 2 massive objects is inversely proportional to the square of the distance between the 2 objects. Without going thru the integration of
F * dr from infinity to a distance r from the massive object you can see
that the work done will be inversely proportional to r when the force is
inversely proportional to r squared. Hope this helps.

 

[Saketh]

We know the law of gravitation:
$$F = \frac{Gm_1 m_2}{r^2}$$
We also know that:
$$U = - \int_{x_1}^{x_2} F(x) \,dx$$
In order to determine gravitational potential energy, we have to think about how much work it takes to get a mass from an infinite distance away to the target distance, [itex]r[/tex]. This is different from previous equations, which have you go from zero distance to the target distance. Once you understand this, you'll get the equation.<br /> <br /> Using our two above expressions and the method in the previous paragraph, we can write:<br /> $$U_{grav} = - \int_{\infty}^{r} \frac{Gm_1 m_2}{r^2} \,dr$$ <br /> Which simplifies to (with much cancellation of negative signs):<br /> $$U_{grav} = -\frac{Gm_1 m_2}{r}$$<br /> <br /> The only confusing things are the limits and the idea of going from infinity to the target distance. You'll do a similar thing if you get into electric potential energy with point charges.[/itex]