# Gravitational Potential Energy formula

1. Aug 13, 2006

### 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.

2. Aug 13, 2006

### 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.

3. Aug 14, 2006

### 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.

Using our two above expressions and the method in the previous paragraph, we can write:
$$U_{grav} = - \int_{\infty}^{r} \frac{Gm_1 m_2}{r^2} \,dr$$
Which simplifies to (with much cancellation of negative signs):
$$U_{grav} = -\frac{Gm_1 m_2}{r}$$

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.