# Gravitational Potential [Moved from Academic Guidance]

1. Oct 16, 2008

### kushal18

hey friends why is gravitational potential negative?

2. Oct 16, 2008

### D H

Staff Emeritus
Any potential energy (gravitational, electrical, ...) has an arbitrary constant term. Given some conservative force field F(x), any function U(x) for which

$$\nabla U({\boldsymbol{x}}) = -\,{\boldsymbol{F}}({\boldsymbol{x}})$$

is a potential energy function U(x) of the force field F(x). Adding a constant to U(x) yields another function Uc(x)=U(x)+c whose gradient is the force field. Bottom line: You can pick any value you want for that constant. One obvious choice for gravitational potential is to make the potential vanish as $||\boldsymbol x||\to\infty$, in which case the potential for any finite x will be negative.

Last edited: Oct 16, 2008
3. Oct 16, 2008

### Naty1

Gravitational is negative by convention. The potential of a body is free space without gravity is taken to be zero...hence near a gravitational mass,say a plant or star, since work must be done to move the body from a strong gravitational influence to free space where it's zero, we say gravity imposes a negative potential...

An analogous situation is on the surface of the earth...say on a beach where we take gravitational potential to be zero....climb out of a hole in the sand to reach zero potential...again gravitational potential is taken to be negative in the hole....

I've not come across a clear explanation as to whether this convention is significant or just convenient...I think idea this matches DH post above....

4. Oct 16, 2008

### D H

Staff Emeritus
First things first: I corrected a sign error in my previous post.

Only in the case of a body in free space.

Suppose I want to do elementary physics near the surface of the Earth. I'll choose coordinates such that the x and y axes are parallel to the surface and z is positive upwards. With this convention, the gravitational force is nearly constant:

$$\boldsymbol{F} \approx -mg \hat \boldsymbol z$$

The potential functions that generate this constant force field are of the form

$$U=mgz+C$$

Here, the "obvious" choice for a constant is C=0. In other words, u=mgz=mgh, which is what you were taught in elementary physics. Now potential is positive above the surface. So gravitation is not always negative by convention.

5. Oct 23, 2008

### kushal18

thanks for the help.