Gravity: signs of the equations?

[21joanna12]

Homework Statement

Hi all,
I have spent many hours now trying to figure out the signs for equations when dealing with gravity...
The first equation I am having trouble with is
F=GmM/r²
Should there be a minus sign? My textbook does not have a - sign, but I have seen online that many sources do put it in. On the one hand I think maybe there should not be one because in the equation you haven't assumed the object you are considering and your positive direction, but on the other hand it will always be inwards towards each object?
Then Thinking about the force per unit mass around Earth i.e. g=GM/r², this should have a - sign because it is always directed towards the Earth (acts in the opposite direction to increasing r, so negative)

Then The gravitational potential, which is the work done per unit mass to move the object from infinity to a point, is negative ebcause work is done on the object.
However when I try to find it by integration , if I assume that the force is negative, I will get

V=\int_{\infty}^{r}-\frac{GM}{r^{2}}dr=\frac{GM}{r}

So I get a positive answer... I'm sure the issue stems from the order in which I put my limits, but I don't understand why t would be the other way.

I would also think that the potential graddient: dV/dr, is positive because less work is done on an object to bring it from infinity to a larger r than to a smaller r, so the object further away has less negative potetial energy, and so if you move an object from a smaller r to a larger r, the potential becomes less negative so V increases with increasing r. This also makes sense from taking the derivative of V because you end up with a positive.

But here I get confused again- You found V by integrating g (which was negative) and then you take the derivative to find g again but now g is positive and not negative? My textbook also says that g=-∇v/∇r...

As you can see, I'm really confused about positives and negatives and which way round the limits of integration go... There are more questions I had about this before I started writing, but now I've forgotten them all. I apologise for my semi-incoherent ramblings. I just really don;t get this/...

Thank you for any help! >_<

 

[olivermsun]

Homework Statement

Hi all,
I have spent many hours now trying to figure out the signs for equations when dealing with gravity...
The first equation I am having trouble with is
F=GmM/r²
Should there be a minus sign? My textbook does not have a - sign, but I have seen online that many sources do put it in. On the one hand I think maybe there should not be one because in the equation you haven't assumed the object you are considering and your positive direction, but on the other hand it will always be inwards towards each object?

If you're dealing with the magnitude of F, then it doesn't matter. The force is inward for both bodies as you say.

If these are signed quantities then you have to be careful and choose the sign of the force on one body such that it points at the other body.

Then The gravitational potential, which is the work done per unit mass to move the object from infinity to a point, is negative ebcause work is done on the object.
However when I try to find it by integration , if I assume that the force is negative, I will get

V=\int_{\infty}^{r}-\frac{GM}{r^{2}}dr=\frac{GM}{r}

So I get a positive answer... I'm sure the issue stems from the order in which I put my limits, but I don't understand why t would be the other way.

The reason may be slightly confusing. The potential is the work done to move the object within the gravity field, not by the gravity field. Another way to say this is the object loses gravitational potential exactly equal to the work that gravity actually does on the object.

The "rigorous" way to get the right sign dependence is to look at the vector calculus definition of the potential, is shown at Wikipedia under Conservative forces.