# Dot product in the Gravitational Potential Energy formula

1. May 22, 2015

### sawer

This is the gravitational potential energy formula
$$U = -\int_\infty^r\vec{F}_\text{field}\cdot d\vec{r}$$
If r vector's direction is form infinity to r, then it means it has same direction as Gravitational Force. So cos0=1
But after multiplication there is a negative sign here: "-GMm"

$$U = -\int_\infty^r \frac{-GMm}{r^2} dr$$

Why is there a negative sign here(before GMm)? Doesn't gravitational force vector and r vector(from infinity to r) have same direction?

2. May 22, 2015

### Noctisdark

Let name a force F, we can express it a gradient of some potential Φ
Mathematically : F = ∇Φ
Gravity is an attractive force, it's tends pull and bring you to itself, but that mean F = - ∇Φ, if you consider a force field that is pushing you away, then you can say that F = ∇Φ,
The minus sign just tells that the force increase whenever the potential decreases .

3. May 22, 2015

### sawer

Thanks but I am not talking about the negative sign before the integral.

I am asking the negative sign that was added after that operation: $$\vec{F}_\text{field}\cdot d\vec{r}$$

Here F is gravitational force, right? So it must be toward the origin, so is r vector, right? So the angle is 0. And there mustn't be a negative sign there. But there is.

Why is negative sign added there?

4. May 22, 2015

### nasu

The r vector is not "from infinity to r". (whatever that means).
It is from the origin to the given point. If you write the gravitational force as you did, the origin is at the center of a spherical body or at a point mass.
The r vector has a direction opposite to that of the force.
.

5. May 22, 2015

### Noctisdark

Because F = -GMm/r^2, the direction of force is opposite to the direction the angle isn't 0 it's pi (180 degree )

6. May 22, 2015

### sawer

I am confused here.

dr is a differential displacement vector and it shows direction of movement right?

So we set infinity to zero potential, make calculations from infinity to point r. Here we are bringing the object from infinity to point r. So r vector is from infinity to point r. Gravitational force vector is also toward origin. So displacement and force vectors have same direction.

Why did you say: "The r vector has a direction opposite to that of the force"?

7. May 22, 2015

### nasu

Because it does. The r vector is from origin to the mass. The force is from the mass to the origin, the mass is attracted to the origin. This is for the vector r.

.

8. May 22, 2015

### Noctisdark

Look at what nasu just said !

9. May 22, 2015

### Staff: Mentor

F points towards the origin, and the unit vector r points away. So F is some negative scalar multiple of the unit vector r. It has nothing to do with the dot product. The sign is in the force law itself.

This doesn't make sense.

Last edited: May 22, 2015
10. May 22, 2015

### sawer

As you answered my other thread, there mustn't be any sign there. Because it is "magnitude" of the gravitational force. I think there is no unit vector there. It must be displacement vector.

And I still don't understand, if we integrate from infinity to r point which means bringing object from infinity to point r, how does displacement vector r points from origin to point r, not from infinity to point r???

11. May 23, 2015

### sawer

Then why is r called "displacement vector"?

12. May 23, 2015

### theodoros.mihos

You are right but displacement mean:
$$\delta{\mathbf{r}} = \mathbf{r}_f - \mathbf{r}_i$$
rf and ri are parallel for direct central movement.

Last edited: May 23, 2015
13. May 23, 2015

### sawer

I looked up Halliday's Fundamentals of Physics book, in section 13-6, he called r as a displacement vector.

He gave an example about shooting a baseball directly away from Earth and integrate from R to infinity(not infinity to R) and said: "The integral contains the scalar (or dot) product of the force F(r) and the differential displacement vector dr along the ball’s path" and integrate
$$\vec{F}_\text{r}\cdot d\vec{r}$$= F(r)drcosϕ

where ϕ is the angle between the directions of F(r) and dr. When we substitute 180° for ϕ"

Why did he say this?

Because F is toward earth and dr is directed away from earth. So F and r vectors have opposite direction.

But here, F is gravitational force so, it is directed to earth and the ball is bringing from infinity to point R but according to examples dr is still pointing infinity from earth.

14. May 23, 2015

### theodoros.mihos

See what does vector minus vector mean:
$$\mathbf{A}-\mathbf{B} = \mathbf{A} + (-\mathbf{B})$$ is another vector than starts from the end of B and end to the end of A.

15. May 23, 2015

### robphy

I think there is a confusion in the use and/or interpretation of $d\vec r$ as an infinitesimal element of a directed path
and as an increment of the $r$-coordinate. Let's use $d\vec s$ for the infinitesimal element of a path.

$$U = -\int_{P}^{Q}\vec{F}_\text{field}\cdot d\vec{s},$$
where $P$ is the start-point (which will be taken at infinite-radius from the origin) and $Q$ is the end-point (which will be taken at finite-radius $R$ from the origin.

Now let us evaluate this symbolic line-integral in polar-coordinates. Keeping only the radial-components (since $\vec F$ is radial),
$$U = -\int_{P}^{Q}\left(-\frac{GMm}{r}\hat r\right) \cdot \left( dr\ \hat r\right),$$
where I have left the endpoints of the path symbolic. Note the sense of the path to be taken hasn't been used at this point. Note: $dr\ \hat r$ is the direction of increasing-$r$.
Simplifying,
$$U = -\int_{P}^{Q}\left(-\frac{GMm}{r} dr\ \right),$$
Now, choosing $P$ and $Q$ according to the definition:
$$U = -\int_{r=\infty}^{r=R}\left(-\frac{GMm}{r} dr\ \right).$$

16. May 23, 2015

### sawer

Thank you.
If we integrate from Q to P, will $dr$'s direction change?

Can you please give more comment about transforming from ds to dr.

17. May 23, 2015

### robphy

No, $dr\ \hat r$ (the direction of increasing r) always points radially outwards.
In polar coordinates, an infinitesimal displacement is $d\vec s= dr\ \hat r+rd\theta\ \hat\theta$.

18. May 23, 2015

### sawer

And Gravitational Force always points inwards. So for gravitational force, the result of dot multiplication is always negative. Then what is the point of dot multiplication.

How can we determine gravitational force make positive or negative work???

Thanks for help...

19. May 24, 2015

### Cecil Tomlinson

Gravitational potential energy is defined the way it is so that it can have real values. Radius = 0 cannot be used because at r=0, by the force formula, the force of gravity is infinite, and any object at a radius greater than zero would then have infinite potential energy relative to r=0. But at r=infinity, the force of gravity is zero and any object at a radius less than infinity can easily compute a real potential energy relative to infinite radius. This potential energy then has to be defined as negative so that when computing the potential energy difference between two radii, the higher radius will have a less negative potential energy and the difference in energy is positive when looking upward. So, it takes positive energy to climb upward.

20. May 24, 2015

### Staff: Mentor

Clearly the the work has to have the same magnitude but opposite signs when we switch the direction of travel (i.e. from P to Q versus from Q to P). In the work integral, we can accomplish this either by switching the limits of integration or by changing the sign on $\hat r dr$. But if we do both of these, there is no net effect. The convention is to switch the limits of integration, because that is consistent with what we expect from ordinary single-variable integration. (At least, I think it's simply a convention in line integration. If someone can give a more substantial reason, I'd be happy to see it!)