Physics with Calculus/Mechanics/Gravity

It may seem obvious to us now that gravity is both the force that makes things fall and makes planets orbit, although it was not obvious at all. People may claim to know what gravity is, but really, we have almost no idea. There are theories about fundamental particles called gravitons, and general relativity is the study of gravity, but essentially we do not know what really causes gravitational effects. It is just a fundamental property of matter that mass attracts other mass, much like charge.

Newton discovered that gravity follows, to an excellent degree of precision,

$F=-\frac{GMm}{r^2}$

in the direction connecting two bodies of mass M and m. G is a fundamental constant, equal to $6.67\times 10^{-}11N{m}^2/k{g}^2$The minus sign merely indicates that gravity is attractive. It is extremely odd that the gravitational mass is exactly the same as the inertial mass. That is, massive objects always have large gravity (general relativity provides a nice explanation for this).

Gravity is a conservative force, which means that the work done moving a small mass from one point to another depends only on the end points, and not the path taken. I will demonstrate this with a single point mass. With this result, it is almost obvious that the force from any arrangement of mass will also be conservative. To see this, note that the gravitational force is the sum of the parts and that work done by one force is the sum of the work done by its parts. Thus, when at test particle is moved through an arrangement of particles exerting gravitational force, if the work done by each of them depends only on the end points, then the sum depends only on the endpoints, which is what we wanted to show. Now, consider a single particle of mass M at the origin, and a test particle of mass m moving from A to B. To simplify calculations, assume that the path connecting A and B lies in the xy plane, although the argument will easily but rather tediously extended to 3 dimensions. Now, it is easier to work in polar coordinates.

When the particle moves a small distance ds, the work done is done only by the force acting in the r direction and thus only movement in the r direction matters. Since this holds exactly for small distances, it holds for large distances and the work done can only depend on how the particle moves in r. That is, we may neglect $\theta$. Now, if the particle moves a small distance dr one direction then a little while later, at the same radius, in the other direction, the work cancels out because both the force and displacement are the same. That means that work is independent of the path and gravity is conservative!

Since gravity is conservative, you can see that we can define a function of position such that the work done from moving from a to b is just this function at a minus the function at b. Such a function is called the potential energy -- the same potential energy you already know! If instead we remove it one step, and divide by the mass m, then we have a function that is independent of the particle in question and we call it the potential. Now, it would be useful to see some properties of potential.

First of all, the defining quality of potential, $\varphi (𝐫)$ is that if you move a particle of mass m from a point a to a point b along any path, then the work done is

$m\varphi (a)-m\varphi (b)$.

Now, to find the other important property of potential, consider moving a particle of mass m a little bit, $d𝐫=(dx,dy,dz)$ in a force field F. The work done by going in the x direction is

$-m(\varphi (x+dx,y,z)-\varphi (x,y,z))=F_{x}dx$

with similar expressions for y and z.

Dividing by dx and letting dx go to zero, we have,

$F_x=m\frac{\partial \varphi }{\partial x}$ which just means take the derivative of $\varphi$ treating y and z as constants.

If we do the same thing for y and z, we have

$F=m(\frac{\partial \varphi }{\partial x},\frac{\partial \varphi }{\partial y},\frac{\partial \varphi }{\partial z})=\mathrm{\nabla }\varphi$.

The upside down triangle is just shorthand for the first expression.

This is amazing -- all the information about the force, a vector, is contained in a scalar. Three components for the price of one. The reason we can do this is because we know that the force is conservative, as you can see, that greatly restricts the number of possible fields. In the case of one dimension, you can see that the force is the derivative of the potential energy. In fact, the gradient (the name for taking the partial derivatives with respect to each variable and putting them into a vector) is one generalization of a derivative to many dimensions.

It is now easy to see one final important property of potential, that it is unique up to an additive constant. If you added a function depending on any of the variables, then at least one of the partial derivatives would not be zero and you would get a different force. But if you add any constant, you get the same force because the derivative of a constant is zero.