Gravity

Gravity is very simple under Newtonian mechanics. Every object with mass attracts every other object with mass. The size of this force depends on the masses of the objects and the square of the distance between them. The equation looks like this:

F = G(mM/r^2)

F = Force
G = Big G
M = mass of 1st object
m = mass of 2nd object
r = distance between objects

G (called 'big gee') is Newton's gravitational constant, with a value of approximately:

 G=6.67*10^-11 m^3 kg^-1 s^2

This is an extremely small number, so gravity is very weak on most scales. Pairs of objects have to be very massive or very close together (the scale of micrometres) before the gravitational attraction is appreciable.

So what's the force between the Earth and the Sun? The relevant data are:

Distance between the Earth and the Sun: 1.4960×10^11 m
Mass of the Earth: 5.9736×10^24 kg

Mass of the Sun: 1.9891×10^30 kg

Plug these numbers into the equation and we get the force as being about 3.53*10^22 N.

This is a pretty large number, but what do we get for things on a more comprehensible scale? The same process for two people of mass 70kg standing 1 metre apart yields a force of about 10^-7 N. Hardly anything at all, which is why we almost never notice gravitational attraction between objects on the earth.

If you divide the both sides of this equation by m then you get an expression for 'Force per unit mass'. I.e. how much force a 1kg object would experience at this distance from the other object. This quantity is called g (aka 'little gee'), the 'gravitational field strength'.

g = F/m = G(M/r^2)

At the surface of the Earth we get a value of about 9.81 N/kg from this equation.

Fields

Newton's law of gravitation predicts that every objects pulls on every other object. The gravitational field of an object is a way of describing the way it pulls on other objects. The field can be represented by field lines, where the direction of the line indicates the direction in which an object would feel a force and the relative density of lines indicates how strong the force is.

NB: A field line diagram gives only a qualitative analysis of the gravitational field. It can only tell you that 'the field is stronger here than there'; it can't tell you 'the field is precisely this strong here'.

Planetary bodies which are approximately spherical can be thought of as being 'point-like' with all their mass concentrated at their centre. This is because for any part of the planet which isn't at the centre, there's another part on the other side of the planet. The resultant gravity of these two objects can be more simply thought of as coming from a single object of double the mass halfway between them. Do this for every part of the planet and you get it all averaging out to a single object at the centre of the planet with the same mass as the entire planet.

For a point like object the field around it is 'radial' with field lines coming in from all directions and getting further apart the further they are from the object. This is what you'd expect: the further away an object is, the smaller the gravitational field strength gets because we are dividing by r squared.

Now imagine zooming in on that picture to an area about the size of your house. From this perspective, the field lines look like they're equally spaced and all going up because we are looking at a scale where the depth and curvature of the Earth are essentially negligible. Field lines being equally spaced and unidirectional is called a 'uniform field' because the value of g is the same everywhere.

Whether you consider the Earth's gravitational field to be uniform or radial is ultimately a question of perspective and scale: if you're considering objects bouncing around close to the surface of the Earth, then it's fine to approximate the field as uniform. If you're considering things on an astronomical scale like orbiting planets and satellites then it will be better to think of it as radial.

Potential

You may remember gravitational potential energy: the energy put into something when you work against gravity to move it up away from the Earth and the energy you get back when it falls down again. Potential is defined as the gravitational potential energy per unit mass, given the symbol V. This is similar to gravitational force: F is the size of the force on a particular object and g is the general force per unit mass of any object. GPE is the potential energy of a particular object and V is the potential energy per unit mass of any object.

The main use of the idea of potential is this: as objects move between points of different potential, they gain or lose energy. This means that if you know the potential at two points in space you can work out how much energy it will take to move a certain mass between them. If it comes out negative then your object will gain energy, probably speeding up as a result. Think about an object falling towards the Earth. As it descends, the force of gravity from the Earth acts on it, speeding it up. Another way to think about this is that it has moved from a point of one potential to a point of lower potential, so it must have gained energy, in this case the kinetic energy of it plummeting downwards.

It is worth remembering that there is no absolute standard of potential or potential energy. This is to say that where the point of 'zero' potential is is ultimately arbitrary as it is only the change in potential from point to point that is of any physical significance. This means that it is fine to have negative potentials, as long as you know where your zero-point (often called the datum) is. Two commonly used conventions are:

• That the zero point is at an infinite distance away from the objects concerned such that all potentials are negative. This one is usually used when considering radial fields.
• That the zero point is at the surface of the Earth such that points above it have positive potential. This is most useful when considering the roughly uniform field near the surface of the Earth.

Recognize that under both conventions potential goes up as one gets further from Earth. This makes sense as to get further away from an object you need to do work against its gravitational field, so your potential energy, and hence the potential at that point, must increase.

Potential for a uniform field

In a uniform field the value of g is the same everywhere, so to raise an object up all we need to do is oppose gravity with a force of the same size. For an object of mass m, this force will have to be of magnitude mg. The work done by this force raising the object through a distance of h will be mgh. So the potential energy of an object of mass m is mgh. To find the potential energy per unit mass we just divide this by m. So the potential of a point at a distance h from the zero-point in a uniform gravitational field is given by:

V = gh

V = potential

g = gravitational field strength

h = distance from datum

Potential for a radial field

With this important result:

V = G(M/r)

V = potential

G = big G

M = mass of object to which gravitational field is due

r = distance from that object

If you're using the convention of zero potential at infinity then you get

This is a bit of a fudge; strictly speaking one should consider the integral of all the little changes in potenial from the centre of the planet to the distance r. However the result is identical so let's just get on with it.

You now know how to calculate the potential due to the gravitational field of point like objects. What if you want to calculate the potential due to two objects? You don't have to for A level. Which is good.

Because it's one complicated motherfucker.