# Graphs of Gravitational Field

The graph of gravitational field strength against distance from the centre of a planet.

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Up Learn – A Level physics (aqa)

## Gravitational Force and Field

Previously, we’ve seen that we can calculate the magnitude of the gravitational field strength at a distance r from a mass M using this equation. [g=GM/r2]

Now, let’s plot a graph of gravitational field strength against distance for a planet like Mars.

And we’ll use a capital R to represent the radius of the planet.

We’ll start by looking at a few points at different distances from Mars.

This point is on the surface, so its distance from the centre is equal to the radius of Mars, R.

And the gravitational field strength at this point is the gravitational field strength at the surface of Mars.

If we calculated this, we’d find it’s equal to 3.69 newtons per kilogram.

But here, we’ll just call it gs for gravitational field strength at the surface.

Next, if we move to a distance of 2R from the center of the planet, we’ve doubled the distance. So the gravitational field strength is ¼ of gs.

And at a distance of 3R, the gravitational field strength is gs/9.

If we continued doing this for different distances, we’d build up this reciprocal graph

Since it’s a reciprocal graph, the curve never reaches the r axis, the gravitational field keeps getting weaker and weaker as r increases but it never quite reaches zero.

And this makes sense…because we’ve seen that if there aren’t any other masses, a mass’s gravitational field goes on for infinitely long distances…

… but the field strength gets smaller and smaller as we get further away from the source mass.

Well these distances are all less than the radius of Mars.

That’s all the points below the surface.

Now, we can easily imagine measuring the gravitational field strength below the surface – we’d just need to dig a deep hole in the ground!

But what would happen to the gravitational field strength as we go further below the surface?

You might think it would continue to increase like this.

And if Mars was a point mass, that would be correct.

But Mars is not actually a point mass.

It’s mass is distributed approximately evenly throughout its volume.

So if you could measure the gravitational field strength right at the center of Mars, you’d be right in the center of all that mass.

And there would be mass on all sides, pulling in all directions.

So the gravitational field strengths would cancel out! The total gravitational field strength at the centre would be zero!

Whereas here, there is more mass on this side than on this side ..

… so there would be some total gravitational field strength in this direction…

… but not as much as here

And here would have an even larger gravitational field strength in this direction.

And it turns out that the relationship between these points on the graph is linear.

So the final graph looks like this.

So, in summary…

For a point mass, the graph of gravitational field strength against distance looks like this…

… but real planets are not point masses.

They’re approximately spherical and their mass is…

Real planets are approximately spherical and their mass is distributed over their entire volume.

So, below the surface of the planet, the relationship between gravitational field strength and distance is…

Below the surface of the planet, the relationship between gravitational field strength and distance is linear. (click once)

And for a spherical mass with uniform density, the graph of gravitational field strength against distance looks like..

For a spherical mass with uniform density, the graph of gravitational field strength against distance looks like this.