Gravitational Fields

Gravity Fields

A field is a region in which a force is felt.

Gravity is a very mysterious force.  Nobody knows why objects have this attractive force between them, even if they are far apart.  This attraction occurs for any object with mass, however small.  The force is very small, and always attractive.  We never get a repulsive gravitational force.

Newton found that all objects accelerate towards the Earth at 9.81 m/s².  He also worked out the moon has an acceleration towards the Earth of 2.72 m/s².  Therefore:

            Acceleration of Moon = 2.72 × 10^-3 m/s² =   1   =   1

            Acceleration of apple          9.81 m/s²      3610   (60.1)²

He also worked out the distances between the moon and the centre of the Earth and the distance between the apple and the centre of the Earth.  Their ratio also turned out to be 60.1:1.  This formed the basis of Newton’s Inverse Square Law of Gravitation.

Every particle of matter in the Universe attracts every other particle with a gravitational force that is proportional to the products of the masses and inversely proportional to the square of the distance between them.

We can write this in code form:

We can now add the constant of proportionality:

The Gravitational Constant, G, has a value of 6.67 ´ 10 ^–11 N m² kg^-2.  The minus sign tells us that gravity is an attractive force.  In gravity problems, we always assume that the masses are point masses.

Gravity is a very small force, but it exists between all objects.  There is a tiny gravitational attraction between you and the student sitting next to you in your physics lesson.  You can't feel it because it's negligible.  The only reason we feel gravity is that the Earth is a very large object.  Remember that as well as the Earth attracting us towards it, we are attracting the Earth towards us.  The tiny attraction has important implications for dust particles coming together in space to form stars and planets.

Gravitational Field Strength

We are all familiar with the magnetic field of a bar magnet.  The field lines show up as areas of attraction and repulsion at the poles and the concentration of field lines at any point.  The closer we are to the magnet, the stronger the lines of force.

We can do a similar exercise with a gravity field; only this time there are attractive forces involved and no repulsive forces. 

The Earth, in common with many other planets, is very nearly a perfect sphere (relatively smoother than a billiards ball).  Its gravity field is radial with the pull being directly towards the centre.  The closer in we get, the stronger the pull.

The concentration of gravitational field lines is an indication of the gravitational field strength at any point, which is formally defined as:

The gravitational force per unit mass at that point.

So we can write that statement in code as:

            g = gravitational force = F  [Units – newtons per kilogram (N/kg)]

                        mass                 m

You will have met the expression F/m in the context of a = F/m, so it doesn’t take a genius to see that gravitational field strength is the same thing as acceleration.  A gravitational field strength of 9.81 N/kg causes an acceleration of 9.81 m/s².

From our derivation of gravitational field strength and Newton’s Law of Gravitation, we can derive an equation to tell us the value of the field strength at a distance r from the centre of the Earth, mass M.

            F = -GmM                  Þ g = F/m  Þ g = -GmM = -GM

                     r²                                                     m r²       r²

Gravitational Field strength is a vector because it has a direction.  Remember always to take the radius of the Earth into account.

Worked Example

If we draw a graph of the gravity field strength going to the centre of the Earth, we would not find a simple inverse square relationship, as if the Earth were a point mass.  Instead we would see a more complex variation.  The Earth is not a uniform sphere, but has a dense core and a less dense crust. The value of g can be calculated from seismic surveys and the graph above shows its variation with distance.  Note that the graph is drawn as a negative function as the field is attractive.  You may see it drawn as a positive function.

At the centre of the Earth, g = 0, because matter will be pulled equally in all directions so that the overall force is zero.

A bear-trap

When we work out g for distances away from the Earth's surface, we MUST add on the radius of the Earth.  Suppose a satellite were flying 1.0 × 10⁶ m above the Earth's surface the distance r would NOT be 1.0 × 10⁶ m but (1.0 × 10⁶ m + 6.37 ´ 10⁶ m) = 7.37 × 10⁶ m.

Gravitational Potential

The gravitational potential of a point is defined as:

The work done on a unit mass in moving it to that point from a point remote from all other masses.

In other words, this means the work done to move a unit mass from infinity to the point under consideration.

The zero point for gravitational potential is at infinity, so as we are moving towards the Earth, we are getting work out of the system.  Therefore gravitational potential is negative.  If we moved the object away to infinity, we would have to do a job of work on the object.  We work out gravitational potential with the formula:

[G = 6.67 ´ 10^-11 Nm² kg^-2; M = mass of Earth or other planet; r = distance from centre.]

If we go back to our graph of gravitational field strength against distance we get:

We see that the gravitational potential is the area under the graph:

• since work = force ´ distance moved. 

• work done per unit mass = force per unit mass ´ distance moved.

If we plot V against r, we get the following graph:

From this graph we can see that the gradient gives us the gravitational field strength, g.  So we can say that:

We can go on to use the idea of gravitational potential to find an expression for the potential energy.  We have defined potential as energy per unit mass, so we can work out the total energy for any mass by multiplying the potential by the mass:

Another bear trap

E_p = mgDh is only true when we are very close to the Earth’s surface.  So do NOT use it for objects out in space.  Instead use

Worked Example