Episode 404: Energy and gravitational fields

In this episode, students will appreciate the changes to the ways that energy is stored as a body moves in a gravitational field. They have seen this concept before, for a uniform gravitational field, in the form

change in energy stored gravitationally, or change in gravitational potiential energy = m x g x Δh

where Δh = change in height, but this will be generalised to non-uniform fields around point or spherical masses. They will then be introduced to the concept of gravitational potential and its uses, before finally making the link between that rate at which gravitational potential changes from place to place and the field strength.

Note that the mention of infinity often gets students’ minds racing, and puzzling over seeming paradoxes such as “If the gravitational field is infinite in extent, what does it mean to be at infinity where the field is zero?” It’s best to approach this pragmatically; by infinity, we really mean as far away from all masses in the universe as we need to be to make the gravitational forces negligible in whatever context we are looking at.

This is quite a long episode, and worth spending the time on to give a good grounding in these ideas. They will be important again with electric fields.

Lesson Summary

• Discussion: Work and energy (5 minutes)
• Worked examples: Energy in a constant gravitational field (10 minutes)
• Discussion: Energy in a non-uniform field (10 minutes)
• Discussion: Potential (5 minutes)
• Worked examples: Energy and potential (15 minutes)
• Student questions (15 minutes)
• Spreadsheet exercise (20 minutes)
• Discussion: Equipotentials, potential gradient and field strength (10 minutes)
• Spreadsheet exercise: Potential gradient and field strength (20 minutes)

Discussion: Work and Energy

This draws on what students should already know about work and energy.

What is work? (Work is done when a force moves its point of application through some distance in the direction of the force. In the absence of frictional forces, this work done is stored as the energy of the body on which the force was acting.)

If I lift an object up from the floor to above my head (demonstrate it!), have I done work on it? (Yes. I applied an upwards force on the object which itself moved upward too.)

What happened to the work I did on the object? (It is stored as energy of the object. We say that the energy is stored gravitationally. 'Gravitationally' because we have to do work against gravity to lift the object. If you have something breakable (but safe and cheap!) they will really see this energy being realised.)

Worked examples: Energy in a constant field

Now, students have come across these ideas before at pre-16 level in the form:

Change in E_G = mgh.

It is as well to revise this with a few worked examples now.

Episode 404-1: Energy in a constant field (Word, 26 KB)

Discussion: Energy in a non-uniform field

Now you can extend these ideas to energy in a non-uniform field.

All the previous examples involved changes to the energy stored gravitationally near the surface of the Earth. What would be the problem if we wanted to use the same equation to work out changes to the energy stored gravitationally for, say, a rocket launched to the moon? (The gravitational field strength is not constant – the value of g changes.)

We therefore need another way of calculating changes to the energy stored gravitationally in non-uniform fields. The full treatment of how we arrive at this formula requires off-syllabus calculus that would actually be accessible to more able students. We find that we can calculate energy of a mass m at a point distance r from a (point or spherical) mass M by:

E_G =   −  GMmr

There are several very important points to note about this equation:

1. We know that the further you get from an object, the greater the energy stored in the gravitational field. (As something must have done more work against gravity to get you there). Thus when you are infinitely far away, you have as high an energy relative to it as possible. We choose (arbitrarily) to make the value of energy of all bodies at infinity zero. Then since this is the highest value of energy, all real values of energy stored gravitationally (closer than infinity) must be negative. Therefore the minus sign in the equation is not optional; it must always be included and all values of energy stored in a gravitational field are negative. (This is not the case when we come on to electric fields, because they can be repulsive too).
2. Note that we have written energy here, and not Change in energy. By defining a point relative to which all energy is measured, we can now talk about absolute values of energy rather than just changes. This point is at infinity (see note 1 above).
3. Note that energy follows an inverse proportion law (1r) and not an inverse square law (1r²)

Discussion: Potential

The weight of an object in the Earth’s gravitational field depends upon the mass of the object (as well as the mass of the Earth). However, as we have already seen, the field strength at a point is independent of the object placed there (because it is defined as force per unit mass of the object). Thus we can think of field strength as a property of the field at a point, and not the particular object placed there.

Similarly, the energy stored in the gravitational field by moving a body to a place relative to the Earth depends upon the mass of the object (as well as the mass of the Earth) and the distance between them. How do you think we can get a quantity related to energy in the field at a point, which does not depend upon the object placed there? (By looking at the energy stored per unit mass of the object, thus removing the dependence on the mass of the object just as we did with field strength.)

We define the potential at a point in a field as the energy per unit mass stored by placing 1kg at that point in the field. We can get equations for potential using this definition. For a field due to a (point or spherical) mass M, we have:

E_G =  −  GMmr

And so the potential, V , is given by:

V =  −  GMr

A few points to make:

1. This only relates to the field due to a (point or spherical) mass M.
2. V is measured in J kg^-1. It follows an inverse proportion law (1r ) not inverse square (1r²).
3. Just as with the equation for energy, the minus sign is not optional. All real potentials are negative, and the zero of potential is at infinity (since all objects store no energy gravitationally at infinity).
4. Potential, like field strength is a property of the field at a point, and is independent of the object placed there. Two objects with different masses at the same point in the field are subject to the same potential, but have different energies.
5. For uniform fields (e.g. close to the surface of the Earth), we can use

change in potential = gh

(since change in E_G = mgh and potential = E_Gm).

(Potential will be returned to again when we study electric fields. There, differences in potential (or potential differences, pds) are what we often call voltages.)

Worked examples:Energy and potential

Episode 404-2: Energy and potential (Word, 30 KB)

Student questions

These questions relate to a spacecraft travelling from the Moon to the Earth.

You may wish to omit questions 3–5 inclusive if your students have not covered the topic of momentum.

Episode 404-3: Gravitational field between Earth and moon (Word, 74 KB)

Spreadsheet exercise

You can extend the spreadsheet activity from Episode 402 by including the idea of gravitational potential.

Episode 402-3: Data from the Apollo 11 mission (Word, 85 KB)

Episode 404-4: Analysing Apollo 11 data, gravitational potential (Word, 29 KB)

Discussion: Equipotentials, potential gradient and field strength

Equipotentials join points of equal potential. They are very simple in the cases of uniform fields (close to the surface of the Earth) and radial fields (for point and spherical masses):

How far apart are the equipotential lines in the first diagram? (Since change in potential for a constant field is simply gh (where h is change in height), we have gh = 9.8 J kg^-1 }. Therefore h = 1 m – the lines are 1 m apart.)

What shapes are equipotentials in the real world? (Equipotentials are surfaces rather than lines in the real 3-dimensional world (i.e. horizontal planes rather than lines close to the surface of the Earth, and concentric spheres rather than concentric circles about a point/spherical mass), but we can only capture a slice of them on paper.)

Equipotentials are always perpendicular to field lines. Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour maps give us information about geographical heights.

What does it mean on a contour map if the contours are very close together? (On a contour map, the contours may be marked off at, say, 5 m intervals. Therefore, if they are close together, it means that the land on which they lie must be very steep.)

What do you think it therefore means if equipotential surfaces are close together? (An educated guess would be that it means that the gravitational field is very strong there. That would be correct – if the equipotentials are close together, a lot of work must be done over a relatively short distance to move a mass from one point to another against the field – i.e. the field is very strong. Hence on the drawing above of the equipotentials around point or spherical masses, the equipotential surfaces get further and further apart as the field strength decreases with distance.)

In fact, the field strength is given by the negative of the gradient of the potential:

g = −  Δ V Δ r

For students that might struggle with a derivative as above, it could be introduced as:

g = − change in potentialchange in distance.

Spreadsheet exercise: Potential gradient and field strength

In this exercise, students will manipulate (calculated) raw data on the variation of potential with height above the Earth’s surface. By the end, they will calculate the change in potential per metre (i.e. the potential gradient) and compare it to field strength and find that the two are equal. When they calculate the change in potential between say 200 km and 300 km, and divide it by the 100 km distance to get a field strength, they are actually getting an average field strength between 200 km and 300 km, which is likely to be very close to the field at 250 km (though not exactly because the variation in potential is non-linear). Hence the field strengths on sheet 2 being given at 50 km, 150 km, etc.

They are expected to have used spreadsheets before and are asked to do some fairly simple spreadsheet manipulations.

Episode 404-5: Potential gradient and field strength (Word, 28 KB)

Episode 404-6: Excel file potential gradient and field strength (Word, 53 KB)