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2.6.4 Gravitation and Energy

By the end of this lesson you can:

  • Calculate gravitational potential energy between two masses
  • Relate energy changes to moving between orbits
  • Calculate escape velocity
Find your course: NSW Year 11NSW Year 12: Module 5, IQ3QLDVICAP Physics 1: Unit 6IB: D.1 Gravitational fields
🎥 Watch📝 Practise🕹 Explore🎯 Exam ready🍎 Teachers
🎥Watch
Learn the idea: a short explanation, then a check.
Previously we have discussed energy as the ability to do work. Another way of expressing this is that when we do work, we change the energy of the system.
So it naturally follows that when a satellite is in orbit, it has gravitational energy by way of  it position in a gravitational field. Work had to be done on it to get it there. Likewise, if it has a velocity it has kinetic energy
In this section we will explore the concept of energy in the gravitational context.

Gravitational Potential Energy

​Gravitational potential energy (U), or simply Gravitational energy, is due to the work done by a gravitation field.
Previously we have discussed this using the formula

U = mgΔh

Unfortunately this formula is a poor model here since it assumes 'g', the gravitational acceleration, is constant. It is not.


It suffices for small scale situations on earth, but when we look at larger situations, it breaks down.

​

​This video discusses this concept, allowing for change in gravitational fields and shows, in a simple way how the formula is determined. All this in the context of a teddy bear.

In a rush?
Short on time?
Here is a brief review of gravitational potential energy.
Otherwise , move one

📝Practise
Worked examples first, then problems to try, then worksheets.
Problems to try

Constants used: G = 6.674 × 10-11 N m2 kg-2; Earth: mass 6.0 × 1024 kg, radius 6.37 × 106 m; Mars: mass 6.39 × 1023 kg. Gravitational energy is negative, U = -GMm / r.

1. An Earth satellite has a mass of 100 kg and is at an altitude of 2000 km. What is its gravitational energy?

Given: m = 100 kg, altitude h = 2000 km = 2.0 × 106 m

Step 1 (Distance from the centre of the Earth): r = R + h = 6.37 × 106 + 2.0 × 106 = 8.37 × 106 m

Step 2 (Gravitational energy): U = -GMm / r

U = -(6.674 × 10-11 × 6.0 × 1024 × 100) / (8.37 × 106) ≈ -4.78 × 109 J

Answer: -4.78 GJ
2. The gravitational energy of a Russian Mars orbiter is 4.1 × 1012 J. It orbits Mars at a distance of 4.8 × 103 km from the centre. What is the mass of the orbiter?

Given: |U| = 4.1 × 1012 J (the energy is really negative, so the size is used), r = 4.8 × 103 km = 4.8 × 106 m, MMars = 6.39 × 1023 kg

Step 1 (Rearrange): |U| = GMm / r → m = |U| r / (GM)

Step 2 (Calculate the mass):

m = (4.1 × 1012 × 4.8 × 106) / (6.674 × 10-11 × 6.39 × 1023) = (1.97 × 1019) / (4.26 × 1013) ≈ 4.6 × 105 kg

Answer: 4.6 × 105 kg
🎥Watch
Learn the idea: a short explanation, then a check.

Total Energy

A satellite in orbit is in motion. This means it has both gravitational potential energy (U) and kinetic energy (K). So how are they related? IF an object increases its gravitational energy , does that mean it will lose kinetic energy? Well that depends - is it in a new orbit, is their work done on it? This video explores the key concepts.
 
📝Practise
Worked examples first, then problems to try, then worksheets.
Sample Problem
We are now ready to try a sample problem
Below is a sample problem with a video that explain how to solve it. It is suggested you try the problem beforehand, as this actually aids understanding, even if you are unsure if you are correct.

A 50 tonne satellite is in a circular orbit at an altitude of 500 km. What is the total energy of the satellite?

🕹Explore
Test your predictions in a simulation.

Interactive

This animation will allow to move satellites into higher or lower orbits. Firing the rockets will therefore alter the total energy of the satellite and hence the orbit.

🎯Exam ready
Check you can do it under exam conditions.

8. How well do you know gravity?


​Test your understanding of gravity. So do the Quiz and try to get full marks

Then check your understanding if necessary, with the video