Orbits Cambridge International AS & A Level Physics revision
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In plain words
A satellite is falling all the time. It is also moving sideways so fast that the ground curves away beneath it as quickly as it falls, so it never gets any closer. Gravity supplies exactly the centripetal force the circle needs.
Three things to know
- For a circular orbit, the gravitational force is the centripetal force: GMm ÷ r² = mv² ÷ r, so v² = GM ÷ r. The satellite's own mass cancels out.
- Putting v = 2πr ÷ T gives T² = 4π²r³ ÷ GM. The period depends only on the radius of the orbit and the mass at the centre. Bigger orbits are slower and take longer.
- A geostationary satellite stays above the same point on the Earth. Its orbit must have a period of 24 hours, be directly above the Equator, and go from west to east, the way the Earth spins.
Worked example
Find the speed of a satellite in a circular orbit 400 km above the Earth's surface. (For the Earth, GM = 3.98 × 10¹⁴ N m² kg⁻¹ and the radius is 6.37 × 10⁶ m.)
- Radius of the orbit r = 6.37 × 10⁶ + 0.40 × 10⁶ = 6.77 × 10⁶ m.
- v² = GM ÷ r = 3.98 × 10¹⁴ ÷ 6.77 × 10⁶ = 5.88 × 10⁷.
- v = 7.7 × 10³ m/s.
Tips and tricks
- Start every orbit question by writing "gravitational force = centripetal force".
- Astronauts in orbit feel weightless because they and their spacecraft are falling together, not because gravity has gone.
It lands in your notebook with its questions as flashcards.
Orbits: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
A satellite is moved to a higher circular orbit. What happens to its speed?
v² = GM ÷ r.
What is the period of a geostationary satellite?
It keeps pace with the turning Earth.
Above which part of the Earth must a geostationary satellite orbit?
Only an orbit above the Equator stays over one point.
Two satellites of different mass share the same circular orbit. How do their speeds compare?
The satellite's mass cancels out of the equation.
The radius of a satellite's orbit is increased. What happens to its period?
T² is proportional to r³.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 9 marks. Write your answers on paper, then check them.
Orbits
Cambridge International AS & A Level Physics 9702 · 9 marks · papermunch.org
Name ______________________________ Date ______________
Show that the period T of a circular orbit of radius r round a mass M is given by T² = 4π²r³ ÷ GM.[3]
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GMm ÷ r² = mv² ÷ r, so v² = GM ÷ r. Also v = 2πr ÷ T, so 4π²r² ÷ T² = GM ÷ r, which rearranges to T² = 4π²r³ ÷ GM.
Find the radius of a geostationary orbit. (GM for the Earth = 3.98 × 10¹⁴ N m² kg⁻¹; 24 hours = 86 400 s.)[3]
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4.2 × 10⁷ m. r³ = GMT² ÷ 4π² = 7.5 × 10²² m³.
State three features of a geostationary orbit.[3]
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Its period is 24 hours. It is directly above the Equator. It goes from west to east, the same way as the Earth turns.
Answers: Orbits
- 1. GMm ÷ r² = mv² ÷ r, so v² = GM ÷ r. Also v = 2πr ÷ T, so 4π²r² ÷ T² = GM ÷ r, which rearranges to T² = 4π²r³ ÷ GM.
- 2. 4.2 × 10⁷ m. r³ = GMT² ÷ 4π² = 7.5 × 10²² m³.
- 3. Its period is 24 hours. It is directly above the Equator. It goes from west to east, the same way as the Earth turns.