WACE Physics — Unit 3
Kepler's Laws — Flashcards & Quiz
The WACE Physics Unit 3 syllabus uses Newton's law of universal gravitation to explain Kepler's third law of planetary motion, modelling planets and satellites as moving in uniform circular motion. You need to derive the relationship T²/r³ = GM/4π² and apply it to orbital periods and radii. Kepler's first and second laws (elliptical orbits and equal areas in equal times) are useful background but are not listed in the syllabus.
Key Points
- Kepler's third law: for all bodies orbiting the same central mass, T²/r³ is constant.
- Derivation: gravitational force provides the centripetal force, GMm/r² = mv²/r, and with v = 2πr/T this gives T²/r³ = 4π²/GM, or r³/T² = GM/4π².
- The constant depends only on the central mass M — the orbiting body's mass cancels out.
- Use r = distance from the centre of the central body: for a satellite, r = Earth's radius + altitude.
- Comparing two orbits around the same body: T₁²/r₁³ = T₂²/r₂³, so units cancel if both use the same units.
- Background only: Kepler's first law (elliptical orbits) and second law (equal areas in equal times) are not listed in the WACE syllabus.
Common Mistakes to Avoid
- Using the same constant for orbits around different central bodies — T²/r³ depends on the central mass.
- Using altitude instead of orbital radius — add the central body's radius.
- Forgetting to convert the period to seconds before using G in SI units.
- Including the satellite's mass in the calculation — it cancels when gravity is set equal to the centripetal force.
- Mixing up r³/T² and T²/r³ when rearranging.
Exam Strategy
Kepler's third law questions usually give an orbital period or radius and ask for the other, or ask you to derive the relationship. Method: (1) state that gravity provides the centripetal force, (2) write GMm/r² = 4π²mr/T² and cancel m, (3) rearrange for the unknown, (4) check r is measured from the centre of the central body and T is in seconds, (5) for two orbits around the same body, use the ratio form so the constant cancels.
Sample Flashcards
Q1: State Kepler's third law of planetary motion.
For all bodies orbiting the same central mass, the square of the orbital period is proportional to the cube of the orbital radius: T² ∝ r³, so T²/r³ is the same constant for every orbit around that body. In WACE Physics the orbits are modelled as uniform circular motion, so r is the radius of the circular orbit measured from the centre of the central body.
Q2: Derive Kepler's third law from Newton's law of universal gravitation.
The gravitational force provides the centripetal force: GMm/r² = mv²/r. For uniform circular motion, v = 2πr/T, so GMm/r² = m(4π²r²/T²)/r = 4π²mr/T². Cancel m and rearrange: T²/r³ = 4π²/GM, or equivalently r³/T² = GM/4π². The right-hand side contains only the central mass M, so the ratio is the same for all orbits around that body.
Q3: Why does the constant in Kepler's third law not depend on the mass of the satellite?
The satellite's mass m appears on both sides of GMm/r² = 4π²mr/T² and cancels. The result, T²/r³ = 4π²/GM, depends only on the mass of the central body. A small satellite and a large space station in the same orbit around Earth therefore have the same period, but the constant for orbits around the Sun is different from the constant for orbits around Earth.
Q4: How do you use Kepler's third law to compare two orbits around the same body?
Because T²/r³ is the same for both orbits, T₁²/r₁³ = T₂²/r₂³. Rearranged: T₂ = T₁ × (r₂/r₁)^(3/2). The units cancel as long as both periods and both radii use the same units, so you do not need G or M. Example: a satellite at four times the orbital radius of another has a period 4^(3/2) = 8 times longer.
Q5: How can Kepler's third law be used to find the mass of a planet?
Measure the period T and orbital radius r of a moon or satellite orbiting the planet, then rearrange T²/r³ = 4π²/GM to M = 4π²r³/(GT²). Use SI units: r in metres, T in seconds, G = 6.67 × 10⁻¹¹ N m² kg⁻². This works because the orbiting body's mass cancels, so only the central mass is found.
Sample Quiz Questions
Q1: T²/r³ is the same for objects orbiting different central bodies.
Answer: FALSE
The constant depends on the central body's mass M.
Revision Tip
Build a Revizi deck that pairs the derivation with a few worked satellite and moon examples, then practise rearranging for T, r and M until it is automatic.
Related Concepts
Last updated: 9 October 2026 · 6 sample flashcards · 1 sample quiz questions