Key Idea: Topic 4.1 is about gravity as a field — how a mass fills the space around it with a pull, and what that pull does to planets, satellites and rockets.
Field
- F = GMm/r², g = GM/r²
- Inverse square: r ×n → g ÷n²
- r from the centre
Orbit
- Gravity is the centripetal force
- v = √(GM/r): mass cancels
- Acceleration to the centre
Kepler
- Ellipse, Sun at a focus
- Equal areas in equal times
- T² ∝ r³
It ties together three ideas: how strong the pull is (Newton's law F = GMm/r² and the field strength g = GM/r²), how things orbit (Kepler's laws, with T² ∝ r³) and why a satellite stays up (gravity is the centripetal force). It is examined on Paper 1A (inverse-square reasoning, recognising T² ∝ r³, the direction of a satellite's acceleration) and on Paper 2 (substitute into g = GM/r², compare two orbits with T²/r³, 'weigh' a central body with M = 4π²r³/(GT²), find a satellite's height from its period).
Key formulas
The first block is given in the data booklet (D.1). The orbit results after it are built from the given equations, so know how each one comes about.
- gravitational force between the masses (N)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻²
- the two masses (kg)
- distance between their centres (m)
- gravitational field strength (N kg⁻¹), also the free-fall acceleration
- the small mass placed in the field (kg)
- mass of the planet or star making the field (kg)
- orbital speed (m s⁻¹)
- orbital period — time for one full orbit (s)
Not in the booklet: build them from the ones above.
- orbital speed (m s⁻¹)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻²
- mass of the central body (kg)
- orbit radius from the centre of the central body (m)
- orbital period — time for one full orbit (s)
Which equation, and when?
The most-tested decision in this topic: is the question about the field at a point, or an orbit's timing or speed?
| What the question wants | Equation to reach for | Watch out for |
|---|---|---|
| Field strength g (or force F) at a distance r | g = GM/r² (or F = GMm/r²) | Inverse-square: r ×n → g ÷n² |
| Compare two orbits round the same body | TA²/rA³ = TB²/rB³ | G and M cancel — you never need them |
| Orbital speed (no period given) | v = √(GM/r) | The orbiting mass cancels — it is not in it |
| Link a period to a radius, or weigh a body | T² = 4π²r³/(GM) | Rearrange to M = 4π²r³/(GT²) to find M |
The big results — what each depends on
| Quantity | Formula | Grows with… | Does NOT depend on |
|---|---|---|---|
| Field strength g | g = GM/r² | central mass M | the falling object's mass — so all masses fall equally |
| Orbital speed v | v = √(GM/r) | central mass M (smaller for big r) | the orbiting body's mass |
| Orbital period T | T² = 4π²r³/(GM) | orbit radius r (T² ∝ r³) | the orbiting body's mass |
Kepler's three laws at a glance
| Law | What it says | Exam use |
|---|---|---|
| 1st (shape) | Orbits are ellipses, with the Sun at one focus | State the law; the Sun is offset from the centre |
| 2nd (speed) | Faster near the Sun, slower far away (equal areas in equal times) | Explain why a planet's speed / kinetic energy changes |
| 3rd (timing) | T² ∝ r³ — bigger orbit, longer period | Ratio two orbits with TA²/rA³ = TB²/rB³ |
The inverse-square shortcut
| Move the distance to… | Field strength g = GM/r² becomes… | Why |
|---|---|---|
| 2r (twice as far) | g ÷ 4 | g ∝ 1/r², so ÷2² = ÷4 |
| 3r (three times as far) | g ÷ 9 | ÷3² = ÷9 — the classic trap is dividing by 3 |
| ½r (half as far) | g × 4 | ×(½)⁻² = ×4 — closer in is much stronger |
IB-style worked examples
A planet has mass 5.4 × 10²⁴ kg and radius 6.0 × 10⁶ m. (a) Find the gravitational field strength at its surface. (b) State the field strength at a point three times as far from the centre. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
Two moons, P and Q, orbit the same planet. Moon Q's orbital radius is 4.0 times that of moon P, and moon P's period is 1.5 days. Find moon Q's orbital period.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A planet orbits a star in a circle of radius 1.8 × 10¹¹ m with a period of 1.6 × 10⁷ s. Determine the mass of the star. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A satellite orbits Earth (mass 6.0 × 10²⁴ kg, radius 6.4 × 10⁶ m) at a height of 6.0 × 10⁵ m. (a) Find its orbital speed. (b) Find its period. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: In every gravity equation r is the distance from the centre. For a satellite at height h above a planet of radius R, use r = R + h.
Quick self-check
Tap each card to reveal the answer.
Move three times farther from a planet — what happens to g? It is divided by 3² = 9 (inverse-square law). Dividing by 3 is the classic mistake.
Two different masses are dropped at the same point — do they accelerate the same? Yes — the acceleration is g = GM/r², which does not depend on the falling mass.
Plot T² against r³ for moons of one planet — what shape? A straight line through the origin, because T²/r³ is a constant (T² ∝ r³).
Does a heavier satellite at the same radius orbit faster? No — v = √(GM/r) has no satellite mass in it, so both orbit at the same speed.
What provides the centripetal force for an orbit? Gravity from the central body: GMm/r² = mv²/r.
Where is the Sun in a planet's elliptical orbit? At one focus, not the centre (Kepler's first law).
Highest-yield exam reminders
Exam tips
- g = GM/r² and F = GMm/r² are both inverse-SQUARE: multiply the distance by n and you divide g (or F) by n². Three times farther → one ninth, not one third.
- Field strength g is also the free-fall acceleration, and it does not depend on the falling mass — so all masses dropped at one place accelerate equally. N kg⁻¹ and m s⁻² are the same unit for g.
- For two orbits round the SAME body, use TA²/rA³ = TB²/rB³ — the constant 4π²/(GM) cancels, so you never need G or M. Watch the powers: T is squared, r is cubed.
- Gravity is the centripetal force for an orbit (GMm/r² = mv²/r). No period given for a speed? Use v = √(GM/r). Linking T and r, or finding the central mass? Use T² = 4π²r³/(GM), i.e. M = 4π²r³/(GT²).
- Orbit radius r is measured from the CENTRE of the planet — a satellite's height above the surface is r minus the planet's radius. A geostationary satellite has a 24 h period over the equator, so it stays above one fixed point.
- Kepler's second law: the line from the Sun to a planet sweeps equal areas in equal times, so a planet is fastest when nearest the Sun.