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
Energy
- V = −GM/r, Eₚ = −GMm/r
- Orbit: E = −GMm/2r
- Escape: v = √(2GM/r)
It ties together four 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), why a satellite stays up (gravity is the centripetal force, giving v = √(GM/r)), and the energy of being in a gravity well (the negative potential energy, and the escape speed). It is examined on Paper 1A (quick MCQs — inverse-square reasoning, recognising T² ∝ r³, the direction of a satellite's acceleration, why escape speed ignores the rocket's mass) and on Paper 2 (substitute into g = GM/r², compare two orbits with T²/r³, 'weigh' a central body with M = 4π²r³/(GT²), or use energy conservation for escape and impact speeds).
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)
- gravitational potential — energy per kilogram (J kg⁻¹)
- orbital speed (m s⁻¹)
- orbital period — time for one full orbit (s)
- gravitational potential energy of the object (J)
- escape speed — the launch speed needed to escape (m s⁻¹)
- change in gravitational potential (J kg⁻¹)
- work done (J)
Not in the booklet: build them from the ones above.
- orbital period — time for one full orbit (s)
- orbital radius — distance from the central body's centre (m)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻²
- mass of the central body being orbited (kg)
- total energy of the orbiting mass (J)
Which equation, and when?
The most-tested decision in this topic: is the question about the field at a point, an orbit's timing or speed, or the energy of being in the well?
| 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 |
| Potential energy, or a launch/impact speed | Eₚ = -GMm/r, vₑₛc = √(2GM/r) | Eₚ is negative; vₑₛc ignores the object's mass |
| Field strength from a potential graph or map | g = −ΔV/Δr | g is the size of the gradient |
| Energy of an orbit, or a change of orbit | E = −GMm/2r | Higher orbit: less negative E |
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 |
| Escape speed vₑₛc | vₑₛc = √(2GM/r) | central mass M (∝ √M) | the escaping object'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 moon has mass 8.0 × 10²² kg and radius 1.8 × 10⁶ m. (a) Find the escape speed from its surface. (b) State whether a 2000 kg lander or a 500 kg probe needs the greater launch speed to escape. 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.
Why is gravitational potential energy negative? We set it to zero at infinity; anywhere closer in, gravity has already pulled the object 'downhill', so it sits in a well.
Does escape speed depend on the rocket's mass? No — the mass cancels in vₑₛc = √(2GM/r). It depends only on the planet's M and r.
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.
- Eₚ = -GMm/r and V = -GM/r are always negative and zero at infinity — never drop the minus sign. Many 'launch or impact speed' questions are energy conservation: kinetic energy gained = depth of the well climbed.
- Escape speed vₑₛc = √(2GM/r) depends only on the planet (M and r), not on the escaping object — and vₑₛc ∝ √M, so quadrupling the mass at fixed radius doubles it.
- A small air drag lowers a satellite's total energy: its orbit shrinks and it moves FASTER (KE up ΔE, Eₚ down 2ΔE).