The big idea: A satellite or planet moves in a circle because gravity pulls it toward the central body.
Gravity as the centripetal force
- That inward pull is the centripetal force — the single force that keeps any object turning in a circle instead of flying off straight.
- Nothing pushes the satellite forward — it just keeps 'falling' around the central body.
Gravity points inward, toward the centre of the central body M. For an orbiting satellite this inward pull is the centripetal force — it is what curves the path into a circle.
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Define: centripetal: Centripetal means 'toward the centre'. The centripetal force is whatever points inward and bends the path into a circle.
For an orbit, that force is gravity — there is no separate 'orbit force'.
A satellite in a circular orbit feels just one force — gravity — pulling it toward the central body (drawn here pointing down, toward the centre). Its acceleration points the same way, so it keeps turning instead of flying off in a straight line.
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Drawing the velocity and the acceleration
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Because gravity is the centripetal force, we set the two equal.
The gravitational force on the orbiting mass m is its weight in the field (g = GM/r², so the force is mg = GMm/r²), and the centripetal force needed is mv²/r:
- orbital speed (m s⁻¹)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻² (given)
- mass of the central body, e.g. Earth or the Sun (kg)
- orbit radius — centre of the central body to the orbiting body (m)
- gravitational field strength (N kg⁻¹)
- gravitational force (N)
- mass feeling the force (kg)
- gravitational constant (given)
- mass of the central body (kg)
- distance from the centre of the central body (m)
Cancel the m and one factor of r, then make v the subject. This gives the orbital speed — notice the mass of the satellite has vanished, so a heavy and a light satellite at the same radius orbit at the same speed:
- orbital speed (m s⁻¹)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻² (given)
- mass of the central body, e.g. Earth or the Sun (kg)
- orbit radius — centre of the central body to the orbiting body (m)
From speed to period: In one orbit the satellite travels a full circumference 2πr in one period T, so its speed is also v = 2πr ÷ T (given in the data booklet for circular motion).
Putting the two expressions for v together gives Kepler's third law: T² = (4π²/GM) r³ — period squared is proportional to radius cubed.
- orbital speed (m s⁻¹)
- orbit radius (m)
- orbital period — time for one full orbit (s)
A satellite orbits Earth (M = 6.0 × 10²⁴ kg) at a radius of 7.0 × 10⁶ m from Earth's centre. (a) Find its orbital speed. (b) Find its orbital period. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
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How this is tested — orbits appear both as a quick MCQ and as an extended calculation:
Paper 1A
- State that the acceleration points toward the central body (gravity is centripetal).
- Find an orbital speed with v = √(GM/r).
Paper 2
- Show that the gradient of a T² against r³ graph is 4π²/GM, then find the mass of the Sun.
- Or find the height of a satellite from its period.
The classic trap: r is measured from the centre of the planet, so a satellite's height above the surface is r − (planet radius), not r itself.
Weighing the central body: Kepler's third law T² = (4π²/GM) r³ contains the central mass M but not the orbiting mass.
So measuring any orbit's T and r lets you rearrange for M = 4π²r³ ÷ (GT²) — that is how astronomers find the mass of the Sun from the planets' motion.
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A satellite moves at constant speed in a circular orbit around Earth. State the direction of its acceleration, and explain why it has an acceleration even though its speed is constant.
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A planet orbits a star in a circle of radius 2.0 × 10¹¹ m with a period of 2.0 × 10⁷ s. Determine the mass of the star. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
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