The big idea: Let go of your phone and it drops straight down — the Earth is pulling on it, as it pulls on every mass nearby.
Field and field strength
- That invisible pull filling the space around a mass is a gravitational field.
- The field strength g measures how strong the pull is per kilogram (N kg⁻¹).
- It is stronger for bigger masses and closer in.
A gravitational field points inward, towards the mass M, because gravity always pulls. The lines get further apart as you move out — the field is weaker further away.
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Spot it: Gravity is always attractive — the field lines point inward, towards the mass.
The lines spread out as you move away, so the field gets weaker the further out you go.
Drawing field lines
A field line shows the force, not the path: The arrow on a field line shows the direction of the force on a mass, and so of its acceleration.
A moving mass (a space probe) need not travel along the line: its acceleration points along the field line, while its path can curve across it.
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Newton's law of gravitation gives the pull between any two masses. The force grows with the masses and shrinks with the square of the distance between them:
- 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)
When can a body count as a point mass?: A sphere of uniform density pulls on things outside it as if all its mass were at its centre, so measure r from centre to centre.
Any body that is small compared with the distance between the bodies can also be treated as a point.
The gravitational field strength g is the force per kilogram on a small mass placed in the field. Dividing Newton's law by that small mass m gives a neat form that only needs the big mass M and the distance:
- gravitational field strength (N kg⁻¹), also the free-fall acceleration
- gravitational force on the small mass (N)
- the small mass placed in the field (kg)
- mass of the planet or star making the field (kg)
- distance from the centre of M (m)
F = m g. Cover the one you want: m and g side by side → multiply (F = m g); F above g → divide (m = F ÷ g); F above m → divide (g = F ÷ m).
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g is also the acceleration of free fall: Because F = mg and F = ma, the field strength g equals the acceleration a falling mass would have.
That is why two different masses dropped at the same place fall with the same acceleration — g does not depend on the falling mass m.
A planet has mass 6.0 × 10²⁴ kg and radius 6.4 × 10⁶ m. Find the gravitational field strength at its surface. (G = 6.67 × 10⁻¹¹ N m² kg⁻².)
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Where the field is zero
Two moons of masses M and 9M are 40 000 km apart (centre to centre). Find how far from the smaller moon the resultant gravitational field strength is zero, on the line joining them.
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How this is tested — gravitation and field strength turn up across the Fields theme:
Paper 1A
- Quick inverse-square reasoning — how g changes at a different distance (three times farther → one ninth).
- Comparing the accelerations of masses falling from different heights.
- Pick the graph that gives a straight line: F (or g) against 1/r² goes through the origin.
Paper 2
- Calculate g = GM/r² at a planet's surface.
- Or a star's field at an orbital distance.
The classic trap: Forgetting the square — moving three times farther divides g by 3² = 9, not by 3.
The inverse-square shortcut: Because g is proportional to 1/r², you don't always need G and M.
If the distance is multiplied by a number n, the field strength is divided by n². So r ×2 → g ÷4, and r ×3 → g ÷9.
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The gravitational field strength a distance r from the centre of a planet is 8.1 N kg⁻¹. Find the field strength at a point three times as far from the centre (a distance 3r).
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Two small balls, one of mass 2.0 kg and one of mass 5.0 kg, are released from rest at the same point above the planet's surface. Compare their initial accelerations.
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