Gravitational force and field strength
Key idea: H2 Physics lessons on inverse-square gravitation, field strength, gravitational potential, escape speed and satellite orbits.
Continue where you stopped
The core idea
Build the idea
Learn the idea
Big question: How does a mass create a force field around it?
Two point masses attract with F = GMm/r². Gravitational field strength is force per unit test mass, so a spherical mass produces g = GM/r² directed inward outside the mass. Superpose field vectors when more than one source acts, and distinguish the source mass from the test mass.
Apply the inverse-square interaction
Point masses, and spherically symmetric bodies viewed from outside, attract with F = Gm1m2/r². The separation r is centre to centre, not height above a surface.
The forces on the two masses are equal and opposite even when their masses differ. Their accelerations differ because a = F/m.
Check your understanding: Earth pulls an astronaut more strongly than the astronaut pulls Earth. True or false?
False. The forces are equal; Earth's much larger mass gives it a much smaller acceleration.
Turn force per mass into field strength
Gravitational field strength g = F/m gives g = GM/r² around a spherical mass M. A freely falling small body has acceleration equal to local g if other forces are negligible.
Near Earth's surface, small laboratory height changes barely alter r, so constant g is useful. Across planetary distances, the inverse-square variation is essential.
Check your understanding: At height R above a planet of radius R, what fraction of surface g remains?
The distance from the centre is 2R, so g is one quarter of its surface value.
Key ideas to keep
- Distance r is measured centre to centre.
- Field strength exists whether or not a test mass is placed there.
- Gravitational force is attractive, so direction matters even when the equation uses magnitudes.
See the reasoning
Worked example
Find a planet's mass from surface field
Question: A spherical moon has radius 1.74 × 10⁶ m and surface gravitational field strength 1.62 N kg⁻¹. Find its mass. Use G = 6.67 × 10⁻¹¹ N m² kg⁻².
Step 1: Choose the field equation
Why: Surface field uses centre distance equal to the moon's radius.
Working: g = GM/R², so M = gR²/G.
Step 2: Substitute with the squared radius
Why: The inverse-square relationship is a common source of power-of-ten mistakes.
Working: M = 1.62(1.74 × 10⁶)²/(6.67 × 10⁻¹¹).
Step 3: Calculate and check units
Why: N kg⁻¹, m² and G combine to kilograms.
Working: M = 7.35 × 10²² kg.
Answer: The moon's mass is approximately 7.35 × 10²² kg.
Check: Substitution back into GM/R² returns about 1.62 N kg⁻¹.
Another worked model
Question
A planet's field strength is 4.00 N kg⁻¹ at 3.00 × 10⁶ m from its centre. Estimate its mass using G = 6.67 × 10⁻¹¹ N m² kg⁻².
Check the worked solution
From g = GM/r², M = gr²/G = 4.00(3.00 × 10⁶)²/(6.67 × 10⁻¹¹) = 5.40 × 10²³ kg.
Use a hint if needed
Practise with support
Try this
A spherical planet has mass 6.00 × 10²⁴ kg and radius 6.50 × 10⁶ m. Find g at altitude 5.00 × 10⁵ m.
Hint: The inverse-square distance is measured from the planet's centre.
Check your answer
r = 6.50 × 10⁶ + 5.00 × 10⁵ = 7.00 × 10⁶ m. Thus g = GM/r² = 6.67 × 10⁻¹¹(6.00 × 10²⁴)/(7.00 × 10⁶)² = 8.17 N kg⁻¹ towards the centre.
Now work without the hint
Practise independently
Your turn
At a planet's surface, g = 12.0 N kg⁻¹ and radius R = 8.00 × 10⁶ m. Find g at altitude R above the surface.
Check your answer
The new centre distance is r = R + R = 2R. Since g ∝ 1/r², g = 12.0(R/2R)² = 3.00 N kg⁻¹.
Avoid these traps
Common mistakes
Common mistake
Gravitational force is inversely proportional to separation.
What is wrong with this reasoning?
Show better thinking
It is inversely proportional to the square of centre-to-centre separation: doubling r reduces force to one quarter.
Common mistake
The r in GM/r² is altitude above a planet's surface.
What is wrong with this reasoning?
Show better thinking
For a spherical planet, r is measured from its centre. At altitude h, use r = R + h.
Common mistake
A heavier test mass experiences a stronger gravitational field strength.
What is wrong with this reasoning?
Show better thinking
It experiences a larger force, but g = F/m = GM/r² is a property of the source and position, not the test mass.
Write for the examiner
Exam guidance
Sketch the field direction first and keep source mass M separate from test mass m.
Exam-style practice [6 marks]
A planet has surface field strength 12 N kg⁻¹ and radius R. Find the field strength at height 2R above its surface. A 5.0 kg probe is there: find the gravitational force and explain why using g = 12 N kg⁻¹ would be wrong.
Plan before you answer
- Convert height above surface to centre distance.
- Use a ratio to avoid needing G or planet mass.
- Then use F = mg with the local field.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
At height 2R, the probe is 3R from the centre. Therefore g/g0 = R²/(3R)² = 1/9, so g = 12/9 = 1.33 N kg⁻¹. The force is 5.0(1.33) = 6.67 N towards the planet. Using 12 N kg⁻¹ would ignore the large change in centre-to-centre distance.
Come back in three days
Check what stayed with you
Recall question 1
State Newton's law of gravitation.
Check the answer
F = Gm1m2/r², attractive along the line joining the masses.
Recall question 2
How is r measured for spherical bodies?
Check the answer
Centre to centre.
Recall question 3
When does free-fall acceleration equal g?
Check the answer
When gravity is the only significant force.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes. The official topic states no explicit exclusions. Models use centre-to-centre distance, zero potential at infinity, point masses or spherical bodies viewed externally, and ideal circular orbits unless a question states otherwise.
- GCE A-Level H2 PhysicsTopic 8(a) / Topic 8(b) / Topic 8(c) / Topic 8(d) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027