In simple terms
A friendly intro before the formal notes — no formulas yet.
Gravitational force between point masses
Cambridge 9702 Paper 4 - Gravitational force between point masses (13.2). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
13.2 Gravitational force between point masses.
- 2
For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
- 3
A uniform sphere is one where its mass is distributed evenly .
- 4
The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 13.2.1
Understand that, for a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre
- 13.2.2
Recall and use Newton's law of gravitation for the force between two point masses
- 13.2.3
Analyse circular orbits in gravitational fields by relating the gravitational force to the centripetal acceleration it causes
- 13.2.4
Understand that a satellite in a geostationary orbit remains at the same point above the Earth's surface, with an orbital period of 24 hours, orbiting from west to east, directly above the Equator
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
13.2 Gravitational force between point masses.
11 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
11 simulations · 4 to start with
Start herein this order — each one shows a different piece of the topic
- 3JCN PhysicsStart here · 19702 13.2 · 9702 13.1 · IB D.1
Newton's Gravity
Move two masses closer or farther and see how the gravitational force vector scales
Why this one: Move the masses closer and watch the force arrow grow as 1/r²; both arrows are always equal.
Try this
- Move the two masses closer and watch the force vector grow.
- Double the separation and compare the vector length.
Look for The attraction on each mass is equal and opposite and falls with the inverse square of separation.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- The Physics ClassroomStart here · 29702 13.2 · IB D.1
Gravitation
Change the masses and the separation and observe the gravitational force values
Why this one: Double one mass, then double the distance, and check the force values against F = GMm/r².
Try this
- Double one mass and read the force.
- Double the separation and read the force.
- Compare the two changes.
Look for Force is proportional to the product of the masses and inversely proportional to the square of the separation.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- PhETStart here · 3IB D.1 · 9702 13.1–13.2 · 12.2
My Solar System
Set the masses, positions and velocities of up to four bodies and watch their gravitational orbits unfold.
Why this one: Add a third body and see that the net force on each is the vector sum of two attractions.
Try this
- Load the Sun–planet preset — turn on the path and the velocity vector.
- Lower the planet’s starting speed — the orbit becomes an ellipse that dips closer to the Sun.
- Raise the speed step by step until the planet escapes.
Look for A circular orbit needs one exact speed; less gives an ellipse, enough gives escape.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- SimuPhysicsStart here · 49702 13.2 · 9702 13.1 · IB D.1
N-Body Gravity Simulator
Place any number of masses, give them initial velocities and watch the gravitational interactions, slingshots, captures and ejections
Why this one: Place three masses and watch each one pulled by the resultant of the others.
Try this
- Place two masses and set a stable orbit.
- Add a third mass and watch.
- Give a small mass a slingshot past a large one.
Look for Two bodies orbit predictably while three or more generally do not.
Open on SimuPhysicsRuns on their siteSimuPhysics by Mohamed Abdelsalam · Licensed to MarkScheme (site publishes no licence; served with frame-ancestors self)
More simulations7 more on this topic — core ones first
- oPhysicsCore9702 13.2 · 9702 13.3 · IB D.1
Elliptical Orbits & Kepler's 2nd Law
Planet orbiting a sun: set initial speed, distance and masses; see elliptical orbit and equal-area sweeps
Try this
- Set an initial speed that gives a nearly circular orbit.
- Lower the initial speed and watch the orbit become elliptical.
- Increase the sun’s mass with the same starting distance.
Look for The planet moves fastest nearest the sun, and the areas swept in equal times stay equal.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- The Physics ClassroomCore9702 13.2 · 9702 13.1 · IB D.1
Orbital Motion
Investigate elliptical orbits and Kepler's laws by changing the satellite's starting speed and position
Try this
- Launch a circular orbit.
- Increase the starting speed and watch the ellipse form.
- Compare the speed at the near and far points.
Look for A satellite moves fastest at the point of its orbit closest to the planet.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- 3JCN PhysicsCore9702 13.2 · 9702 13.1 · IB D.1
Gravity Force
Change two masses and their separation; read the gravitational attraction between them
Try this
- Double one mass and read the attraction.
- Double the separation and read the attraction.
- Halve the separation and read the attraction again.
Look for F = Gm1m2 / r²: doubling either mass doubles the force and doubling the separation quarters it.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 13.3 · 9702 13.2 · IB D.1
Kepler's Third Law
Change orbital radius and read the period; verify T^2 proportional to r^3
Try this
- Set an orbital radius and read the period.
- Quadruple the radius and read the period again.
- Check T² / r³ for both orbits.
Look for T² is proportional to r³, so quadrupling the radius multiplies the period by eight.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 13.3 · 9702 13.2 · 9702 2.1
Newton's Cannon
Increase a cannonball's launch speed from a mountain until it orbits Earth
Try this
- Launch at a low speed and watch the cannonball fall to Earth.
- Increase the launch speed step by step until the ball completes an orbit.
- Increase the speed further and compare the shape of the orbit.
Look for At the right speed the ball falls toward Earth at the same rate the surface curves away, giving a circular orbit; faster gives an ellipse.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 13.3 · 9702 13.2 · IB D.1
Sun and Earth
Vary Earth's orbital speed around the Sun and see circular, elliptical or escape trajectories
Try this
- Set the orbital speed for a circular orbit and watch one revolution.
- Reduce the speed and compare the shape of the orbit.
- Increase the speed until the Earth escapes.
Look for One speed gives a circle; lower or higher speeds give ellipses and beyond escape speed the path is open.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
Unpacking Newton's Law of Gravitation
Sir Isaac Newton's groundbreaking law provides a precise way to determine the strength of the attractive force between any two objects possessing mass. It states that the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This 'inverse square' relationship is a key feature of many physical laws.
13.2 Gravitational force between point masses.
For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
A uniform sphere is one where its mass is distributed evenly .
The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
An object can be regarded as point mass when a body covers a very large distance as compared to its size .
Radial fields are considered non-uniform fields.
The Concept of a Point Mass and Uniform Spheres
Newton's law is defined for 'point masses' - objects with mass but considered to have zero volume. While no real object is a true point mass, this is an excellent approximation when the distance between objects is much larger than their individual sizes, such as in astronomy. A crucial extension of this concept, also proven by Newton, is that a uniform spherical body (one with evenly distributed mass) exerts a gravitational force on an external object as if its entire mass were concentrated at its geometric centre. This allows us to treat planets, stars, and moons as point masses for calculating the forces between them, greatly simplifying cosmic mechanics.
The Universal Gravitational Constant (G)
The constant 'G' in the equation is the Universal Gravitational Constant. It is a fundamental constant of nature, meaning it is believed to be the same everywhere in the universe. Its extremely small value ($6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}$) explains why gravitational forces are only noticeable when massive objects are involved. The value of G was first measured with reasonable accuracy by Henry Cavendish in 1798 using a sensitive torsion balance, an experiment often called 'weighing the Earth' because it allowed for the calculation of Earth's mass for the first time.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
Calculate the gravitational force between Earth (mass $5.97 \times 10^{24}$ kg) and the Moon (mass $7.35 \times 10^{22}$ kg) if their average centre-to-centre distance is $3.84 \times 10^8G = 6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}
- 1
Identify given values:
Two large spheres with masses kg and kg are placed with their centres 4.0 m apart. A smaller object with mass kg is placed on the line connecting their centres, at a distance of 1.0 m from the 200 kg sphere. Calculate the net gravitational force on the 10 kg object. Use .
- 1
Identify forces and distances: The 10 kg mass (m) is attracted by both and . Let be the force from and be the force from . These forces act in opposite directions along the line connecting the centres.
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- mass affect
The force is directly proportional to the product of the two interacting masses (). This means more mass equals more force.
- distance affect
The force is inversely proportional to the square of the distance () separating the centres of the masses.
- approximate numerical
$6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}$
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
13.2 Gravitational force between point masses.
For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
A uniform sphere is one where its mass is distributed evenly .
The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
An object can be regarded as point mass when a body covers a very large distance as compared to its size .
Radial fields are considered non-uniform fields.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Show that the distance y of point P from the centre of sphere Y is equal to 2x.
State Newton's law of gravitation.
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do 9702/41 · Q5(c)(i) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
Discuss Gravitational force between point masses
Ask, share and discuss with other Physics students