Newton’s Third Law Pairs Calculator
Which Two Forces Are the Action–Reaction Pair?
A book lies still on a table. Its weight and the table’s normal force are equal, opposite, and not a third-law pair — they act on the same body, and they are not even the same kind of force. That one mistake costs more marks than anything else in school mechanics. Name your situation and the pair you think it is, and this page gives the verdict, the genuine pair, and a test you can apply to any of them in five seconds.
Pick a situation and the pair you think it is
A 2.0 kg book lies still on a table. Are its weight and the table’s normal force a third-law pair? Name every force on the book, find each one’s partner, and say which pairs are genuine. These are the page’s own starting selections.
The law, and the three-question test that settles every case
- the law itself
- “To every action, there is always an equal and opposite reaction”. In modern words: if A exerts a force on B, then B exerts an equal and opposite force on A. The NCERT module adds what the wording hides — there is no cause-effect relationship; they are simultaneous and either may be called the action
- test 1: different bodies?
- the question that decides almost everything. A third-law pair never acts on one body. If both your arrows are on the same free-body diagram they are not a pair, however equal and opposite they look. This is the test the book-and-table error fails
- test 2: same kind of force?
- gravity pairs with gravity, a contact push with a contact push, tension with tension, friction with friction. A weight (gravitational) can never pair with a normal force (contact). The partner of a weight is always another gravitational pull
- test 3: name the two objects
- say it as “the force on ___ by ___”. A genuine pair is the same two names in the opposite order: on the wall by the hand and on the hand by the wall. If the names do not swap it is not a pair, and if you cannot name the second object you have not got a force
- why a pair never cancels
- cancelling means adding, and you may only add forces on the same body. The halves of a pair are on different bodies and go into different resultants. Taken over the two bodies together the internal pair does sum to zero, which is where conservation of momentum comes from — see impulse and momentum
- what is NOT a force
- ma, centripetal force (a requirement on a sum — see centripetal vs centrifugal), and any inertial term in a rotating frame. None has an agent, so none has a partner. Having no partner is the test for not being a real force
Worked example
A 2.0 kg book lies still on a table. Are its weight and the table’s normal force a third-law pair? Name every force on the book, find each one’s partner, and say which pairs are genuine. These are the page’s own starting selections.
Forces on the book: exactly two. Its weight W = mg = 2.0 × 9.81 = 19.6 N downwards, exerted by the Earth. And the normal force N = 19.6 N upwards, exerted by the table. They are equal and opposite because the book is in equilibrium — which is Newton’s second law with a = 0, not the third law.
Find W’s partner by swapping the names. W is “the force on the book by the Earth”, so its partner is “the force on the Earth by the book”: 19.6 N upwards on the planet. Gravitational, equal, opposite, different body. A genuine pair.
Find N’s partner the same way. N is “the force on the book by the table”, so its partner is “the force on the table by the book”: 19.6 N downwards on the table top. A contact push, equal, opposite, different body. The second genuine pair.
So why are W and N not a pair? Three reasons, any one enough. They act on the same body. They are different kinds of force. And they are only equal because the book happens to be in equilibrium on a horizontal surface.
Prove the last point with one change. Put the book on a slope. Now N = mg cos θ, smaller than the weight, so W and N are no longer even equal — while each keeps its own partner, unchanged and still exactly equal. A relationship that survives tilting the table is a law; one that does not was a coincidence. Work the slope at the inclined plane page.
And pull the table away. W is unchanged, N is gone, the book accelerates. Neither half of a pair can exist alone, so that settles it: two pairs, four forces, and the book staying still is the second law, not the third.
The genuine pair in each situation, and what gets chosen instead
| Situation | The genuine pair | Kind of force | The usual wrong answer |
|---|---|---|---|
| Book on a table | Book down on table ↔ table up on book; and Earth on book ↔ book on Earth | Contact; gravitational | Weight and normal force — both on the book. |
| Hand on a wall | Hand on wall ↔ wall on hand | Contact | Expecting the wall not to push, because it does not move. |
| Ball in free fall | Earth on ball ↔ ball on Earth | Gravitational | Believing the ball’s pull on the Earth does not exist. |
| Rocket in space | Rocket on gas ↔ gas on rocket | Contact | Thinking the exhaust pushes against the air. There is none. |
| Horse and cart | Horse on cart ↔ cart on horse | Contact (tension) | Cancelling the two and concluding nothing can move. |
| Swimmer and water | Swimmer on water ↔ water on swimmer | Contact | Forgetting the water has to be pushed somewhere. |
Second law or third law? They are confused constantly
| Statement | Which law | Why |
|---|---|---|
| W and N on a book are equal | Second | Because a = 0. Tilt the table and they stop being equal. |
| Hand on wall and wall on hand are equal | Third | Always true, whatever either body is doing. |
| The resultant force equals ma | Second | ma is the consequence of the forces, never one of them. |
| A pair never cancels | Third | The two forces are on different bodies, so they are never added. |
| The lift accelerates, so N is above mg | Second | N still pairs with the passenger’s push on the floor, equally. |
One force, one partner — and why the book on the table fools everybody
Start with the sentence that fixes it. Every force is a force on something by something else, and its third-law partner is the force on that something else by the first. Say it that way and the law becomes bookkeeping: on the book by the Earth pairs with on the Earth by the book. The two names swap, and equal size, opposite direction and same kind of force all follow.
So why does the book on the table catch almost everyone? Because its two forces are equal and opposite, and that is what the law’s wording emphasises. But they are equal and opposite for a completely different reason: the book is not accelerating, so by the second law the forces on it must add to zero. The equality is a consequence of equilibrium, and it is fragile. Tilt the table and the normal force becomes mg cos θ while the weight stays mg, and the two are no longer equal. Each one’s genuine partner stays exactly equal through all of it, because a third-law pair does not care what either body is doing.
Three tests, and the first does nearly all the work. Are the two forces on different bodies? If they are on the same body, stop: not a pair. Are they the same kind of force? A weight can never pair with a normal force. And can you name both objects in “the force on ___ by ___”? If you cannot name the second, it is not a force at all — which disposes of ma, of centripetal force and of every inertial term in a rotating frame. The NCERT module gives the same three conditions, and says in terms that action and reaction act on different bodies and not on the same body.
A pair can never cancel, and that is not a technicality. Adding forces is only meaningful for forces on the same body, because that is the body whose acceleration you are computing, and the halves of a pair live on different diagrams. That is the resolution of the horse-and-cart paradox: two diagrams, two resultants, no contradiction.
Equal forces, unequal accelerations. A dropped ball pulls the Earth upwards exactly as hard as the Earth pulls it down. Forces equal; accelerations not, because a = F/m and the Earth is about 1025 times heavier. The same arithmetic explains why a small car hitting a lorry comes off worse and why a rifle’s recoil is gentler than the bullet’s flight. The pair is always equal; what the two bodies do with it depends on their masses.
Where this goes wrong in exams. Four ways. One: W and N on a resting body offered as a pair — the error this page exists for. Two: a pair “cancelled out” to prove nothing can move. Three: ma, or a centripetal force, drawn as an arrow and then given a partner. Four: the right pair with the wrong bodies, as in “the table pushes the book up and the book’s weight pushes it down” — the first error in different words. The fix for all four is five seconds of work: write each force as on ___ by ___ and swap the names.
Frequently asked questions
Why are the book’s weight and the table’s normal force not a pair? They are equal and opposite.
They are, and they are still not a pair, for three independent reasons. They act on the same body, the book — which is why you may add them. They are different kinds of force, one gravitational and one contact. And they are only equal because the book is in equilibrium on a horizontal surface: tilt it and N becomes mg cos θ, smaller than the weight, while both genuine pairs stay exactly equal. The weight pairs with the book’s pull on the Earth; N pairs with the book’s push on the table.
If the forces are always equal, how does anything ever accelerate?
Because the two act on different bodies and so never appear in the same sum. To find out what a body does, add the forces on that body. The horse’s pull on the cart is a force on the cart; the cart’s pull on the horse is not, it is a force on the horse. Each body gets its own diagram and its own resultant, and either can be non-zero while the pair between them stays perfectly equal.
Does the falling ball really pull the Earth upwards?
Yes, with exactly the same force in newtons. A 1 kg ball is pulled down with about 9.8 N and pulls the Earth up with about 9.8 N. The Earth’s acceleration is that divided by some 6 × 1024 kg: far too small to measure, and not zero. Nothing in the third law ever promised equal accelerations.
Which one is the action and which is the reaction?
It does not matter, and the NCERT module says so: the two forces are simultaneous and there is no cause-effect relationship between them, so either may be called the action. The words mislead, because “reaction” sounds like a response that comes afterwards. There is no afterwards. The wall pushes your hand at the same instant your hand pushes the wall.
Is centripetal force part of a third-law pair?
No, because it is not a force. It is the resultant that circular motion requires, mv²/r towards the centre, supplied by whatever real force is available — tension, friction, gravity. Those have partners: the string pulls the ball inwards, the ball pulls the string outwards. The requirement itself has no agent and so no partner, which is also why the centrifugal term in a rotating frame is not a real force.
Does the third law hold when things are accelerating?
Always, and that is the difference between it and the equalities the second law produces. A pair is exactly equal at every instant whatever either body is doing: in a lift going up, in a crash, in orbit. Anything that stops being equal when the body accelerates — a weight against a normal force, a driving force against friction — was never a third-law pair.
Related calculators
References
- CIET, NCERT, Physics-01, Module 3: Third law of Motion (Module ID Keph_10503_eContent), Unit 3, Chapter 5 “Laws of Motion” of Physics Part I, Class XI. Read 12 November 2026. It states the law as “To every action, there is always an equal and opposite reaction” and then, in the lines this page is built on: “Forces always occur in pairs”; “Force on a body A by body B is equal and opposite to the force on the body B by body A”; “The terms action and reaction in the third law mean force”; “But there is no cause-effect relationship implied in the third law”; and “Action and reaction forces act on different bodies, and not on the same body”. On the block resting on a table it says in terms that the weight and the normal force are equal and opposite but that “these are not third law pair of forces, because here both the forces are acting on the same object (block)” — the sentence this calculator exists to make unmissable. Its test for a valid pair: same kind of force, two different objects, equal and opposite.
ciet.nic.in/moocspdf/Physics01/Unit03/ - PHYSICS (042) Curriculum, Class XI–XII, Central Board of Secondary Education, session 2025–26. Unit III, Laws of Motion, is where this material sits; the line this project holds verbatim from that unit reads “Dynamics of uniform circular motion: Centripetal force, examples of circular motion (vehicle on a level circular road, vehicle on a banked road)”. Quoted to establish what is in scope at this level, never to claim any endorsement by the board.
- BIPM, The International System of Units (SI), 9th edition (2019), Table 4: the newton is kg m s−2 and the joule is kg m² s−2. That is the whole reason a force times a distance is an energy, and it is the dimensional check that catches a kilojoule written where a kilonewton was meant.
- NIST, Fundamental Physical Constants — Complete Listing, 2022 CODATA adjustment, Standard Reference Database 121. The line both of these pages lean on is “standard acceleration of gravity 9.806 65 (exact) m s−2”. The word exact is the point: gn is a conventional value fixed by the 3rd CGPM in 1901, so it carries no uncertainty and it is not the g where you are standing.
physics.nist.gov/cuu/Constants/