AP PHYSICS C: MECHANICS › UNIT 2, FORCE AND TRANSLATIONAL DYNAMICS › TOPIC 2.3
Newton’s Third Law
Every force is half of something. The other half is the same size, points the other way, and is exerted on the other object, which is the detail that makes the law useful and the detail that everybody drops.
2.3.A Describe the interaction of two objects using the third law, draw the paired forces on the correct diagrams, and explain why internal forces never move a system.
The whole topic on one sheet, the same one handed out in class. Click it to open the full-size version, which prints cleanly on a single page.
1. Forces come in pairs, and the pair is one interaction
Topic 2.2 said that a force is something two objects do to each other. The third law is what follows from taking that seriously: if A pushes B, then B pushes A, with the same size force in the opposite direction, at the same instant, and of the same kind.
FA on B = −FB on A
There is no delay, no winner, and no dependence on which one is heavier, faster, harder or more alive. A mosquito hitting a windshield is pushed by the glass exactly as hard as it pushes the glass. What differs afterward is not the force. It is what a mosquito and a truck each do when given that force, and that is the second law, a different topic entirely.
Two skaters push off each other on ice. Change the mass of the left one and watch the two gold arrows: they never come apart. The speeds below do, because the same force divided by different masses gives different accelerations.
2. The two halves live on two different diagrams
This is the part that goes wrong, and it goes wrong in one specific way. The two forces in a pair are exerted on different objects, so they can never appear on the same free-body diagram, and therefore they can never cancel each other. Cancelling is something two forces do when they act on the same object.
A book resting on a table has two arrows on it, the pull of the Earth and the push of the table, and they happen to be equal because the book is not accelerating. Those two are not a third-law pair. The partner of the table pushing up on the book is the book pushing down on the table, and that arrow lives on the diagram of the table.
The same book and table, drawn four ways. Start with the forces on the book, then add the table, then isolate the one interaction they share, then see what happens if you put both halves of it on a single diagram.
3. Why a system cannot push itself along
Take any two objects inside one system boundary. Every force between them turns up twice, equal and opposite, and both copies are inside the loop. Add them up and they contribute nothing. That is why the internal forces of Topic 2.1 could be ignored, and now you know it is not an approximation: it is the third law, applied to every internal interaction at once.
So the center of mass of a system can only be accelerated from outside. A car accelerates because the road pushes its tires forward, not because the engine pushes the car. Sitting in the driver seat and pushing the dashboard changes nothing at all, however hard you push.
4. Tension, and what a string really is
A string under tension is a long chain of third-law pairs. Each short segment pulls on the segments either side of it, and each of those pulls back with the same force. What we call the tension is the size of that pull.
An ideal string has negligible mass and does not stretch, and it is worth being precise about what that buys you: the tension is then the same at every point along it. Give the string real mass and that stops being true, because the upper part now has to hold up the lower part as well as whatever hangs on the end. The tension in a heavy hanging chain is largest at the top, where the most chain hangs below.
An ideal pulley has negligible mass and turns without friction. It changes the direction of a tension without changing its size, which is exactly why it is worth drawing.
In this course the segments are infinitesimal, and that word is doing work. A rope of linear density λ hanging from a ceiling has a tension at height y equal to the weight of everything below that point, so FT(y) = λgy for a uniform rope: zero at the free end and λgL at the ceiling. AP Physics 1 may only describe that qualitatively. Here it is a calculation, and for a rope whose density varies it is an integral over the same dm = λ dℓ you met in Topic 2.1.
5. Reading it wrong
The tug-of-war objection. If both teams pull with equal force, why does anyone win? They do pull on each other equally. The rope is not the system that moves; the teams are, and each team is also being pushed by the ground. The team that wins is the one the ground pushes harder, which is why the shoes matter more than the arms.
The horse and the cart. The horse pulls the cart forward and the cart pulls the horse back, equally. Those two arrows are on different objects, so they do not cancel. Draw the horse alone and the forward arrow comes from the ground, not the cart.
Equal forces, so equal effect. Never. The third law fixes the forces, and the second law decides what each object does with the force it got.
Check yourself
1. A 1200 kg car collides head on with an 8000 kg truck. Compare the forces on each and the accelerations of each, and say which of the two statements a passenger is more likely to notice.
The forces are equal and opposite, always. The accelerations are not: the car takes roughly 8000/1200, about seven times the acceleration of the truck, because the same force is divided by a smaller mass. A passenger notices acceleration, not force, which is exactly why the third law feels wrong here and is right anyway.
2. A book rests on a table. Name the third-law partner of the table pushing up on the book, and explain why the weight of the book is not that partner.
The partner is the book pushing down on the table: same interaction, same size, opposite direction, exerted on the table. The weight of the book is the Earth pulling on the book, a different interaction with a different second object. It is equal to the normal force here only because the book is not accelerating; put the whole thing in an elevator and the two stop being equal while the true pair stays equal.
3. A student says that a car cannot accelerate, because the engine pushes the car forward and the car pushes back on the engine with an equal force. Find the error.
Both of those forces are internal to the car, so they cancel in the system and nothing about them could ever accelerate it. The forward force on the car comes from outside: the road pushes the tires. The engine turns the wheels, and the road does the pushing, which is why the same engine on frictionless ice moves nothing.
4. A heavy chain hangs from a ceiling. Describe how the tension varies along it and why, and say what changes if the chain is replaced by an ideal string.
The tension is largest at the top and smallest at the bottom, because each point has to support the weight of everything hanging below it, and that is greatest at the top. An ideal string has negligible mass, so there is nothing extra to support and the tension is the same everywhere along it. This is a qualitative comparison, which is all this course asks of a string with mass.
5. Two teams pull on a rope. Both pull with 900 N. One team wins. Explain, naming the system you are analyzing and the external force that decides it.
Analyze one team, not the rope. The forces on that team are the rope pulling them forward toward the middle and friction from the ground pushing them backward, away from the middle. The team wins when the ground can push it harder than the rope pulls, which is a question about shoes, surfaces and how the team leans, not about how hard they pull. The two teams pull on each other equally whatever happens, and the rope tension being the same everywhere is exactly why that is guaranteed.
Topic 2.4, Newton’s First Law. You can now draw every arrow on an object and find the partner of each. The next question is the simplest one available: what happens when they all add to nothing.