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5.1 Newton's third law by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

a / Two magnets exert forces on each other.

Newton created the modern concept of force starting from his insight that all the effects that govern motion are interactions between two objects: unlike the Aristotelian theory, Newtonian physics has no phenomena in which an object changes its own motion.

Is one object always the “order-giver” and the other the “order-follower”? As an example, consider a batter hitting a baseball. The bat definitely exerts a large force on the ball, because the ball accelerates drastically. But if you have ever hit a baseball, you also know that the ball makes a force on the bat --- often with painful results if your technique is as bad as mine!

d / A swimmer doing the breast stroke pushes backward against the water. By Newton's third law, the water pushes forward on him.

How does the ball's force on the bat compare with the bat's force on the ball? The bat's acceleration is not as spectacular as the ball's, but maybe we shouldn't expect it to be, since the bat's mass is much greater. In fact, careful measurements of both objects' masses and accelerations would show that`m_"ball" a_"ball" ` is very nearly equal to `?m_"bat"a_"bat"`, which suggests that the ball's force on the bat is of the same magnitude as the bat's force on the ball, but in the opposite direction.

Figures a and b show two somewhat more practical laboratory experiments for investigating this issue accurately and without too much interference from extraneous forces.

b / Two people's hands exert forces on each other.

In experiment a, a large magnet and a small magnet are weighed separately, and then one magnet is hung from the pan of the top balance so that it is directly above the other magnet. There is an attraction between the two magnets, causing the reading on the top scale to increase and the reading on the bottom scale to decrease. The large magnet is more “powerful” in the sense that it can pick up a heavier paperclip from the same distance, so many people have a strong expectation that one scale's reading will change by a far different amount than the other. Instead, we find that the two changes are equal in magnitude but opposite in direction: the force of the bottom magnet pulling down on the top one has the same strength as the force of the top one pulling up on the bottom one.

c / Rockets work by pushing exhaust gases out the back. Newton's third law says that if the rocket exerts a backward force on the gases, the gases must make an equal forward force on the rocket. Rocket engines can function above the atmosphere, unlike propellers and jets, which work by pushing against the surrounding air.

In experiment b, two people pull on two spring scales. Regardless of who tries to pull harder, the two forces as measured on the spring scales are equal. Interposing the two spring scales is necessary in order to measure the forces, but the outcome is not some artificial result of the scales' interactions with each other. If one person slaps another hard on the hand, the slapper's hand hurts just as much as the slappee's, and it doesn't matter if the recipient of the slap tries to be inactive. (Punching someone in the mouth causes just as much force on the fist as on the lips. It's just that the lips are more delicate. The forces are equal, but not the levels of pain and injury.)

Newton, after observing a series of results such as these, decided that there must be a fundamental law of nature at work:

Forces occur in equal and opposite pairs: whenever object A exerts a force on object B, object B must also be exerting a force on object A. The two forces are equal in magnitude and opposite in direction.

Two modern, high-precision tests of the third law are described on p. 783.

In one-dimensional situations, we can use plus and minus signs to indicate the directions of forces, and Newton's third law can be written succinctly as `F_"A on B"=?F_"B on A"`.

*self-check:*

Figure d analyzes swimming using Newton's third law. Do a similar analysis for a sprinter leaving the starting line.

(answer in the back of the PDF version of the book)

e / Newton's third law does not mean that forces always cancel out so that nothing can ever move. If these two figure skaters, initially at rest, push against each other, they will both move.

There is no cause and effect relationship between the two forces in Newton's third law. There is no “original” force, and neither one is a response to the other. The pair of forces is a relationship, like marriage, not a back-and-forth process like a tennis match. Newton came up with the third law as a generalization about all the types of forces with which he was familiar, such as frictional and gravitational forces. When later physicists discovered a new type of force, such as the force that holds atomic nuclei together, they had to check whether it obeyed Newton's third law. So far, no violation of the third law has ever been discovered, whereas the first and second laws were shown to have limitations by Einstein and the pioneers of atomic physics.

The English vocabulary for describing forces is unfortunately rooted in Aristotelianism, and often implies incorrectly that forces are one-way relationships. It is unfortunate that a half-truth such as “the table exerts an upward force on the book” is so easily expressed, while a more complete and correct description ends up sounding awkward or strange: “the table and the book interact via a force,” or “the table and book participate in a force.”

To students, it often sounds as though Newton's third law implies nothing could ever change its motion, since the two equal and opposite forces would always cancel. The two forces, however, are always on two different objects, so it doesn't make sense to add them in the first place --- we only add forces that are acting on the same object. If two objects are interacting via a force and no other forces are involved, then *both* objects will accelerate --- in opposite directions!

f / It doesn't make sense for the man to talk about using the woman's money to cancel out his bar tab, because there is no good reason to combine his debts and her assets. Similarly, it doesn't make sense to refer to the equal and opposite forces of Newton's third law as canceling. It only makes sense to add up forces that are acting on the*same* object, whereas two forces related to each other by Newton's third law are always acting on two *different* objects.

Mnemonics are tricks for memorizing things. For instance, the musical notes that lie between the lines on the treble clef spell the word FACE, which is easy to remember. Many people use the mnemonic “SOHCAHTOA” to remember the definitions of the sine, cosine, and tangent in trigonometry. I have my own modest offering, POFOSTITO, which I hope will make it into the mnemonics hall of fame. It's a way to avoid some of the most common problems with applying Newton's third law correctly:

? A book is lying on a table. What force is the Newton's-third-law partner of the earth's gravitational force on the book?

Answer: Newton's third law works like “`B` on `A`, `A` on `B`,” so the partner must be the book's gravitational force pulling upward on the planet earth. Yes, there is such a force! No, it does not cause the earth to do anything noticeable.

Incorrect answer: The table's upward force on the book is the Newton's-third-law partner of the earth's gravitational force on the book.

This answer violates two out of three of the commandments of POFOSTITO. The forces are not of the same type, because the table's upward force on the book is not gravitational. Also, three objects are involved instead of two: the book, the table, and the planet earth.

? A person is pushing a box up a hill. What force is related by Newton's third law to the person's force on the box?

? The box's force on the person.

Incorrect answer: The person's force on the box is opposed by friction, and also by gravity.

This answer fails all three parts of the POFOSTITO test, the most obvious of which is that three forces are referred to instead of a pair.

If we could violate Newton's third law, we could do strange and wonderful things. Newton's third laws says that the unequal magnets in figure a on p. 154 should exert equal forces on each other, and this is what we actually find when we do the experiment shown in that figure. But suppose instead that it worked as most people intuitively expect. What if the third law was violated, so that the big magnet made more force on the small one than the small one made on the big one? To make the analysis simple, we add some extra nonmagnetic material to the small magnet in figure g/1, so that it has the same mass and size as the big one. We also attach springs. When we release the magnets, g/2, the weak one is accelerated strongly, while the strong one barely moves. If we put them inside a box, g/3, the recoiling strong magnet bangs hard against the side of the box, and the box mysteriously accelerates itself. The process can be repeated indefinitely for free, so we have a magic box that propels itself without needing fuel. We can make it into a perpetual-motion car, g/4. If Newton's third law was violated, we'd never have to pay for gas!

g / Example 3. This doesn't actually happen!

? Solved problem: More about example 2 — problem 20

? Solved problem: Why did it accelerate? — problem 18

? When you fire a gun, the exploding gases push outward in all directions, causing the bullet to accelerate down the barrel. What third-law pairs are involved? [Hint: Remember that the gases themselves are an object.]

? Tam Anh grabs Sarah by the hand and tries to pull her. She tries to remain standing without moving. A student analyzes the situation as follows. “If Tam Anh's force on Sarah is greater than her force on him, he can get her to move. Otherwise, she'll be able to stay where she is.” What's wrong with this analysis?

? You hit a tennis ball against a wall. Explain any and all incorrect ideas in the following description of the physics involved: “According to Newton's third law, there has to be a force opposite to your force on the ball. The opposite force is the ball's mass, which resists acceleration, and also air resistance.”

5.1 Newton's third law by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.