Author: Lou Bloomfield

If forces are always equal but opposite, how can a hammer drive a nail into a wall? Don’t the forces on the nail cancel?

Although forces always appear in equal but oppositely directed pairs, the two forces in each pair act on different objects. The nail and hammer experience one of these force pairs—the hammer pushes on the nail just as hard as the nail pushes on the hammer. Because the nail’s force on the hammer is the only force that the hammer experiences, the hammer accelerates away from the nail and the wall. The nail and wall experience the other force pair—the wall pushes on the nail just as hard as the nail pushes on the wall. The nail thus experiences two horizontal forces: the hammer pushes it toward the wall and the wall pushes it away from the wall. As long as all the forces are gentle, the two forces on the nail cancel and it doesn’t accelerate at all. But if you hit the nail hard with the hammer, the wall can’t exert enough support force on the nail to prevent it from enter the wall. The two forces on the nail no longer cancel and it accelerates into the wall.

If the net force on an object is zero and it has no acceleration, then what causes it to have velocity? Doesn’t a force give it velocity? And doesn’t this make the gravitational and support forces unequal? – EH

The great insights of Galileo and Newton were that an object doesn’t need a force on it to have a non-zero velocity. Objects tend to coast along at constant velocity when they are free of forces, or when the net force on them is zero. Inertia keeps them going even though nothing pushes on them. While it takes an acceleration and thus a non-zero net force to get an object moving in the first place, it will continue to move even if the net force on it drops to zero. So while I was lifting the bowling ball upward at constant velocity, the net force on the bowling ball was truly zero—it was coasting upward because its weight and the support force from my hand were canceling one another. However, to start the bowling ball moving upward, I had to push upward on it harder than gravity pushed downward. For a short time, the bowling ball experienced an upward net force and it accelerated upward. After that, I stopped pushing extra hard and let the bowling ball coast upward at constant velocity.

What would it be like if Newton’s third law weren’t true? Can we imagine that?

Many strange things would happen. For example, suppose that you pushed on your neighbor and your neighbor didn’t push back—you wouldn’t feel any force pushing against your hand so you wouldn’t even notice that you were pushing on your neighbor. Your neighbor would feel you pushing on them and they would accelerate away from you.

Among the many consequences of such a change would be that energy wouldn’t be conserved—you would be able to create energy out of nowhere. To see how that would be possible, imagine lifting a heavy object and suppose that as you pushed upward on it, it didn’t push downward on you. As you lifted it upward, you would do work on it—you would exert an upward force on it and it would move upward. But it wouldn’t do negative work on you—it would exert no force on you as your hands lifted it upward. As a result, its energy would increase but your energy wouldn’t decrease. Energy would be created. In fact, you wouldn’t even notice that you were lifting it because it wouldn’t push on you as you lifted it.

Would a small-mass hammer that accelerated rapidly exert more horizontal force on a nail than a large-mass hammer that didn’t accelerate very much?

Yes. Since the only horizontal force acting on the hammer is that exerted on it by the nail, the hammer’s acceleration is entirely determined by that force. The force on the hammer is equal to the hammer’s mass times the hammer’s acceleration (Newton’s second law). If both hammers experienced the same acceleration, then the large-mass hammer would have to be experiencing the larger force from the nail and would therefore be exerting the larger force on the nail. But because the small-mass hammer is experience a larger acceleration, the force that the nail is exerting on it may be quite large. If the small-mass hammer’s acceleration is large enough, the force on it may exceed the force on the large-mass hammer.

You said that if I push on a friend they will push back (even if they are asleep). But if I push hard enough, they will fall to the ground, whereas I will not. Therefore, I don’t see how the reaction is equal. Can you please explain this? – JK

Newton’s third law only observes that the forces two objects exert on one another are equal in amount but opposite in direction. The law doesn’t make any statement about the consequences of those forces on the objects involved. Moreover, it doesn’t say that those forces are the only forces on the objects. When you push on an awake friend, your friend will obtain additional forces from the ground or a nearby wall, and will manage to avoid falling over. Even though you push your friend away from you, your friend will see to it that the ground pushes them toward you. As a result, they will probably stay in one place. But when your friend is asleep, they won’t be able obtain the additional forces necessary to compensate for the force you exert on them and they may accelerate away from you or fall over.