The Physics Philes, lesson 7: Mutual Attraction
In which a trilogy is finished, diagrams are drawn, and the confusion is remedied.
Congratulations. You made it. You made it to the last installment of The Newton’s Laws of Motion trilogy. If you’re just joining us you can catch up here and here. For the rest of us, let’s dive into Newton’s third law of motion.
The third law is my favorite of the laws of motion because it doesn’t make any sense. At least, it doesn’t at first. But don’t worry. I’m going to try to explain it.
OK, so when a force acts on a body, that force is always the result of the body’s interaction with another body. As a result, forces always come in pairs. Whenever those two bodies interact, the two forces they exert on each other are always equal in magnitude and opposite in direction. These two facts combine to form Newton’s third law of motion:
If Body A exerts a force on Body B (an “action”), then Body B exerts a force on Body A (a “reaction”). These two forces have the same magnitude but are opposite in direction. These two forces act on different bodies.
Or, if you’re more comfortable reading math:
F(A on B) = -F(B on A)
(Apologies for the sloppy math. The subscript won’t work, so I’m just going to use parentheses for this post to indicate when something should be a subscript. Trust me, it is an issue that causes many headaches. But I can’t fix it so this will have to do for now.)
Notice the parentheses in the definition of the third law of motion. We have an “action” and a “reaction.” These two forces form an action-reaction pair. An action-reaction pair is just the two opposite forces in Newton’s third law. It’s important to remember that these forces act on different bodies. If two forces act on the same body, those forces cannot be an action-reaction pair.
The name – action-reaction pair – is a little misleading. It’s not meant to imply that there is a cause and effect relationship. If you’re someone as steeped in words as I’ve been for my entire life, this can be a little confusing. As long as you can remember that these are just opposite forces, I think you’ll be fine.
So far this doesn’t sound too hard to grasp. I mean, we’ve probably known about this law since we were in grade school. For every action there is an equal and opposite reaction. But consider this: When you drop a ball, the ball accelerates toward the Earth and the Earth accelerates toward the ball.
What? Is? This? Magic?
I know it’s not magic, but this is why the third law of motion blows my mind. Every time my elbow pushes my phone or the remote control off the sofa onto the floor, the Earth is also moving toward my phone or the remote at the same magnitude. This is crazy. Who needs a religious explanation of the universe? Real science is mind-blowing enough!
Enough gushing. Let’s try some examples. This first problem deals with objects at rest.
You’ve got a broken car, so you have to push it to the garage. How does the force you exert on the car compare with the force the car exerts on you?
We just got done discussing this. The force you exert on the car has the same magnitude, but is in the opposite direction, of the force the car exerts on you. It’s the law.
I bet you got that right. But what about this one: How do those same forces compare to the forces exerted when you push the car at a constant speed?
The answer is the same. The force exerted on the car is the same magnitude, but in the opposite direction, of the force the car exerts on you. This is true whether the object is at rest, accelerating, or moving at a constant speed.
This is still a weird concept that takes me a while to wrap my head around. The car exerts a force on me? But how does it know? It doesn’t know, exactly. Think about it this way. These interactions are between atoms on the surface of your hand and the surface of the car. Like miniature strings, and these compressed springs exerts and equally strong force at both ends.
That makes a little more sense, I guess. Except now I think of atoms as little spring, so I don’t know how helpful that’s going to be in the long run. Oh well! Onward! This next example deals with objects in motion.
There is a mason (like a stonemason, not like, you know, the Masons) pulling a block via a rope. The block might be in equilibrium; it might not be. What are the action-reaction pairs?
We should probably draw a diagram for this one.
This is what we’re working with. The mason (M) pulls on the rope (R), which is attached to the block (B). Where are the action reaction pairs. MOAR DIAGRAMS.
Now the forces are easier to identify! According to the third law, the force the mason is exerting on the rope is equal to the opposite of the force the rope is exerting on the mason. Same goes for the block and the rope. These are action reaction pairs.
Be careful, though. Did you think that the force exerted by the mason on the rope and the force exerted by the block on the rope was an action-reaction pair? That is wrong. Remember, the forces in an action-reaction pair never exert a force on the same body. In this case, each force is exerted on the rope, thus those forces cannot be an action-reaction pair.
Plus, the Mason on Rope force and the Block on Rope force are not necessarily of equal magnitude. We know this by applying the second law of motion to the rope. Check it out:
∑F = F(M on R) + F(B on R) = m(rope)a(rope)
If the rope and the block are accelerating, the rope is not in equilibrium and the Mason on Rope force must have a different magnitude than the Block on Rope force.
That isn’t to say that those forces will never be equal and opposite, it just won’t be because of Newton’s third law of motion.
If you’re like me, at this point you’ve got a clump of hair in your hands and screaming, “But…how does anything ever move?!” If all forces have an equal and opposite partner, how can I throw a ball or push a toy car across a table? It doesn’t make any sense!
Oh but it does. There are important distinctions between the first and second laws and the third law. The first and second laws relate to forces that act on a body. The vector sum of all forces acting on a body determines how the body moves, if at all. The third law, on the other hand, deals with forces that two different bodies exert on each other. The third law says nothing about the motion of those bodies.
Let’s look at the mason example to prove the point. The block will start moving if the magnitude of the force the mason exerts on the rope is greater than the magnitude of the friction force the floor exerts on the block. The net force would be toward the right, therefore acceleration of the block would also be toward the right. Once the mason gets the block going, she doesn’t have to work as hard to keep it moving. All she has to do is balance the force she exerts with the friction force exerted by the floor. The block will continue moving to the right, per Newton’s first law. So not so mysterious or paradoxical as it first appears.
Now you know everything I know about Newton’s laws of motion. But here are some helpful hints to remember if you ever want to use your new knowledge to impress your friends and influence people:
1. Newton’s first and second laws apply to a specific body. In order to solve the problem, you need to identify what body is being referred to.
2. Only forces acting on the body matter. Remember, the net force is the vector sum of all forces acting on the body. Don’t let yourself get confused by the forces exerted by the body on some other body. Those don’t matter. Only the member of the action-reaction pair that acts on the body in question counts.
3. For the love of Thor, draw a free body diagram! The diagrams above are free body diagrams. It’s basically just a drawing that shows the chosen body by itself, free of its surroundings, with vectors drawn in to show the magnitudes and directions of the forces applied to the body. Free body diagrams will help you identify relevant forces.
Remember, both forces in the action-reaction pair must never show up in the same free body diagram. Action-reaction pairs never act on the same body and free body diagrams only deal with one body at a time. Furthermore, a force a body exerts on itself is never included in a free body diagram because that force cannot affect the body’s motion. Finally, you know you’ve drawn a good free body diagram when you can account for where every force is coming from.
You have all the tools you need to solve Newton’s laws problem like a boss. Newton achievement unlocked! Go forth! Impress your friends with your mastery of motion!
Featured image credit: Wonderlane