Newton’s Third Law

31 Newton’s Third Law

In addition to Newton’s second law, there is one more important bit we must learn about forces. That is, forces come in pairs. To illustrate, consider the following thought experiment.

Exercise 31.1: Pushing off a wall

You place a skateboard against a wall, and climb on top. You then place your hands in the wall, and push. Clearly, the skateboard will start rolling away from the wall. But why?

When you push on the wall, it is the wall that should accelerate. So what’s going on?

Newton’s third law takes the conclusion from our thought experiment, and promotes it to a law of physics:

If A pushes on B, then B pushes on A.

The forces “A pushing on B” and “B pushing on A” are referred to as force pairs, and are often described as “equal and opposite” in that they are equal to each other in magnitude, but point in opposite directions. Let’s look at some examples.

Exercise 31.2: Force Pairs

A book is resting on a table. What is the force pair of the normal force pushing upwards on the book?

The force pair of the normal force on the book is the normal force of the book pushing on the table. Every force-pair question will come down to repeating our mantra: if A pushes on B, then B pushes on A.

Let’s try a few more.

Finally, let’s try a conceptual question. When you paddle a canoe, you move the oar backwards to make the canoe move forwards. Explain why the canoe accelerates forward when you move the oar backwards.

Warning: Consider the force diagram shown at right for a box resting on the floor. The single most common mistake when it comes to Newton’s 3rd law is that people want to call these two forces a force pair. This is incorrect.

The forces are equal and opposite, but that is because of Newton’s 2nd law: the box is not accelerating, so these two forces must be balanced.

What about the third law? Let’s go through it:

$mg$ is “the Earth pulling on the box.
Therefore, the force pair for $mg$ is “the box pulling on the Earth” (sentence A).
The normal force is “the floor pushing on the box” (sentence B).

Sentence A is not the same thing as sentence B. For instance, in sentence A, the thing being pushed is “the Earth.” In sentence B, the thing being pushed is “the box.” Bottom line: the normal force is not the force pair of the weight.

Finally, let’s solve a problem that requires using Newton’s 3rd law to arrive at the answer.

Exercise 31.3: Using Newton’s 3rd Law

A small block of mass $m=3\ {\rm kg}$ sits a top of a larger block of mass $M=5\ {\rm kg}$ . The static frictional coefficient between the blocks is $\mu=0.2$ . The larger block rests on top of a frictionless table (this is just to make the algebra easier).

You attach a rope to the larger block, and use it to pull horizontally with a force $F$ .

Evidently, there is a threshold $F_{\rm max}$ such that if you pull with a force $F\leq F_{\rm max}$ , the two boxes will move together. Pull any harder, and you will yank the bottom box from under the small box. Find $F_{\rm max}$ .

We start in the usual place: what’s the story? This story is a bit unusual: it doesn’t have a clear beginning and end. Instead, try to explain in English why there is a maximum force $F_{\rm max}$ .

Before getting to our drawing, answer the following questions; it will help you make a good drawing.

Now for the picture. In this problem we care about how both boxes move, which could result in very crowded force diagrams. To avoid this, you should “pull out” one of the boxes, and use that to draw the forces on that box (see below). Finish up my drawing, and then compare to my solution.

If your solution doesn’t look like mine, or you’re simply feeling wobbly, I have included below a video of how I make my picture.

Now that you have the drawing in place, find relations and solve unknowns. If you get stuck, I go through the full solution in the video below.

You now know how the “tablecloth” trick works!

Key Takeaways

Complete the mantra:

31 Newton’s Third Law

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