**Chapter 4: Newton’s Laws of Motion**

We’re getting to more fun stuff. So far, we’ve learned how things behave once in motion. Now, we get to think about why they behave that way.

The crucial concept here is called “force.” We’ll get into force in depth in a bit, but first off – if you haven’t studied it before, your common sense notion will screw you up royally here. I think a lot of students get confused about what force really means in the realm of physics.

We get to a mathematical framework in 4.2, but for now, I want to give you this sense: Force is what causes something to change velocity. An object experiencing no net force will experience no change in velocity. An object experiencing any amount of net force, no matter how little, will change in velocity.

Corollary to that is that if an object is not changing velocity, it is experiencing no net force. So, for example, you are currently being pulled down by Earth’s gravity. Why don’t you fall through the floor? Think about it for a second.

Got it?

You don’t fall through the floor because the floor is pushing back up at you. In fact, it’s pushing back up with a force exactly equal to the force of gravity pulling you down. If it pushed less, you’d fall through the floor. If it pushed more, you’d pop up, like on a trampoline.

The more mathy way to think of it, that I prefer, is this: Force is the derivative of momentum with respect to time. That is, force is what causes a body with mass to change velocity over time.

But, we’re getting ahead of ourselves.

**4.1 Force and Interactions**

The book gives you a bunch of different types of force: contact force, normal force, friction force, tension force, and long-range force.

I don’t really like separating this stuff out, because it confused me at first. All force does the same thing – change momentum. On a day to day basis, you deal with two force: electromagnetic and gravity.

Gravity is simple: It brings stuff toward the surface of the Earth. It’s also handy for you, since it guides the planets around the sun. Beyond that, it’s not doing much for you. Pretty much everything else is electromagnetism.

For example, your body doesn’t hold together due to gravity. Gravity holds your body to the ground, but your body is held together by the electromagnetic interactions among the molecules of your body.

Why am I getting into all this stuff when it’s not in this section? Because all those forces above (and, indeed, pretty much all force in this book) involves one or the other.

When you deliver contact force to a ball it’s because the stretching and tightening of your muscles delivers your hand to the ball. That’s force form EM.

Friction, tension (and whatever other simple forces you want to make up) work along similar lines.

Weight actually is gravity. Weight is a measure of the force you (or some object) exerts on the surface of Earth. Actually, you of course have a weight on other bodies on which you find yourself. So, more accurately, you might say weight is a measure of the force between you and the nearest large body.

Normal force is confusingly located in the middle of that list of types of force. “Normal” is a mathematical word that basically just means orthogonal to the surface in question. That is, the “normal” refers to the direction of the force with respect to the place where the force is being exerted. When you on a flat surface, you exert force that is normal to the surface you stand on. When you push on a ball in the direction of its center, you are exerting force that is “normal” to the surface of the ball at the point of exertion.

For this chapter, I believe normality will come up most often in reference to a body on a flat surface.

Fun fact: In English, the word to use when talking about things getting back to how they usually are is “normalcy,” not “normality.” So, if your teacher says “I’m looking forward to having normality in my life,” you can correct by saying “you’re ALREADY normal to the surface of the Earth, you piece of shit.”

All these force can be expressed in terms of vectors. In the same way that an acceleration vector indicates change in velocity over time, a force vector indicates change in momentum (mass times velocity) over time. That might seem little, since mass doesn’t change, but it produces big results.

Think of it this way: If you stick an acceleration vector on something you can simply predict its future – it’ll move at this speed in this direction at this time. However, it’s not very practical to go around sticking acceleration vectors on things because (as you know from real life) it’s harder to acceleration something with a lot of mass.

If you know the FORCE acting on something, you can predict its future regardless of how much mass it has. If 10 units of force act on a bowling ball, it’ll move at a certain rate. If 10 units of force act on a ping pong bal, it’ll move at a higher rate. So, knowing the force allows you to deal with any object.

Of course, things in reality can be a bit more complex, but that basic principal is true. If you know the force, you can predict the kinematics for an object of any mass. Neat, eh?

So, that’s the first concept – that there’s this thing called force and it can be expressed vectors.

The second concept involves combining these vectors.

As you’ve probably gathered from the above and from real life, force is cumulative. This is called the “principle of superposition.” Basically, what this means is that if you wanna be able to predict the action of that bowling ball from a couple paragraphs ago, you need to know ALL the forces acting on it because they all add up. So, if you’re pushing 5 units left and 5 units right on the bowling ball, it won’t move. Why? Because 5 – 5 = 0. Although 10 units of force are being exerted on the ball in SOME direction, in this case, the directions cancel out. If you then apply 5 units of force downward, the ball will go down.

As with all other vectors, a particular force vector can be split up into vectors. And, lots of different vectors can be turned into a single vector.

This is conveniently simple. If you push a car forward with 5 units of force and you also have a magnet push it forward with 5 units of force, it still gets pushed with 10 units of force forward. There is no need to account for the different types of force. In fact, even fundamentally different forces, like gravity and EM, are still cumulative in this nice simple manner.

In practice, this means in your physics class one of the big games you’ll play is figuring out all the forces acting on an object. This is called a “free body diagram.” Basically, when you want to figure out the net force on an object, you draw the object and figure out ever single force acting on it. You express each force as a vector then sum the vectors. That’ll give you a single nice vector that tells you the net force acting on an object.

**Next stop: Newton’s First Law**

Literally the only thing that I don’t love about your walkthroughs in Physics and Calc is that you’ve only gotten to chapter 4.1 and we just finished chapter 8 in my AP Physics (Mechanics) course.

Ha! Yeah, it’ll be a bit before I get there. But, when you get to finals, there may be some useful refreshers in here for you. I’m also tryyyying to provide some insight on things that confused me the first time around.

“Force is the derivative of momentum with respect to time. That is, force is what causes a body with mass to change velocity over time.”

If force is fundamental, then wouldn’t it be better to say that momentum is the integral of force wrt time? I know that integrals are a little harder than derivatives, but we’re going there anywhere.

Huh, that’s interesting. I always found the idea of the derivative of momentum to be very intuitive, though. It also comes in handy when people get confused about why it is that when you push something small, it gets pushed off more than you do despite the whole “equal opposite reaction” thing.

In some sense, you are correct. One perspective on F=ma is to say that it defines mass. This is because acceleration is something we can measure with rods and clocks, and F is really a placeholder for ‘insert some specific model of the world here’–implicit in writing ‘F’ is the promise that I will also give you supplemental equations that will lay out in detail how different forces work. So from this perspective, what F=ma says is, take some object, act on it with your known force law, and measure its acceleration. There will be some proportionality constant, that number is what I mean by mass.

From this point of view, what you are saying makes a lot of sense. In the case of nonconstant forces or accelerations, you will need to do an integral to find the mass. That is a good observation.

But F=ma is just an equation, it doesn’t force us to look at it in just one way. More practically speaking, we know the masses of objects, and we are not interested in using F=ma to define mass. Another interpretation is that F=ma is an equation of motion that tells you how the object moves in response to a force law. If you like, we only need to use F=ma once to define the mass of an object, then we can use it to figure out the motion of the object as many times as we like after that.

The point is that F=ma (or really F=dp/dt) is more frequently used the second way. The ‘standard mechanics problem’ is to take a bunch of point masses, specify some forces between them, set them up in some initial configuration (with initial positions and momenta), and then watch them evolve. Writing F=dp/dt emphasizes this ‘watch them evolve’ point of view.

Typos:

That’s force _form_ EM

When you __ on a flat surface

That might seem little ___

it’s harder to _acceleration_ something

ping pong _bal_

can be expressed __ vectors