Two perspectives on Newton's second law

Newton’s laws of motion are often the first things peopole learn about physics. In spite of the fact that the theory is rarely used in academic physics research, physicists keep teaching it to people, and I think we are absolutely correct to do so. Newton’s second law might be the most beautiful equation in physics. The equation is a rigorous mathematical framework for describing motion, but every piece of it can be directly attached to statements most people would find intuitive. The theory also hides a dark secret. I want to focus on Newton’s second law for right now, which is usually written \(\vec{F}=m\vec{a}\).

An intuitive explanation of Newton’s second law

Let’s take a look at the equation in a less traditional equivalent form:

\[\vec{F}=m\frac{d}{dt}\vec{v}\]

Newton’s second law is used to describe a single object. The symbols refer to physics quantities which are understood to correspond to that single object. I’ll explain all of the terms later, but the force is a force on the object. The mass and velocity are the mass and velocity of the object.

The first symbol we find in this equation is \(\vec F\), which represents force. A force can be thought of as a push or a pull. If you touch something, then you put a force on it. If you pull on a spring, you can feel it pulling back on you, which means that the spring is exerting a force away from you. Note that \(\vec F\) has a funny arrow over it. That means that force is represented by a mathematical object called a vector. Vectors have both a size and a direction. You can push really hard or not that hard on things, and you can push them in different directions.

The second symbol is \(m\), which represents mass. Mass is a number which tells you how much stuff is in an object. If you have two pieces of wood from the same tree, the one which is bigger than the other will have a bigger mass than the other. If we have a long plank of wood and measure its mass, then we would expect to measure half of that mass if we cut the plank exactly in half and measured the mass of one of the halves. If you have a piece of steel and a piece of styrofoam which are each the same size, the piece of steel has a larger mass. There’s more stuff in the piece of steel in a given bit of space than there is in the styrofoam. The steel is denser. If you run over the piece of foam with a truck, it will probably get smaller because you can move the stuff in the styrofoam closer together. The steel won’t change much, because the stuff is already pretty close together. Notice that mass is just a number and not a vector. It has no direction associated with it, and it is never a negative number.

The final thing we see in this equation is \(\frac{d}{dt}\vec{v}\). This is calculus, but don’t panic if you’re not familiar with calculus! We can understand this set of symbols in two parts. I think the easier one is \(\vec{v}\), which represents velocity. Velocity tells you how fast something is moving and what direction it moves in. Notice that it is again a vector! The vector is bigger if a thing is moving faster and smaller if a thing is moving slower. The vector points in the direction the thing is moving. The second part looks like \(\frac{d}{dt}\). This symbol has a formal definition, but it basically means “how much the thing after me will change over time.” So \(\frac{d}{dt}\vec{v}\) represents the change in the speed and direction of motion of an object.

Now we can interpret Newton’s second law. It is a mathematically precise way to say “if you push on something, then you will change how it moves.” If you push in the same direction that something is moving, then it will move faster. If you push in the opposite direction to that in which an object is moving, then the object slows down. If you push in a direction other than those two directions, then the thing will move in a different direction than the one it used to be moving in. Which is to say that you can push an object off-course by pushing sideways relative to the direction of travel. How does the mass play into this? Well, if you have an object with a big mass and you push on it a certain amount, you will get a certain amount of \(\frac{d}{dt}\vec{v}\). If you push on an object with a smaller mass the same amount, you will need a larger \(\frac{d}{dt}\vec{v}\). Basically, if you push on a heavy object, its speed changes less than if you pushed the same amount on a smaller object. If you have a car and a skateboard which are the usual sizes and aren’t moving, the skateboard will move faster than the car after you push on them both as hard as you can. In a sense, this is so obvious that it’s not worth writing down, but remember that the point of physics is to describe reality. Physics should be boring if you’re describing something boring!

So why bother with the math? What do we get by writing down the equation instead of the paragraph you just read? We get precision. It’s way more impressive to say that something will be moving exactly 2 kilometers per hour faster than it is to say merely that it will move faster. Newton’s second law gives us something exact to test. If we exactly double the mass in a car and push it the some amount, it better be moving exactly half as fast as it does with the original amount of mass. Eventually, maybe we can make theories which precisely predict the value of the force exerted by some push. Then we can find objects with a known mass, push on them with that force, and make sure that we see the exact change in speed that Newton’s second law predicts.

You might be asking what Newton’s first law is if Newton’s second law is so great. Newton’s first law says that an object continues moving with the same velocity (or stays at rest) unless acted on by an unbalanced force. It turns out that Newton’s first law is just Newton’s second law in disguise. If you don’t have a force, then \(m\frac{d}{dt}\vec{v}=0\), and because the amount of stuff in an object doesn’t change if you stop pushing on it, that means that velocity doesn’t change if there’s no force. This is a little unintuitive. When was the last time you saw something that moves at the same speed forever? The only thing I can think of is a car with cruise control turned on, and that only works because there is an engine or a motor providing a push. If you turn off the car, it eventually comes to a stop. We can look to Newton’s second law to help explain why. Newton’s second law tells us that if the speed is changing, there must be a force. What could be pushing on the car? A physicist might point to two sources of force. One is all of the air which pushes on the car. The car wants to move forward, but all of the air in front of the car needs to be pushed out of the way as it moves, and this air resists that change by pushing back on the car. The second source of force is all of the machinery in the car rubbing against other bits of the car. Both of these resistive forces pull on the car or pull on the machinery which is connected to the wheels which have to move if the car is going to keep moving, and this pulling slows down the car if you don’t counteract it with another force from the engine or the motor. We find that we need to change Newton’s second law a little bit:

\[\sum\vec{F}=m\frac{d}{dt}\vec{v}\]

The symbol \(\sum\) tells us to add together everything that comes after it. So this equation says that the sum of all the forces equal the change in speed times the mass. Going back to the car with cruise control active, we should be able to take the forward force which is coming from the engine or the motor and subtract all of the resistance force from the air and the stuff rubbing together and get a total force of zero.

These sorts of complications are inevitable once you examine reality in sufficient detail. We formalized the obvious statement that if you push on something then its motion changes. We see that our theory implies that if you don’t push on something, then its motion should not change. Suddenly we find ourselves in the middle of a tangle of pushes and pulls we never really had to think about much before. We try to make a simple theory, but if we take it seriously, it leads us to complications. However, it would surprise few to hear that we live in a complicated world, and physics describes reality. Physics should be complicated if you’re describing something complicated.

A cynical path toward discovering Newton’s second law

In this section, I argue that the equation above contains almost no actual information about how the universe works, but it provides a framework for describing motion. Let us begin.

We want to describe how objects move, so we start off with the easiest theory to write down: nothing moves.

\[\vec{v}=0\]

This is very simple, but it defeats the purpose of writing a theory of motion. Let’s try a slightly more complicated theory: objects can move, but they move in one direction at the same speed forever. In order to describe this theory in math, we just say that the change in velocity is zero:

\[\frac{d}{dt}\vec{v}=0\]

This is still pretty simple, but it’s not what we actually see. If I throw a ball forward, it does not go straight forward forever. Its path curves down until it hits the ground. If I pull a box along the floor and let go, it doesn’t keep going in a straight line forever, it slows down to a stop eventually. Clearly we can’t let the change in velocity be nothing all the time, so let’s set it equal to something:

\[\frac{d}{dt}\vec{v}=\vec F\]

Now whenever we see something not moving in a straight line at the same speed, we just say there’s a force making it move differently. You throw a ball upward and it turns around and comes back down? Say there’s a force everywhere on Earth pointing downward; call it gravity. Push a box along the ground and it stops? Say there’s a force which opposes motion between two objects which are touching each other; call it friction. We know that we can push on things to make them move faster, so we have to say that pushing on things exerts a force on those things. We can even start feeling like this pretty simple theory has some relation to our reality! We have a lot of experience pushing and pulling things, and we can associate that with forces. That gravity thing must push things to the ground just like I push shopping carts around the grocery store.

We have a problem though. If we set our pushes directly equal to the change in motion, then that implies that every object should have the same change in motion for a given push, but this is not actually what we see! What do we see? I can put a car in to a neutral gear and push as hard as I can on it while it is on a flat surface, and it will slowly start moving. If I push as hard as I can on an empty shopping cart, I will soon be chasing it through a grocery store. There seems to be something which makes pushing a car result in less of a speedup than pushing on an empty grocery cart.

Taking this into account, we need to find a way to mathematically sweep this issue under the rug. What if we come up with another thing in our equation which is different for every object and says that some objects are harder to push around than others. Let’s call it mass, and because it’s slightly easier to do algebra on things which don’t have sums on them, we can just divide the force by the mass and still have a pretty easy equation to deal with:

\[\frac{d}{dt}\vec{v}=\vec F/m\]

OK, so now we have a thing called mass. To figure out how much the change in velocity is, you divide the force by the mass. This implies that some things are implicitly harder to make move or to steer than other things, but that seems true to me. When I’m moving furniture around my house, it sure seems like I don’t have much trouble moving kitchen chairs around and I have a lot of trouble moving my couch around. If one of my kitchen chairs was falling down the stairs toward me, I would catch it and stop if without much trouble, but if my couch was falling down the stairs toward me, I would get out of the way because I know I’d have trouble stopping it. (We theoretical physicists are not well-known for our physical strength. Falling couches scare me.) It actually seems reasonable to just say the couch is inherently harder to move around than a kitchen chair, and we might as well specify that by saying that there is a big mass associated with the couch but a small mass associated with the chair.

We couldn’t quite keep the simplest theory of motion that we could think of, but we tossed in a couple of other numbers and ended up with something which is still pretty simple. Notice that it’s just Newton’s second law, but written funny. The only thing we assumed to write this down was that some things have an inherent property which makes it harder to steer them in a different direction if they’re already moving and also harder to change the speed that they are moving at. Everything else we said was a thing you could have said in any universe where things don’t always move in a straight line at the same speed forever. The thing I want to get across here is that Newton’s second law is not a thing which describes our universe, but it is rather a framework in which we are able to describe our universe.

Reverse engineering forces

To wrap up, let’s look at some examples of how Newton’s second law can help us design theories which explain any change in motion which we see.

Gravity

Suppose you wake up in a world where stuff falls down if you hold it up above the ground and let go of it. I did in fact wake up in a world like that this morning. If you very carefully measure how quickly things fall when you let go of them (and you let go of them in a vacuum chamber so that there isn’t any air in the way), then you will see that all objects fall at the same rate. It doesn’t matter what their mass is. Every second, the speed of a falling object increases by a little less than 10 meters per second close to Earth’s surface. If you start off with an object moving upward, its speed decreases by about 10 meters per second every second until it stops and starts falling downward. If an object is moving sideways at all, it keeps moving sideways at the same speed while its downward speed increases by 10 meters per second per second.

Suppose we have an object with mass \(M\). How should we explain this change in motion? If we boldly plug in the \(10~m/s/s\) downward acceleration we observe, we find that a force appears:

\[F_g=M(10~m/s/s)\]

We will call this force gravity. Our theory becomes that there is an omnipresent downward facing force which is bigger the more mass you have. It is now up to us to decide whether that makes sense. According to Newton’s second law, if you want a thing to not move downward, the sum of all of the forces needs to be zero. So if there is a force of gravity downward, you have to exert a force upward if you don’t want want the thing to start falling downward. If I imagine holding the skateboard from earlier in my outstretched arms, then I expect that my arms will get tired after a while. If I don’t want the skateboard to fall down, I have to push upward. If I imagine trying to hold up a car with my outstretched arms, I imagine failing and probably getting very badly hurt. It sure seems reasonable to imagine a force that pushes down on the skateboard and the car which pushes harder on the car. We mentioned earlier that the relative difficulty of pushing the car around and pushing the skateboard around implies that the car has a larger mass than the skateboard. It seems like our theory that there is an omnipresent downward force which is bigger for things with more mass matches reality.

Contact forces

It is an observed fact that solid objects do not move through each other. Gravity makes a ball fall toward Earth, but it stops once it hits the surface. A car moves forward with constant speed, but it will (violently) stop if it hits a very massive wall. These are changes in velocity, so Newton’s second law demands a corresponding force. You are allowed to move next to an object, so whatever that force is, it seems to push directly away from the surface that an object would otherwise move through. Newton’s second law does not tell us the numerical value of these forces, but we can use the law to reverse engineer the numerical value of the contact force from an object’s acceleration. This is best shown by example.

Consider an object with mass \(M\) at rest on the ground. We know that gravity pulls the object downward, but solid objects cannot move through solid ground, and so we know that the object’s velocity must be constantly zero relative to the ground. Assuming that we’re working in the ground frame, a constant velocity means zero acceleration and thus zero net force. Because we know that the force of gravity is \(M(10~m/s^2)\) downward, the contact force must be equal and opposite.

Now consider an object with mass M sitting on the floor of an elevator which is accelerating upward at \(1~m/s^2\). By Newton’s second law, the net force on the object must be \(M(1~m/s^2)\) upward, and if gravity is \(M(10~m/s^2)\) downward, than the contact force which stops the object from passing through the floor of the elevator must be \(M(11~m/s^2)\) upward.

Static friction

It is an observed fact that if you lightly push a heavy object with a flat bottom which is sitting on a flat surface, the object does not start moving. According to Newton’s second law, that means that there is an equal and opposite force to whatever the force is that you push on the object is. We call this static friction. Eventually you can push hard enough on something that it starts moving, so we formalize the static friction as a response force which negates all forces on an object that would otherwise cause the object to move relative to the surface it is touching up to a certain threshold, after which the object slips relative to the surface it is touching and no longer is in the domain of static friction.

One theory of static friction that we could make is that every object has a maximal static friction associated with it just like it has a mass associated with it. This theory does not work. If you take a given table and try to push it along a steel floor, it will take a noticeably smaller amount of sideways force to do that than if you try to push it along a rubber floor. We must at least say that the static friction is a function of both the object you are trying to move and the type of surface that you push it against. This is not a parsimonious theory, and even if it worked, then the only thing we learn about the universe is that resistance to slipping is consistent over multiple trials. We can improve the theory by noticing that it takes more sideways force to make the table start moving if you pile a bunch of stuff on top of the table so that gravity pushes it down more. This leads us to the form of static friction on an object that is taught in introductory mechanics: \(F_{fric} < \mu_s N\), where \(N\) is the contact force on the object in question from the surface it is sliding against and \(\mu_s\) is the coefficient of static friction for the combination of materials which make up the surface and the object. Note that the vector signs are gone, so this is a statement about the magnitude of forces rather than the forces themselves, which must have a direction.

This theory says that the maximum amount of force a surface can put on an object to resist its slipping along the surface is equal to the force the surface is exerting outward on the object multiplied by some number, which we call the static friction coefficient. We still need to go out and measure the static friction combination for every combination of two surface types which can touch each other, but that is far fewer numbers than we would have to measure for every combination of surface type and object like we did before. We don’t have to measure a separate number for “table” and “table, but now I’ve piled some books on it.” We just measure one static friction coefficient for “wood in table leg against wood in floor” and multiply it by the weight of the table or the weight of the table with books on it (which by the previous section is the contact force) to get the maximum amount you can push the table sideways before it starts slipping. We can even use the same coefficient for a chair we want to push around assuming that the chair legs and the table legs are made of the same material. In fact, this is one good experimental test for the theory. First pile stuff on the chair until it shows the same weight on a scale as the table does. Then confirm that it takes the same amount of force to make the chair start moving as the table. Repeat for other flat-bottomed things which are each made of the same material.

This is still not a parsimonious theory. We need to keep track of a static friction coefficient for every element of the square matrix whose columns and rows are the types of solid substances you can make things out of. It does tell us a bit more about the universe than our earlier nonfunctional theory though. It says that there are consistent properties of materials, and we don’t need to make new physics for every new object we find so long as we have seen another object made out of the same thing.