If you have ever been near a train when it starts to move, you see (and hear) something interesting. The engine car at the front starts to move and in doing so, you get this wave of compressing couplings between all the cars. This was the source of a recent Car Talk puzzler. Here is the actual puzzler as stated on Car Talk.

Basically, the idea is that a train tried to start with the caboose brakes stuck on. After releasing the caboose, the train still could not start. The problem was that when the train attempted to start with the caboose brake on, it stretched all the inter-car couplings so that the whole train was just like one big car. At this point, the friction from the engine train wheels was not enough to get the whole thing going. Instead, you need to just get one car moving at a time - this is why there is space between the couplings.

I think there is some interesting physics here. In particular, there is something curious about the difference between static and kinetic friction. First, let me make some observations and assumptions.

The train has a big engine in it. Right? This engine makes the wheels turn to pull the rest of the cars. If we consider the train and wheels as the system, the force that changes its momentum is the static friction force between the wheels and the rail. Right? Yes, right. What about the cars? They also have wheels. However, these are not driving wheels, they just roll but they also have friction. I will assume that the frictional force is in the axle of the wheels. For these rolling cars, the friction is kinetic friction and not static.

What is the difference between kinetic and static friction? Static friction is the model for the frictional force between two surfaces that are at rest relative to each other. This would be the case of the engine car's wheels. Even though the wheels are rolling, the point of contact with the rails is at rest with respect to these rails. Kinetic friction is the model to use when the two surfaces are moving relative to each other - like the car's axle and the rest of the car.

This leads to two models for the magnitude of the frictional force:

These two models look similar, but here are the differences.

The kinetic friction (I wrote that as F fk ) is equal to the normal force (the force two surfaces are pushed together) multiplied by some constant called the coefficient of kinetic friction.

) is equal to the normal force (the force two surfaces are pushed together) multiplied by some constant called the coefficient of kinetic friction. For the static friction, it is less than or equal to the product of the static coefficient of friction and the normal force. This means that the static friction force is whatever value it needs to be to prevent the two surfaces from sliding - up to a point of maximum static friction.

In general, the coefficient of kinetic friction is less than the coefficient of static friction. This means it would take less force to slide something at a constant velocity than it would to get it moving.

Remember, this is just a model for friction. There are some cases where this model doesn't really work. I'm pretty sure that it works here.

Stretched Couplings ——————-

Consider a train in which all the cars have stretched couplings. This would make it just like one big rigid object. I will just draw the engine car and one car along with the forces on it (while at rest but trying to move).

I am representing all the cars as just having a mass of Nm. I hope that's not too confusing. Also, I left off the force the train pulls on the cars and the force the cars pull on the train. Although they are labeled the same, there are actually two different static frictional forces. The static frictional force on the train is between the wheels and the track. The frictional force on the cars is between the axle and the wheels (so, I cheated a little bit here). But here is the important part. As long as the frictional force on the train is greater than the frictional force on all the cars, the whole system can accelerate.

If I call the coefficient of static friction for the cars μ cs and the coefficient for the train μ ts , then this would be the equation for the forces in the horizontal direction.

For the case where it accelerates, I can solve for the maximum number of cars the train can get moving.

I don't know values for these two coefficients of friction, but it seems crazy to think that the train's friction coefficient is 10 times more than the cars. How could a train get a series of cars moving? The only way would be to just overcome a large frictional force would be to get one car moving at at time. Once a car is moving, the axle-wheel interaction changes to kinetic friction with a lower coefficient.

Modeling a Starting Train ————————-

This is really what I wanted to do - make a model that shows these cars starting to move. Ok, let me tell you how I am going to cheat to model this train to car coupling force. My first idea was to use a spring, but I decided against that (not sure why). My plan is to just have a constant coupling force. If the distance between cars is greater than some value, there is a force pulling it forward. If the distance between cars is too small, there will be a force pushing them apart. It's that simple.

Of course, I need to add a frictional force also. For the cars, there will be some maximum static frictional force to keep it stationary. After it starts to move, this will be replaced with a constant kinetic friction.

Before I start, I have to pick some values for things. I don't know why, but I decided to model this as a small train model. I don't think it really matters too much. Also, I have the coefficient of friction on the driving wheels at 0.5 and the kinetic friction on the car wheels at 0.09 with 5 cars.

Here is what it looks like:

That is in slow motion so you can see the different cars moving at different time. Here is a plot of the position of each car relative to its starting position.

In this model, the train just keeps on accelerating. Really, I should put a velocity dependent drag force on the train engine so it looks more realistic. However, there is something pretty cool in the above plot. Look at the time difference between each car starting. It looks to be evenly spaced out in starting times. This seems to agree with the sound of a real starting train.

Homework ——–

This starting train problem is one of those weird things. You begin looking at it and modeling it and then you realize there are all sorts of cool things to explore. Since I clearly left a lot of unanswered questions on the table, I will let you explore them. But wait! I'm not going to leave you empty handed. Here is my VPython code that I started with. I tried to add comments so you could figure out what is happening, but remember that I am a sloppy coder. I don't always do things the most optimal way (and neither should you).

Change some stuff and see what happens. Try playing with the coefficient of static friction on the engine wheels and the car wheels. Try changing the kinetic friction on the car wheels. Is there a point at which the train engine can just barely get the train moving?

What about mass of the cars? Try making random masses for the different cars (within a reasonable range). What does that do?

Add something to the program that makes the train reach a constant speed.

Find a way to measure the different start times for the trains cars. Are they actually evenly spaced? If you change something, does the start time spacing change?

See if you can reproduce this same situation with actual cars. I would start with something like the PASCO low friction cars and use a fan on the first car for the engine. I know it's not the same thing, but it's a start.

Here's one more train picture, just for fun.

Image: Rhett Allain

Homepage Image: Doug Wertman/Flickr