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 μcs and the coefficient for the train μts, 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.
