# Circular Motion, Train Engine

1. Dec 22, 2016

### Kernul

1. The problem statement, all variables and given/known data
A train engine of mass $m$ is chugging its way around a circular curve of radius $R$ at a constant speed $v$. Draw a free body/force diagram for the train engine showing all of the forces acting on it.
Evaluate the total vector force acting on the engine as a function of its speed in a plane perpendicular to its velocity $\vec v$.
You may find the picture of the train’s wheels useful. Note that they are notched so that they fit onto the rails – the thin rim of metal that rides on the inside of each rail is essential to the train being able to go around a curve and stay on a track!
Draw a schematic picture of the wheel and rail in cross-section and draw in the forces using the force rules we have learned so far that illustrate how a rail can exert both components of the force needed to hold a train up and curve its trajectory around in a circle.
Discussion: What is the mechanical origin of the force responsible for making the train go in a curve without coming off of the track (and for that matter, keeping it on the track in the first place, even when it is going “straight”)? What would happen if there were no rim on the train’s wheels?

2. Relevant equations
Newton's Second Law
Circular Motion

3. The attempt at a solution
This is the free body diagram I drew:

But besides this, I don't know what are the forces that are actually acting on the train. Can someone guide me through the exercise?
And about drawing a schematic picture of the wheel and rail in cross-section, I didn't quite get how I should do that.

2. Dec 22, 2016

### jbriggs444

That drawing shows an x and a y axis. What are x and y? Are we looking down at the track from above so that x is east and y is north? Or are we looking at the train from the rear so that x is right and y is up?

The drawing shows a leftward F. What force is that?
The drawing shows an upward "v", "q" or backwards "g" with a minus sign in front. What force is that?

The drawing shows R. A free body diagram is supposed to show forces. What forces act on the train engine?

3. Dec 22, 2016

### Kernul

So the problem actually asked me to draw the curve seen from above.

That's the net force pointing to the center of the circular curve.

It's a "v", since the problem was talking about that, and it's a velocity.

The "R" is the radius of the circular curve.

I wrote these last two just to show how I see the problem.

4. Dec 22, 2016

### jbriggs444

A "free body diagram" is a diagram that shows all forces acting on a body. The problem does not ask for a curve seen from above.

5. Dec 22, 2016

### Kernul

Oh my... you're right. What was I even trying to do there?

Anyway, from what prospective should I draw the free body diagram?
If I write it from the top of the train, wouldn't I have just one force pointing to the center of the curve?
If I write it from the rear of the train, I would have the gravity force $mg$ going down and the normal force $N$ going up, with, again, a force $F$ going to the side because of the centrifugal acceleration. (For example the negative side of the x-axis)

6. Dec 22, 2016

### jbriggs444

I like the way you are thinking here.

As you say, if you look down from the top, you will not be able to present the downward force of gravity or the upward normal force. If you look forward from the train's rear you will be able to present both of those forces easily.

A free body diagram is mainly an aid to understanding. It is a graphical presentation that forces you do identify all of the forces on an object. Let us go back to the problem statement and see what forces will be important...

As I read it, the free body diagram is going to be a rough outline which gets made more specific by this schematic. So you need the free body diagram to be from the same perspective as the schematic. A "cross section".

That is to say, a view from the rear, looking at engine and rails end-on.

7. Dec 22, 2016

### Kernul

So I have to draw the rails and wheels and write the forces acting between the two? All this in the same drawing looking at the train engine from the rear?
Does it stays on track because the rails exert a normal force on the wheels, specifically on the rim of metal?

8. Dec 22, 2016

### jbriggs444

For the free body diagram, concentrate on the engine. All you are concerned with is the forces acting on the engine. You would not want to get into details of the wheels. When you get to the schematic, you are asked to focus in more on the details.

I see it as two drawings. One crude and one with more details.
You use the term "normal force". What do you mean by that in this context? How would you say that it explains how the train stays on the track?

9. Dec 22, 2016

### Kernul

This would be the diagram from the rear.

This is what I mean:

This happens anytime the train engine turns. Or, to better say, when the rails change course, and so they "curve". When they curve, they basically exert a force on the rim of metal at the end of the wheel while at the same time the wheel exerts a force on it too.

This case, instead, makes the train engine derail because of no opposition of the rail. (No rim of metal, no force on the wheel, and so train goes on the side without any force opposing it)

10. Dec 22, 2016

### jbriggs444

On the free body diagram, you showed a vertical force of rail on engine. On the detailed diagram, that force is absent. The "normal" force you depict is not perpendicular to anything. Where is the vertical supporting force?

On the final drawing, you speak of a "force applied by the wheel but without opposition of the rail". What does Newton's third law have to say about that concept?

11. Dec 23, 2016

### CWatters

I think your FBD is slightly too simple. Can I point out that there are two rails so I would expect more than one vertical force acting on the train (unless the train was going dangerously fast!).