So, this is your online class to make up for class on July 5. Just to be clear.
I have two problems for you to work on. For these problems, I will first give them to you in a plain format. Then I will give you some supplemental questions in case you can’t get started. Finally, I will have video solutions to these problems as a last resort. Only use these in cases of emergencies.
Before you try these problems, you need to read. Read the section on centripetal acceleration and friction. Those two links are to the html version of my book. So there is no excuse (although it looks better if you use the ebook reader version). If you haven’t looked at this material, don’t go forward. You will just be wasting your time.
Problem 1: Find the speed limit
In Slidell, I-12 intersects with I-59. If you are going east on I-12 to North I-59, there is this nice sharp turn on-off ramp. Here is a picture of that ramp from google maps.
The question: what should the speed limit be for this path? I like this question. It seems like a reasonable thing to ask and it doesn’t really show you how to answer the question. So, let me break this down into some questions that you can ask yourself.
1. Why does the car move in a circle? Can you draw a free body diagram for the car? Let me give you hint. Draw the diagram as the car is moving towards you. Here is a car to get you started.
If your picture doesn’t look as awesome as mine – don’t worry. You will get better with practice.
2. Your free body diagram should have three forces. There is one long range force and one thing touching the car. The ground touches the car, but it really can be thought of as two separate forces. One perpendicular to the ground (normal force) and one parallel to the ground (friction).
3. So, what things would you need to know to figure this problem out? Do you need to know the color of the car? Do you need to know who is driving the car? Probably not. But you do need some things. Write down all the things you think you might need to know and give them variables. For instance, if I thought I needed to know the color of the car I might assign the variable c to that.
You don’t need to find numerical values for these variables just yet. You should just set up the problem with your variables. I know everyone hates to do this, but trust me here.
4. Write down the momentum principle in the acceleration form for both the x- and y-directions. Here, let me get you started.
5. What do you want to solve in these equations? Can you do that? Do you have too many variables?
Ok. No more questions. Below is just some hints.
What is the radius of the circular path? In the picture above, I measured the diameter and get a radius of approximately 57 meters. If you want to look at other ramps, just find them on Google Maps and then click the small ruler button in the lower left of the screen. This will let you measure distances on the map.
What about the coefficient of friction between the tires and road? You can search for this, but I suggest a value of 0.4. This is the value I found for the coefficient between tires and a wet road. Why wet? Well, if I want to set a speed limit, it would be nice if it worked int he worst conditions. You know, for safety.
You should use the coefficient of static friction. Even though the car is moving, the tires are not sliding on the road.
Here is the model you should use for the frictional force (this is just the magnitude of the frictional force):
Why should you use = instead of less than or equal to? Because you want to find the worst case. You want to find the case where the car is just about to start sliding.
What about accelerations? What is the acceleration in the y-direction? What about the x-direction? Since it is moving in a circle, the acceleration in the x-direction should be:
Ok, here is the video. Like I said, you should only watch this as a last resort. Please don’t think that watching me work through this will make you understand what is happening.