Lesson - Car on a Banked Road

This lesson uses the applet Car on a Banked Road to simulate the motion of a car turning a corner.

Preamble

This lesson will teach you about the the physics involved when a car negotiates a corner.

The applet should be open. The step-by-step instructions in this lesson are to be carried out in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.

Contents

What keeps your car on the road when turning a corner? Why do you skid when you hit an icy patch? Why are the corners of most highway interchanges sloped or banked? These are some of the questions that will be explored in this lesson. In particular you will learn about:

Getting to Know the Applet - Car on a Banked Road

Spend a little time "playing" with the applet. Try adjusting the speed of the car, coefficient of friction and angle of the roadway (the "bank angle"). Observe under what conditions the car will either skid inward or outward and what you need to do to counteract this.

Once you have run the applet a few times you should explore the forces acting on the car. Toggle the force button and show the free-body diagram. Again observe how the forces change with varying speed, coefficient of friction or bank angle. When you are ready, please answer the following:

  1. Explain why the free-body diagrams only show three forces acting on the car (We are ignoring air resistance here)



  2. What force or forces cause the car to round the corner and in what direction do these forces act?




  3. During an outward skid, does a force cause the car to skid outward? If so, identify this force?

Car on a Level Corner

The simplest case to consider involves a car turning at a normal, flat intersection. Here the bank angle is zero degrees so make sure that the angle slider on the applet is set at 0°.

As you learned earlier, there are three forces that are important in understanding the motion of a car as it rounds a corner:

It is easy to see in this case that the frictional force is the only force that can act inward to supply the force needed to deflect the car from its original direction of motion into a circular turn. Let's investigate this a little closer by looking at the force needed to deflect the car moving at 15 m/s around a corner of radius 80 m. Consider these steps:

  1. Get the acceleration : Remember that the acceleration of a body moving in a circular path is given by the expression for centripetal acceleration: . Find the magnitude of the acceleration that the car will undergo and specify the direction in which the car accelerates.


  2. If we assign a mass of 1000 kg to the car, calculate the size of the force needed to deflect the car in this circular path (Hint: Use Newton's 2nd Law)






  3. In question #5 you calculated the size of the force needed to deflect the car in a circular path. This force is produced by the force of friction between the tires and the road. The road "pushes" the car inward, allowing it to deflect. As long as we can develop enough frictional force we can safely turn the corner. Experiment with the applet and adjust the coefficient of friction. Note how the frictional force changes as you do this. What is the smallest value (approximately) that the coefficient of friction can be to prevent skidding?






  4. Since we know that the force of friction is responsible for deflecting the car inward can equate the force of friction and the inward force. This gives us: and since on a level surface , we get. This can be simplified to give . Calculate the minimum coefficient of friction for which the car can safely turn the corner. Verify this using the applet.



  5. For normal, dry road conditions m = 0.80. Under these conditions what is the fastest safe speed for turning a corner of radius 60.0 m? Does the mass of the car matter?





Car on a Vertical Corner!

What happens if, instead of no bank angle you enter a corner banked at 90°! This is definitely in the "don't try this at home" category but is a situation sometimes seen at carnivals and exhibitions!

The following figure is a screen capture of the applet with the angle slider set at 90°. Please adjust the applet now. Set the angle at 90°, the coefficient of friction at 0.50 and leave the radius of turn at 60 m. Note the relationship between the vectors and particularly between the Weight vector and the Frictional Force vector.

 
  1. Draw a Free Body Diagram in the space provide on the right and carefully label the forces.


  2. Which force supplies the inward force needed to propel the car around the vertical corner?





  3. Look carefully at the figure shown above for the car rounding a 90° turn. Would the car be able to turn without skidding for the case shown here? Explain your answer.

 

 

 

 

 

 

 

put diagram here

 

 

 

 

 

 

 

 

 
  1. It is clear from both the figure shown above and your FBD that the Frictional Force must be equal to the Weight of the car to prevent sliding. We can write this as: . We also know that the Normal force must be responsible for the inward force. Show that the minimum velocity the car must have to safely (without skidding) round the corner is given by.

 

 

Car on a Banked Corner (optional)

Now consider the case when the corner is not flat but tilted or banked at a small angle. To help visualize this set the applet with the following values:

Also, turn on the Free Body Diagram options so that you ca see the FBD in "real time".

  1. Press the play button. The car (speed = 0 m/s) just "sits there". Explain why the Frictional Force appears as it does in the FBD. In particular, why does it point in the direction that it does?
    Now begin to increase the velocity of the car and note how the frictional force changes. Does the direction of the frictional force ever change? If so, why does this happen?

 
  1. Adjust the speed of the car to equal 30.0 m/s. In the space on the right, draw a FBD. Carefully label the forces and assume that the ordinary x-y axes define the reference frame for this diagram but choose the direction toward the center of the turn as positive x.

  2. Draw in all components that act in either the x or y directions (use the applet to assist you in visualizing them). It's tricky but carefully label where angle q is on your FBD. As a hint you should observe that the normal force tips away from vertical by the same amount that the curve is banked.




  3. How big is the inward force needed to deflect the car around the corner? Calculate this, show your work and confirm it using the applet.

 

 

 

 

 

put diagram here
  1. Show that the following equation correctly represents all relevant forces in the x-direction:




  2. Show that the following equation correctly represents all relevant forces in the y-direction:
  3. Show that this leads to the following condition: which can be rearranged to give you and expression for the fastest possible speed you can have before skidding. Find this speed and verify this result by using the applet.









  4. Show that there is a speed for which the car would be able to travel around the banked curve even is there was no friction. Calculate the size of this speed and try to show this using the applet.