Relationship between tangential and angular properties.
Pre-requisite skills: Basic vector algebra, reference frames.
Approximate completion time: 30 minutes.
Provide sufficient detail to verify that the assignment was completed in a meaningful manner.
Applet by B. Surendranath Reddy
A. Rolling Without Slipping.
Set both the linear and angular velocities equal to 1 and press the Start button and watch what happens. Velocities are always measured with respect to a particular reference frame. The relative velocity of the moving frame with respect to the still frame is represented by the yellow vector in this applet, which we define as V. (You could also say that the yellow vector represents the velocity of the center of mass of the wheel with respect to the applet frame.)
1. Describe which reference frame is chosen for the red and green velocity vectors, vred and vgreen. What do these velocity vectors represent physically? (If you are having trouble answering this question, set either one of the velocity or angular velocity equal to 0 and observe.)
2. By examining the directions and magnitudes of the three vectors at some arbitrary time during the rotation, write down the vector equation that describes how V, vred, and vgreen relate to each other.
3. What is the speed associated with vgreen when the point on the rim of the wheel is at ground level? What does this mean physically?
4. Now consider the situation where the point on the rim is at its highest. How are the speeds related to each other in this situation?
B. Rolling With Slipping
Now change the linear velocity to 2 and press Start again. When finished, reset the linear velocity to 1 and change the angular velocity to 2. Press Start.
1. Do the definitions of the vectors described above change? Does the vector relationship you expressed in your answer to Question A-2 change?
2. Consider the situation where the point on the rim is at ground level. In this case, is the magnitude of vgreen in both instances the same as in Question A-3?
3. Now consider the situation where the point on the rim is at its highest. Are the relationships between the speeds the same as in Question A-4?
C. General Considerations
1. From your answers to Questions A-3 and B-2, can you now describe at least one characteristic of rolling without slipping?
2. What is the most important characteristic of rolling without slipping? Does this applet do a good job illustrating this characteristic? If not, what changes could be made to improve the effectiveness of this applet?
Helpful Resources
- Rolling with Friction by Angel Franco Garcia.
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