Lab 5: Rotational Motion II

Transcription

1 Lab 5: Rotational Motion II Written October-November 1986 by Nancy Bronder '86, Tom Budka '89, Bob Hamwey GS, D. Mook revised by Mook, K. Muenchinger '93 and E. Pleger '94 July-November 1991 and by Melissa Wafer '95 May-June revised Thorstensen 1995 As you are probably well aware, rotational dynamics can be one of the most difficult parts of introductory mechanics. The behavior of rotating systems can defy intuition. The mathematics behind them involves vector cross products, so you cannot avoid working in three dimensions, and it's difficult to draw three dimensions on a flat piece of paper. The following exercises have been created to give you practice at applying the difficult concepts behind rotational mechanics to systems that you can really play with. The central intent of this laboratory is to provide you with an opportunity to predict the behavior of rotating systems using your knowledge of rotational dynamics and then to check to see if your predictions are correct. Suggested Reading: You may want to consult pages 80 to 81 of the book The Way Things Work (on reserve in Kresge Library) before coming to this laboratory session. Other students have found these pages to be worthwhile reading. GOAL: For you to use your knowledge about rotational systems to predict their behavior and to then play with them until you are comfortable explaining why they behave the way they do. PART 1: THE GYROSCOPE Start the gyroscope spinning using the motor provided. Position the gyroscope frame so that the rotation axis is horizontal. To represent the gyroscope's motion in your notebook, draw a picture like this:

2 where the vertical oval represents the wheel of the gyroscope and the horizontal line represents the rotation axis. A. Draw the suggested representation of the gyroscope in your notebook and show which way the wheel is spinning. Then put an arrowhead on the axis to indicate the direction of the angular momentum vector. How do you know this is the correct direction for the angular momentum? B. Next imagine that you apply a force to the rotation axis by tipping the end with the arrowhead either upward or downward, which will generate a torque. Draw that imaginary force on another diagram of the gyroscope and predict the direction of the torque that will result. (Hint: Draw the position vector of the point at which the force is applied.) From your predicted torque predict how the axis of the gyroscope will move in response to the applied force. What is your reasoning behind this prediction? C. Now we are going to work backwards and do the opposite of what we did in the previous part. Figure out how you would make the end of the rotation axis with the arrowhead point upward by applying a torque to the gyroscope. (You can do this by gently rotating the base.) Draw another diagram that shows what you would do, and explain why it should work. D. Now go back and test your predictions in parts B and C. Do they work? Comment on any misconceptions you may have had and any realizations you may have made. Play with the gyroscope until you feel comfortable predicting in which direction the gyroscope will move for any given applied force. Record any observations you make or conclusions you draw in your notebook. Now on to BIGGER things... In class you saw a large gyroscope made from a bicycle wheel used to demonstrate the effect of torque on a rotating object. E. Start this large gyroscope rotating with its axis horizontal. Now hang a mass from one end of its axis and observe the result. Draw a diagram in your notebook and explain how the behavior of the gyroscope is consistent with Newton's Second Law for rotational motion. F. Why is the rim of the bicycle wheel in the large gyroscope loaded with lead? What is term used to describe the effect that the lead has and what does it mean? G. Why does it take a greater torque to change the rotation rate of the large gyroscope than to change the rotation rate of the small gyroscope? Use an equation in your explanation.

3 PART 2: ROTATING STOOL One lab partner needs to sit on the rotatable stool while holding a rotating bicycle wheel by one end of the axle. Hold on tightly! A. Start by holding the axle horizontally and note whether it is spinning clockwise or counterclockwise. Indicate the direction in a picture and draw the angular momentum vector. Does the person on the stool feel anything unusual while holding the spinning wheel? B. In a moment you are going to be asked to exert forces to change the wheel's axle to a vertical orientation. What, if anything, do you think might happen? C. Now go ahead and do it. What happens? What do you feel? Why does this happen? Explain in words and draw a diagram showing the applied force, the position vector to the point at which the force is applied, the resulting torque and the change in angular momentum. (Hint: Be aware of exactly where the force is applied - at what point does the thing exerting the force and the thing feeling the force meet? Between the wheel and the axle, the axle and your hands, your hands and your arms, your arms and your shoulders? This will help you determine the position vector.) Be sure that you take your turn as the person holding the rotating bicycle wheel on the stool. Notice what you feel when you exert a force on the axle of the rotating wheel. D. Have you ever watched an ice skater spinning in circles? Next time you do, watch his or her arms. Ice skaters use their arms when they turn, either extending them outward or pulling them in towards their bodies. What effect do you think this has on their turns? E. Now start a person rotating on the stool while holding a mass in each hand with arms fully extended. To the spinning student: Pull in your arms so that your hands (still holding the masses) are right next to your body. What happens? What do you feel? F. To the stationary student: Use a stopwatch to determine the angular velocity with which the person is initially rotating. Then determine the spinning student's new angular velocity. Use your measurements to estimate the percentage change in the moment of inertia of the person and masses when she/he pulled in her/his hands. Be sure that you take your turn as the person holding the masses on the stool. Notice what you feel when you pull in your arms.

4 PART 3: THE YO-YO A. Wind up a yo-yo and place it on the table so that the string leaves the axis of the yo-yo from a point on the axis between the table and the center of the yo-yo, as shown below: Place the yo-yo in front of you so that it can roll towards you or away from you. Make sure that the string comes out from underneath the axis towards you. What do you think will happen when you pull the string gently towards you? B. Now try it. Does the yo-yo roll towards you or away from you? What is the approximate angle f between the string and the table? Try several different angles and record your data in a table. Is there some "critical angle" at which the yo-yo does not roll at all? Try to figure out what this angle is. string φ parallel to table Table surface C. Draw a free-body diagram of the yo-yo while it is stationary. What tells you that it has the potential to rotate?

5 D. Draw and label three diagrams of the yo-yo like the one above, one for each of the following situations: 1) it rolls forward, 2) it does not roll, 3) it rolls backward. Draw the string at an appropriate angle to the table and draw in the force vector along the string for each. Indicate the range of angles over which each diagram applies. E. In a caption below each diagram, write a mathematical equation to explain what is happening. (Hint: Use the point at which the yo-yo rests on the table as your point of origin. You may then want to add something to your diagrams.) F. Explain in words why the yo-yo behaves as it does in all three scenarios. PART 4: TOPS Try out the various tops in the room. A. THE STANDARD TOP: Draw a free body diagram of a standard top. Then draw a diagram showing the angular velocity, angular momentum, and torques acting on it. Use this diagram to explain briefly why tops precess. THE TIPPEE TOP: Notice that the round-bottomed tops (sometimes called "Tippee Tops") will flip themselves over and spin on their spindles. The sketch on the last page of these instructions is a side view of a spinning Tippee Top with its center of mass shown as a dot. You will need to cut this out and tape it in your lab notebook. B. How can you experimentally confirm that the center of mass of the Tippee Top is near the table top and not at the center of its spherical base? (Hint: Try setting the top so that its spindle is horizontal, as in the second diagram. It will not maintain this position. Why? Could the center of mass be at the center of the spherical section if the top behaves this way?) C. On the first diagram of the tilted top on the last page of these instructions, draw the following: 1) an angular velocity vector, 2) the angular momentum vector, 3) a vector to show in which direction friction between the spinning top and the surface of the table acts on the top, and 4) the position vector r from the center of mass to the point at which the frictional force is applied. (Hint: The frictional force acts because the surface of the top is slipping on the surface of the table top; this produces a kinetic frictional force. The direction of this frictional force will be such as to slow the top's motion. Look carefully at the point of contact between

6 the top and the table and decide on the direction of the frictional force). Then draw 5) the torque produced by this frictional force. Would this torque tend to make the top flip over? D. On the second diagram of the horizontal top, draw in all the things named above plus any other forces acting on the top. Also indicate the torques that the applied forces cause and describe the resulting motion. (Hint: Remember that a position vector goes from a point of origin to the point at which the force is acting. You may have more than one position vector here.) Does the torque due to the frictional force between the top and the surface of the table still tend to make it flip over? Why don't ordinary tops flip over the way Tippee Tops do?

7 The Tippee Top