11a lab 4: Rotation and Gyroscope

Transcription

1 11a lab 4: Rotation and Gyroscope Prelab: Familiarize yourself with the concepts of rotation, angular momentum, torque, and moment of inertia (lecture notes and Giancoli chapter 11). 1. How large is the torque if a 1m long lever arm is exposed to a force of 100N normal to the arm? 2. What is the relationship between torque and angular momentum? For comparison, what are the corresponding variables and equation for linear motion? 3. What is the relationship between angular momentum and rotation speed? For comparison, what are the corresponding variables and equation for linear motion? 4. Two person weighing 60kg each sit on a seesaw, with a total bar length of 4m. Estimate the moment of inertia (Neglect the weight of the bar). 5. Read the lab instructions. In particular, read the section about the gyroscope and make sure you follow the discussion of gyroscopic forces (bicycle wheel). The lab consists of two parts one at which you will work with a rotating chair and explore angular momentum and inertia, and one where you will work with a precision gyroscope (and a bicycle wheel) to explore gyroscopic forces. Half of the groups will start with the rotating chair setup, the other half will work with the gyroscopes first. Rotation and angular momentum In this part of the lab we work with a rotating chair and explore various aspects of rotation and angular momentum. The chair we use is mounted on a ball bearing with low friction, so that it does not immediately slow down its rotation. (Discuss with your TF: Why does a ball bearing have low friction?) You have the following equipment: The rotation chair, a rope, a weight, two stopwatches, and a pulley. How can you measure the moment of inertia of the chair with you sitting on it? Discuss with your TF. Hint: Whenever you measure the speed of rotation, take the time for a full rotation of the chair. That way you average over any periodic variation of the rotation speed that can occur if the chair is not perfectly level. 1) Take all the data you need for the measurement of your moment of inertia on the chair. You probably have seen how ice dancers turn themselves into a fast spinning motion on ice, or how divers do fast summersaults and then nearly stop the rotation before touching

2 the water. How is that possible? How is that consistent with the conservation of angular momentum? Discuss with your TF (don t write anything here). 2) Let s try it out: Sitting on the chair, and using two dumbbells, collect the data needed to measure the maximum increase in your spinning speed by moving the dumbbells from far outstretched to close in. Collect only one dataset, but have everyone in the group try it. 3) It turns out that the rotational kinetic energy is larger when you spin faster where does this energy come from? Using a bicycle wheel mounted on handles you can nicely explore the concept of conservation of angular momentum. Spin up the bicycle wheel as fast as you can (the TF can show you a nice trick to reach highest speeds). Then sit on the chair. 4) Start with the chair at rest and the bicycle wheel vertical. The wheel should be spinning so that the top edge is rotating away from you. What happens as you turn the bicycle wheel to the left or two the right, into the horizontal position? 5) Start with the bicycle wheel horizontal and the chair at rest. Turn the wheel by 180 degree until it is horizontal with opposite orientation. Measure your rotation speed. What happens if you turn it back to the initial orientation? What happens if you turn it the other way around? Does it make sense? Discuss with your TF. 6) If you are the group doing this part of the lab first: Spin up the wheel and explore the weird gyroscopic forces. Hold it at one handle, and explore what happens when you try to tilt the wheel. (See description below in the Gyrsoscope section, and discuss with your TF) Let s analyze the measurements you made. 7) Calculate your moment of inertia as you sit on the chair. Does this value make sense to you? 8) Using your calculated moment of intertia and your knowledge about conservation of angular momentum, can you predict the change in angular velocity when you move the dumbbells? Calculate the measured change in your rotation speed (the mass of a dumbbell is 3.5 kg). How does this compare with what you predicted? What are some likely sources of error? 9) The wheel has most of its weight (2.5 kg) in the rim (r = 0.29 m). Using the speed of rotation you measured on the chair after tilting the wheel, calculate how fast the wheel was spinning. Gyroscope In this part of the lab you will explore gyroscopic forces. You will then use this insight to explore questions like why a bicycle does not tip over, why a Frisbee flies stable without tumbling, and why a spinning top stays upright on its tip.

3 The key to understanding gyroscopic forces is the insight that angular momentum is conserved. Not only is the magnitude of angular momentum conserved, but also the direction of rotation. We can describe the angular momentum with a vector that is oriented along the axis of rotation. Its length denotes the magnitude of angular momentum. Let s think what we expect to happen if you hold a spinning bicycle wheel with a handle in your hand, and try to tilt the spinning wheel upwards (see figure below). The wheel initially has angular momentum of L along the rotation axis for spinning the wheel. If you try to tilt the wheel upwards, you would try to rotate it around the end of the handle. The rotation axis for lifting the wheel is horizontal (see fig. a). The torque along this axis increases the net angular momentum by ΔL. ΔL is a vector along this rotation axis, hence, the new net angular momentum has changed direction in the horizontal plane (see fig. b). The rotation axis of the wheel follows the direction of the angular momentum. Hence, if you try to tilt the wheel upwards, it will not turn upwards, but to the right. This counterintuitive behavior is hard to imagine without trying it out! 10) Explore the gyroscopic forces of the spinning bicycle wheel. Can you predict how the wheel will move if you try to tilt it upwards? Try it out! The TF can show you how to spin the wheel to high speeds. Let everyone in the group explore it, and discuss with the TF. 11) Put a string through the handle. How does the wheel move if you hold it that way? The motion is called precession can you understand it based on the gyroscopic forces? We will now use a precision gyroscope in a gimbal mount. Important: Always leave the gyroscope within the wooden box. If you drop it on the concrete floor, it will be damaged, and you will have to exchange the ball bearings!

4 Figure: Gyroscope in gimbal mount. Here it has a rod attached to the right that causes a precession motion. Assemble the gyroscope in the gimbal mount. This mount allows the gyroscope to freely rotate along two axis of rotation (horizontal and vertical). For now, do not attach the long arm as shown in the figure above. Without the arm, the gyroscope is balanced, i.e. there is no net torque acting on the gyroscope due to gravity. Spin up the gyroscope with the motor. 12) Gyroscopes are used as compasses that do not rely on the earth magnetic field (which can be locally distorted). In fact, every commercial plane has multiple gyroscopes, as a compass, and also as an attitude indicator. How can you use a balanced gyroscope as a compass - how would you initially orient it, so that it always points north everywhere on the globe? Try it out, and carry the gyroscope around the room inside the wooden case (make sure it spins at high speed). You may tilt and rotate the case. Does it stay aligned well? Discuss sources of error. 13) Touch the c-shaped gimbal and try rotating it around the vertical axis. What happens? Can you explain what happens based on adding angular momentum vectors (see bicycle wheel)? 14) What happens if the rotation axis of the gyroscope gets aligned with the vertical gimbal axis? This is called gimbal lock. It occurred on the navigation gyroscope onboard Apollo 11, and nearly caused mission failure.

5 15) Now attach the short screw rod at the screw hole. Use a straw to apply forces to it, and see how the gyro reacts. You once again observe the gyroscopic forces. 16) Attach the long rod (see figure above). Now the gyroscope is clearly out of balance. Via the lever arm, gravity will cause a torque. This torque sets the gyroscope into a rotation around the vertical axis, called precession. Explore qualitatively: Does the precession speed depend on the gyroscope orientation (angle between rod and vertical axis?) How can you change this angle? What happens if you attach a weight to the end of the lever arm? 17) Spin up the gyroscope to max speed and let it precess, with the arm in horizontal orientation. Use a stopwatch to record the time of each full precession, over about 5 minutes. You can apply torque to the c-shaped gimbal arm to keep the orientation of the rod constant. Use LoggerPro to plot a graph of precession time vs time. 18) The precession angular frequency Ω is inversely proportional to the spinning angular frequency ω: Ω=Mgr/(Iω), where M is the mass of the imbalance r the distance of the mass from the center, g the gravity constant, and I the moment of inertia (See Giancoli, chapter 11-7). The LoggerPro worksheet has a calculated column with these parameters entered. Plot the rotation speed, and fit it with an exponential without offset. 19) Only give short answers for the next questions, mostly discuss with your TF: Can you explain why a wheel or a bicycle does not tip over when it rolls? Try it out with the black bicycle wheel (not the fancy silvery one it s bearings might get damaged if it falls onto the concrete floor). How does a Frisbee work? Why does a spinning top not fall over? Can you observe the precession? Have Fun!