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Torque and Angular Momentum Experiment 13 Objective: To investigate the change in the angular momentum of an object due to an applied torque. DISCUSSION: = I L Angular velocity and angular momentum L=I are vectors. The direction of these vectors are given by the right hand rule. The equations describing Newton’s 2nd law for rotational motion are fully analogous to the equations for linear motion. A net torque on an object will change its angular velocity : I , where I is the moment of inertia, and the angular acceleration = d/dt. We may also write the 2nd Law to relate the net torque to angular momentum L. dL , r dt Thus, the direction of the applied torque is always in the same direction as the resulting angular acceleration and the change in the angular momentum. Consider Fig. 1. A torque is applied on a rotating disk by a hanging weight. The net torque on the system is T T I r mg I disk I weight rmg I disk mr 2 mg Fig. 1 1 EXERCISE 1: Torque and Angular Acceleration 1. Setup the signal interface a. Verify that the unit is plugged in and powered. b. Plug the phone jacks from the motion detector into plugs 1 and 2 of the interface c. Connect your computer via the USB cable. 2. Setup DataStudio a. When prompted, choose ‘Create Experiment’ b. Click ‘Add Sensor or Instrument’ c. Choose from ‘Science Workshop Digital Sensors’ d. From the list, select ‘Rotary Motion Sensor’ e. On the list of measurements, check the following Angular Position, Ch 1&2 Angular Velocity, Ch 1&2 f. From the list of Displays, drag Graph to Data list above and drop it onto Angular Position. g. Drag Graph1 and drop it onto Angular Velocity. 3. Assemble the apparatus a. Securely, attach the weights to each end of the bar. Then attach the bar to the axel. Make sure it is balanced. b. Using the string provided, measure the length between the two knots tied on the string. Then wind the length of string between the two knots around the larger of the two pulleys on the apparatus. 4. Perform the experiments a. Attach a 20g mass to the end of the string. In DataStudio, click Start and the drop the mass. The angular position and velocity will be recorded by the computer b. Wind the string around the smaller pulley and repeat the experiment. c. Move each mass about halfway between the middle and end of the bar. Repeat the experiment for each pulley. DATA ANALYSIS 1. Finding the Moment of Inertia a. Using the measured values of the acceleration from each experiment, calculate the moments of inertia of the apparatus using the equations above. b. Now, calculate the moment of inertia from the geometry of the apparatus for each of the two configurations. . c. Find the percent deviation between these values. 2 2. Finding the Torque a. From your experimental value of the moment of inertia and the angular acceleration, calculate the net torque on the apparatus. (Note, this is less than the net torque on the apparatus plus the hanging mass.) b. From the experimental data gathered after the hanging mass fell free, calculate the torque of friction on the apparatus. Is this sufficient to explain the experimental deviation found in the moment of inertia measurements? EXERCISE 2: Gyroscopic Precession A gyroscope which is not spinning and not perfectly vertical will fall over. This is due to the torque of the gyroscope’s own weight. A spinning gyroscope which is not perfectly vertical will undergo a precession motion. This is due to the torque of the gyroscope’s own weight. In both cases, the change in the in the dL gyroscope’s angular momentum is in the dt same direction of the gravitational torque. 1. Observe a non-spinning gyroscope fall over. What is the direction of the torque? L o mg 2. With a quick pull of the supplied string, give the gyroscope a large angular momentum in the red direction. Set the gyroscope on the table with the red end upward and observe the precession. What is the direction of the torque? What is the direction of the precession? (Clockwise or counter-clockwise.) Sketch a diagram of the top showing the direction of the angular momentum, gravitational torque, and precession. 3. With the gyroscope still spinning in the red direction, flip it over so the red end is on the bottom. What is the direction of the torque? What is the direction of the precession? (Clockwise or counter-clockwise.) Again, sketch a diagram.. 3