Worksheet for Exploration 10.1: Constant Angular Velocity Equation

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1 Worksheet for Exploration 10.1: Constant Angular Velocity Equation By now you have seen the equation: θ = θ 0 + ω 0 *t. Perhaps you have even derived it for yourself. But what does it really mean for the motion of objects? This Exploration allows you to explore both terms in the equation: the initial angular position by changing θ 0 from 0 radians to 6.28 radians and the angular velocity term by changing ω 0 from -15 rad/s to 15 rad/s. Restart. Answer the following questions (position is given in meters and time is given in seconds). a. How does changing the initial angular position affect the motion? b. How does changing the initial angular velocity affect the motion?

2 Additional Questions Several measurable quantities are related. Use the equation given above for your settings of initial angular position, and initial angular velocity (ω 0 ) and complete the tables below In the table S means the net displacement around the circle (in meters, not angle), and v tang means the tangential velocity the ball has going around the circle. i. In addition to filling out the table, label an example of what is meant by the initial angular position, angular displacement, and displacement on the figure. ω o = θ o θ t θ t s v tang t1= t2= Select a new initial angular speed and repeat. ω o = θ o θ t θ t s v tang t1= t2=

3 Worksheet for Exploration 10.2: Constant Angular Acceleration Equation By now you have seen the equation: θ = θ 0 + ω 0 *t + 0.5*α*t 2. Perhaps you have even derived it for yourself. But what does it really mean for the motion of objects? This Exploration allows you to explore all three terms in the equation: the initial angular position by changing θ 0 from 0 radians to 6.28 radians, the angular velocity term by changing ω 0 from -15 rad/s to 15 rad/s, and the angular acceleration by changing α from -5 rad/s 2 to 5 rad/s 2. Restart. Answer the following questions (position is given in meters and time is given in seconds). a. How does changing the initial angular position (θ 0 ) affect the motion of the object? b. How does changing the initial angular velocity (ω 0 ) affect the motion of the object? c. How does changing the angular acceleration (α) affect the motion of the object? d. Can you get the object to change direction? i. Try different combinations of initial angular velocity and angular acceleration (which is constant).

4 Additional Questions For your selected values of angular position, velocity, acceleration and a couple of times, complete the following table. θ o = ω o = α= t1= θ t θ t ω s v tang a tang t2= Now select a set of positive values for initial angular position, initial angular velocity, and a negative angular acceleration. Before running the simulation, see if you can predict the following. i. How long does it take until the angular velocity becomes zero? ii. Through how much angle has it traveled? iii. What is the average angular speed for this trip?

5 Worksheet for Exploration 10.3: Torque and Moment of Inertia A mass (between 0.01 kg and 1 kg) is hung by a string from the edge of a massive (between 0 kg and 2 kg) disk-shaped pulley (with a radius between 0.1 and 4 meters) as shown (position given in meters, time given in seconds, and angular velocity given in radians/second). For a, b, and c, set the hanging mass to 0.25 kg, the radius of the pulley to 2 m, and vary the mass of the pulley. a. How does the magnitude of the angular acceleration of the pulley depend on the mass (and therefore moment of inertia) of the pulley? i. As the mass of the pulley increases, the angular acceleration INCREASES or DECREASES b. How does the magnitude of the acceleration of the hanging mass depend on the mass (and therefore moment of inertia) of the pulley? i. As the mass of the pulley increases, the acceleration INCREASES or DECREASES c. How are your answers to (a) and (b) related? i. Give a specific relation between angular acceleration and acceleration. Keep careful track of signs. For d, e, and f set the mass of the pulley to 0.5 kg, the radius of the pulley to 2 m, and vary the hanging mass. d. How does the magnitude of angular acceleration of the pulley depend on the hanging mass? i. As the hanging mass increases, the angular acceleration INCREASES or DECREASES e. How does the magnitude of acceleration of the hanging mass depend on the hanging mass? i. As the hanging mass increases, the angular acceleration INCREASES or DECREASES f. How are your answers to (d) and (e) related?

6 For g, h, and i set the hanging mass to 0.25 kg, the mass of the pulley to 0.5 kg, and vary the radius of the pulley. g. How does the magnitude of angular acceleration of the pulley depend on the radius of the pulley? i. You should note a specific functional relation here. Take measurements at several values for R to determine this relation. h. How does the magnitude of acceleration of the hanging mass depend on the radius of the pulley? i. Again you should get a specific relation. Here this should be apparent after only a couple of measurements. i. How are your answers to (g) and (h) related? For j, k, and l set the mass of the pulley to 0.5 kg, the hanging mass to 0.25 kg, and the radius of the pulley to 2 m. Use measurements from the simulation to answer the following questions. j. Determine the acceleration of the hanging mass and the angular acceleration of the pulley. i. Use measurements of displacement and time to determine this. k. From Newton's second law, determine the (magnitude of) tension in the string. i. If you are not sure what to do, you may want to draw a free body diagram for the hanging mass. l. How much torque does this tension provide to the pulley? i. Now that you know the tension, you can calculate the torque by using the definition (force times lever arm).

7 Additional Question As a last part of E10.3 you may want to see if you can write out a theoretical prediction for the motion of this system. Many texts will derive this for you, but see if you can do it. Here is a brief outline of what you need. Write out an equation for Newton s second law for the hanging mass. Write out an equation for the torque acting on the pulley/disk. Write out a third equation relating angular acceleration, and the acceleration of the hanging mass. Be careful with signs. You should have Tension, the two masses, the radius, acceleration, and angular acceleration. Assume the system properties are known (masses and radius).

8 Worksheet for Exploration 10.4: Torque on Pulley Due to the Tension of Two Strings A is a top view of a pulley on a table. The massive diskshaped pulley can rotate about a fixed axle located at the origin. The pulley is subjected to two forces in the plane of the table, the tension in each rope (each between 0 N and 10 N), that can create a net torque and cause it to rotate as shown (position is given in meters, time is given in seconds, and angular velocity is given in radians/second). Restart. Also shown is the "extended" free-body diagram for the pulley. In this diagram the forces in the plane of the table are drawn where they act, including the force of the axle. Set the mass of the pulley to 1 kg, the radius of the pulley to 2 m, vary the forces and look at the "extended" freebody diagram. a. How is the force of the axle related to the force applied by the two tensions? i. Sketch a head to tail vector addition figure for the two tensions and compare this to the axle force vector. Then discuss how these all relate. b. How do you know that this must be the case? (explain the results from a)

9 For parts c-f set the mass of the pulley to 1 kg, the radius of the pulley to 2 m, and vary the forces. c. What is the relationship between F 1 and F 2 that ensures that the pulley will not rotate? d. For F 1 > F 2, does the pulley rotate? In what direction? e. For F 1 < F 2, does the pulley rotate? In what direction? f. What is the general form for the net torque on the pulley in terms of F 1, F 2, and r pulley? i. Make sure that you include appropriate information about the signs. Note that F 1 and F 2 denote magnitudes of the forces only. The directions are as indicated in the animation. For g set the mass of the pulley to 1 kg, F1 to 10 N, F2 to 5N, and vary the radius of the pulley. g. How does the angular acceleration of the pulley depend on the radius of the pulley? i. Give a specific function of radius here. Check and verify your answer using the animation. Set the radius of the pulley to 2 m, F1 to 10 N, F2 to 5N, and vary the mass of the pulley. h. How does the angular acceleration of the pulley depend on the mass of the pulley? i. Give a specific function of radius here. Check and verify your answer using the simulation. i. Given that the pulley is a disk, find the general expression for the angular acceleration in terms of F 1, F 2, m pulley, and r pulley. i. Make sure that the signs and limiting cases agree with your expression. Limiting cases to consider are letting the mass of the pulley get large or small, likewise with the forces. You should check that your results agree with the simulation.