# Establishing and understanding equations that define rotational motion and how they relate to the equations for linear (translational) motion.

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1 Name Period Date Rotational Dynamics Structured (S) Objectives Investigate factors that define and affect spinning objects by: Defining and exploring the concept of rotational inertia I. Understanding how torque causes angular acceleration. Establishing and understanding equations that define rotational motion and how they relate to the equations for linear (translational) motion. Materials and Equipment For each student or group: Data collection system Rotary motion sensor Mini Rotational Accessory Mass and hanger set, 0.5-g resolution Balance, (1 per class), 400-g capacity Super Pulley with Clamp Large base and support rod Scissors String Meter stick Stainless steel caliper Background Spinning and rotating objects are all around us. Bike tires, your car's drive shaft, and the earth itself, are all examples of spinning objects. How they spin, what makes them spin, and what factors will change the way they spin, are all relevant questions answered by the physics of rotational dynamics. The only way to cause objects to spin is to apply an external torque, which is a force applied at some distance from the axis of rotation. The greater the applied torque, the greater the affect on the object s rate of spin If a hanging mass on a string is attached to a pulley, the falling mass will create a torque that will cause the pulley s rotation to accelerate. The moment of inertia I of the pulley is a product of the pulley s mass and geometry, but is better defined as the pulley s resistance to changes in rotational velocity. The relationship between torque τ, moment of inertia I, and angular acceleration α is governed by the equation: m T T mg 1

2 Rotational Dynamics Torque = Rotational Inertia Angular Acceleration τ = Iα (1) Notice here that torque is measured in Newton meters (not the same as Joules), I is measured in kg m 2, and α is measured in radians/s 2. If the applied torque increases, the angular acceleration should also increase. For a constant applied torque, a rotating object with smaller rotational inertia should have a greater angular acceleration than that of an object with greater rotational inertia. All of the rotational concepts covered are analogous to linear motion properties, with very similar equations. In this lab, you will measure all of these properties and thoroughly explore rotational dynamics to discover these equations. Relevant Equations Σ τ = τ Iα (1) net = This equation states that an unbalanced torque (or nonzero, "net" torque) will induce an angular acceleration α in a body whose moment of inertia is I. This equation is very much like Newton's second law of motion (F = ma), but applied to rotating objects rather than to linear motion. Safety Add these important safety precautions to your normal laboratory procedures: Keep hands and heads clear of the spinning platform while it is moving and use caution when stopping the rotating arm. Be sure objects on the spinning platform are attached or they will fly off. Procedure After you complete a step (or answer a question), place a check mark in the box ( ) next to that step. Note: When you see the symbol " " with a superscripted number following a step, refer to the numbered Tech Tips listed in the Tech Tips appendix that corresponds to your PASCO data collection system. There you will find detailed technical instructions for performing that step. Your teacher will provide you with a copy of the instructions for these operations. Part 1 Varying Torque by Changing Force Set Up 1. Measure the mass of each sliding mass and the rotating arm, and then measure the length of the rotating arm. Record these values in the Data Analysis section. 2 PS-2899

3 Student Worksheet (S) 2. Attach the rotary motion sensor to the large base and support rod at the top of the support rod using the rod clamp on the top of the sensor. 3. Mount the super pulley to the opposite end of the rotary motion sensor (cable end) using the super pulley clamp. 4. Mount the rotating arm to the rotary motion sensor. Note: Invert the pulley on the rotary motion sensor before mounting the rotating arm. Rod guides 3 step pulley Rotating arm Rotary motion sensor Smart pulley Support rod Mass hanger with mass 5. Attach one end of the string to the smallest pulley groove on the 3-step pulley and the other end to the mass hanger. Run the string from the 3-step pulley over the super pulley allowing the mass hanger to hang freely. 6. Adjust the height of the super pulley so that the section of string between the top of the super pulley and the point at which it meets the 3-step pulley is horizontal. 3 step pulley Adjust pulley so string is horizontal Rotary motion sensor 7. Adjust the sliding masses along the rotating arm so the center of each mass is 15 cm from the axis of rotation. 3

4 Rotational Dynamics 8. Use the stainless steel caliper to measure the diameter of the three different pulley grooves on the 3-step pulley. Be sure to measure the part of the pulley that the string encircles, not the outside edge of pulley. Record each value in Table Connect the rotary motion sensor to the data collection system. (2.1) 10. Create a graph display of Angular Acceleration on the y-axis with Time on the x- axis. (7.1.1) 11. Add 25 g of mass to the 5 g mass hanger giving a total of 30 g of mass hanging from the string. 12. Wind the string around the smallest pulley groove on the 3-step pulley until the mass hanger is just below the super pulley and hold it in place. Collect Data 13. Begin data recording. (6.2) 14. Release the hanging mass (from rest), and collect data until the rotating arm has completed 5 full revolutions. 15. Once the arm has completed 5 full revolutions, stop data recording (6.2) and then carefully stop the rotating arm. 16. For consistency, you may want to rename each data run with a definitive description of the setup used (example: "Data Run 2 = 60 g ). (8.2) 17. Change the amount of mass (force) on the hanger: Repeat the previous steps two more times using an additional 30 g each time. 18. Record the 3 different masses used into Table 2. Save your recorded data. (11.1) Part 2 Varying Torque by Changing Pulley Diameter Set Up 19. Remove the string from the small pulley groove and reattach it to the middle pulley groove. 20. Remove mass from the mass hanger until you have a total of 60 g of mass attached to the string. Be sure to keep all other variables the same. Do not change the position of the masses on the rotating arm or the amount of mass attached to the string (60 g). 4 PS-2899

5 Student Worksheet (S) 21. Wind the string around the middle pulley groove until the mass hanger is just below the super pulley and hold it in place. Collect Data 22. Repeat the collect data steps from Part 1. However, in this part you will keep the hanging mass (force) constant and vary the pulley diameter by repeating the collect data procedure one additional time using the large pulley groove. Note: There is no need to repeat the procedure for the small pulley groove since data for that setup was recorded as part of Part For consistency, you may want to rename each data run with a definitive description of the setup used. (8.2) 24. Record the hanging mass, rotational arm length, and pulley groove diameter values in Table Record all of your angular acceleration data onto the data collection system and be sure to save your progress as you go. (11.1) Part 3 Varying Rotational Inertia Set Up 26. Remove the string from the large pulley groove and reattach it to the small pulley groove. 27. Move the sliding masses on the rotating arm so the center of each is 5 cm from the axis of rotation. Be sure to keep all other variables the same. Do not change the pulley diameter or the amount of mass attached to the string (60 g). 28. Wind the string around the middle pulley groove until the mass hanger is just below the super pulley and hold it in place. Collect Data 29. Repeat the collect data steps from Part 1. However, in this part you will keep the pulley diameter and mass constant and vary the rotational arm length. 30. Repeat the collect data procedure one additional time with the center of each sliding mass 10 cm from the axis of rotation. Note: There is no need to repeat the procedure with the sliding masses at 15 cm since data for that setup was recorded as part of Part 1. 5

6 Rotational Dynamics 31. For consistency, you may want to rename each data run with a definitive description of the setup used. (8.2) 32. Record the hanging mass, pulley groove diameter, and rotational arm length values in Table Record all of your angular acceleration data onto the data collection system and be sure to save your progress as you go. (11.1) Data Analysis Part 1 Varying Torque by Changing Force Mass of sliding masses =. Mass of rotating arm =. Length of rotating arm =.. Table 1: Pulley groove diameters Pulley Step Diameter (cm) Large Groove Medium Groove Small Groove Use the statistics tool to determine the average angular acceleration for each trial (9.4), and then record the results in the tables below. Table 2: Varying Torque by Changing Force Distance of masses from axis =. Pulley diameter =. Hanging Mass Value (g) Average Angular Acceleration (rad/s 2 ) Data Run 1 Data Run 2 Data Run 3 6 PS-2899

7 Student Worksheet (S) Part 2 Varying Torque by Changing Pulley Diameter Table 3: Varying torque by changing pulley diameter Distance of masses from axis = Hanging mass value =.. Pulley Diameter (cm) Average Angular Acceleration (rad/s 2 ) Data Run 1 Data Run 2 Data Run 3 Part 3 Varying Rotational Inertia Table 4: Varying rotational inertia Pulley diameter = Hanging mass value =.. Data Run 1 Data Run 2 Data Run 3 Distance of Masses From Axis (cm) Average Angular Acceleration (rad/s 2 ) Analysis Questions 1. For each part of your experiment, list each variable, state whether it was held constant, increased, or decreased, and state how each change affected the angular acceleration. _ 7

8 Rotational Dynamics 2. From your results, can you predict a mathematical relationship between torque τ, rotational inertia I, and angular acceleration α? 3. In which situation did the rotating arm spin faster, with the weights closer to or further from the axis of rotation? Explain why. 4. Indicate any factors that might have caused your data to be incorrect. Also explain how these factors may have been corrected or avoided. 5. Calculate the rotational inertia of the entire rotating arm plus sliding masses (at a distance of 15 cm from the axis of rotation). What shape and corresponding formula did you use for the two masses? What shape did you use to model the rotating arm? 6. How would your results be affected if you held the torque constant but used a much smaller rotating platform? Would angular acceleration be higher, lower, or the same? Justify your answer. 8 PS-2899

9 Student Worksheet (S) Synthesis Questions Use available resources to help you answer the following questions. 1. What angular acceleration would you expect from a rotating object with rotational inertia of kg m 2 that was subjected to a net torque of 4.25 N m? 2. Think of a bicycle sprocket and the gears you use when pedaling. Would the larger pulley wheel on the spinning platform's axle represent a higher gear or lower gear on a bicycle? Justify your answer. _ Note: Keep in mind that the "drive gears" on a bicycle (where the pedals are attached) work in opposite fashion. The larger sprocket is for high gear because the cyclist applies a larger torque, due to the larger lever arm of the big sprocket. This is noticeably harder for the cyclist, but leads to higher torque once the bike is moving. 3. When you release the hanging mass and the arm starts rotating, do you expect the angular acceleration to be constant? Why or why not? _ 4. Would frictional errors affect this lab more or less if you had used a rotating arm with a much smaller mass? Explain your answer. 9

10 Rotational Dynamics 5. Compare the angular quantities you just examined to linear quantities you have measured in the past. Fill in the blanks below with the angular (spinning) quantities that best correspond to their linear motion counterparts. Be sure to include the variable used to denote the quantity, and the SI units used to measure it. Linear Motion Angular Motion Position x (m) Velocity v (m/s) Acceleration a (m/s 2 ) Force F (N) Mass m (kg) 6. For linear motion, kinematics equations are commonly used for 1-dimensional and 2-dimensional motion problems. Use your completed table above to "translate" and rewrite the kinematics equations for angular motion. Multiple Choice Questions Select the best answer or completion to each of the questions or incomplete statements below. 1. Three children are sitting on a merry-go-round while a fourth child is pushing on the outside edge with a constant force. Why does it take less time for the fourth child to speed the merry-go-round up if the three other children all sit in the center rather than half way between the center and outside edge? A. The merry-go-round accelerates faster because there is more torque spinning it. B. The merry-go-round accelerates slower because there is more torque acting on it. C. The rotational inertia is lower, therefore the merry-go-round spins slower. D. The rotational inertia is lower, therefore the merry-go-round spins faster. E. None of the above 10 PS-2899

11 Student Worksheet (S) 2. When you increased the pulley diameter in part 2 of the lab, which statement best describes what happened? A. The time for 5 revolutions decreased because the force got larger and rotational inertia stayed the same. B. The time for 5 revolutions increased because the torque remained unchanged, but the rotational inertia decreased. C. The time for 5 revolutions decreased because the lever arm got larger, and rotational inertia stayed the same. D. The time for 5 revolutions increased because the torque remained unchanged, but the rotational inertia increased. E. The time for 5 revolutions increased because the air resistance was less. 3. When you moved the masses closer to the axis in part 3 of the lab, which statement best describes what happened? A. The time for 5 revolutions decreased because the rotational inertia got larger, and applied torque did not change. B. The time for 5 revolutions decreased because the torque got larger and the rotational inertia decreased. C. The time for 5 revolutions increased because the rotational inertia did not change. D. The time for 5 revolutions increased because the torque remained unchanged, but the rotational inertia decreased. E. The time for 5 revolutions decreased because the rotational inertia got smaller for the same amount of applied torque. 11

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