Rotational Inertia Lab

Computer / LabStream Rotational Inertia A hanging mass is connected to a light string which passes over a pulley and wraps around a turntable. When released, the mass will accelerate downward while angularly accelerating the turntable. The acceleration depends upon the weight of the hanging mass as well as the rotational inertia of the turntable plus any object placed upon it. In this lab, you will determine the rotational inertia of various objects using a Photogate and Ultra Pulley to record the linear acceleration of the string and hanging mass. You can relate this to the tangential and angular acceleration of the turntable. From this information, you can use Newton’s Second Law in Angular Form to determine the rotational inertias of objects placed on the turntable. Figure 1 OBJECTIVES ∙ Use a Photogate to study the acceleration of a falling mass that accelerates a turntable angularly. ∙ Determine the rotational inertias of various objects. MATERIALS Vernier Photogate with Ultra Pulley Attachment Turntable apparatus Meter Stick Objects to measure inertia of Accelerating mass set (170g) Laptop with Logger Pro Labstream One meter long string

Physics with Vernier 1 Computer / Labstream – Rotational Inertia PRELIMINARY EXERCISE 1. Perform the pre-lab exercise given in the pre-lab handout. (last page of this document) 2. You will use this pre-lab work to help you make calculations for your data table during the lab. PROCEDURE (You need to complete through step 15 in class) Part I Measuring the acceleration of the turntable using 4 different accelerating masses. 3. Set up the equipment as shown in Figure 1. Be sure that the hanging mass can fall far enough to rotate the turntable through about one rotation before striking the floor. 4. Measure the circumference of the turntable using the string as a measuring tool. Rap the string around the turntable for one complete revolution. In your data table record the measured length of the string (circumference, C ) obtained using a meter stick. Use C = 2Πr to calculate radius, r . Record this calculation in the data table, too. 5. Connect the Photogate to DIG 1 of LabStream, open Logger Pro on the computer, and choose New from the File menu (if necessary). 6. Set up LabStream in Logger Pro for a pulley with a string that runs in a groove as described below. a. Experiment ► Set Up Sensors ► LabQuest Stream ► (click on picture of) Photogate b. Make sure Motion Timing is checked c. Set Distance or Length ► (drop down menu) Ultra Pulley (10 spoke) In Groove ► OK d. This setting must be confirmed for each new data file!! e. Modify length of data collection by Experiment ► Data Collection ► Duration (set time) 7. Display only the velocity vs. time graph by clicking in the center of undesired graphs then hitting delete. CTL+R will resize your velocity graph to full size. To measure the acceleration of this system, hook the loop at the end of the string on the small nail in the side of the turntable and wind the string around the perimeter of the turntable. Then run the string over the pulley. Start by hanging a 50 g mass from the string, keeping a slight tension in the string to prevent it from falling off the edge of the turntable. Steady the mass so it is not swinging. Start data collection and then release the mass. Stop data collection after the entire string has accelerated the turntable. 8. Examine the resulting velocity graph. The slope represents the linear acceleration of the mass and the tangential acceleration of the turntable. Fit a straight line to the velocity vs. time graph. a. Highlight with the cursor the straight line portion of the graph. b. Analyze ► Linear Fit (you can drag the pop-up box to an ideal location on the graph) c.

cylinder, silver metal tile, round flagstone, etc.) 20. Calculate the percent difference between your experimental value of I object and the theoretical value. (% difference = 100 * (experimental – theoretical) / theoretical). Record your value in your data table. Physics with Vernier 3 Computer / Labstream – Rotational Inertia DATA TABLE Your name TURNTABLE ALONE Lab partners 1. circumference of wheel (m)= 0.94m 2. radius of wheel (m)= 0.15m 3. acc. Mass (Kg) Tang. Acc. (m/s 2 ) alpha (rad/s 2 ) Tension in string (N) Torque from string (Nm) 0.05 0.9752 6.501 0.441 0.066 0.09 1.644 10.96 0.73 0.11 0.13 2.241 14.94 0.98 0.15 0.17 2.749 18.33 1.2 0.18 Object #1 = Wooden Disk mass (Kg) = 0.654 (if needed) dimension a (m) = n/a (if needed) dimension b (m) = n/a (if needed) radius r (m) = 0.15 acc. Mass (Kg) Tang. Acc. alpha Tension in string Torque from string

(m/s 2 ) (rad/s 2 ) (N) (Nm) 0.05 0.524 3.49 0.46 0.07 0.09 0.9574 6.38 0.8 0.12 0.13 1.365 9.1 1.1 0.16 0.17 1.722 11.48 1.37 0.21 Object #2 = Flagstone mass (Kg) = 8.377 (if needed) dimension a (m) = n/a (if needed) dimension b (m) = n/a (if needed) radius r (m) = 0.154 acc. Mass (Kg) Tang. Acc. (m/s 2 ) alpha (rad/s 2 ) Tension in string (N) Torque from string (Nm) 0.05 0.0465 0.302 0.49 0.08 0.09 0.1266 0.822 0.87 0.13 0.13 0.217 1.409 1.25 0.19 0.17 0.291 1.89 1.62 0.25 experimental theoretical % error Moment of Inertia slope = I I object I object TURNTABLE ALONE I table = 0.009 xxx xxx xxx Object #1 = Wooden Disk I total_1 = 0.017 0.08 0.007 12.5 Object #2 = Flagstone I total_2 = 0.11 0.101 0.099 1.98 ANALYSIS 1. Include the three velocity-time graphs with slopes labeled with Logger Pro. Each graph should have four plots corresponding to hanging masses 50g, 90g, 130g, 170g. 2. Include your τ s tring -vs - α plots in your report. Display the best fit lines and equations on the graphs. 3. Discuss possible causes fr the % difference seen.

c. Flagstone:

2. a. Table only b.

1. Assuming that the string does not slip, algebraically determine the equation needed to calculate the angular acceleration ( α ) of the turntable in terms of the radius of the turntable, r , and a . (Here, the falling mass’ acceleration, a , is a T of the turntable). 2. Draw a force diagram of the falling mass, m, and determine the algebraic equation for the tension, T, in the string in terms of m, g, and the acceleration, a, of the mass. 3. Write the equation for the torque produced on the turntable by the string, τ string , in terms of tension, T, and turntable radius, r. 4. Newton’s Second Law for angular motion is ∑ τ = I α . Our net torque consists of two torques – one from the tension in the string ( τ string ) and one from the opposing force of friction ( τ friction ). This leads to τ string - τ friction = I α . 5. Using the equation from step 4, solve for τ string . 6. Look at your equation from step 5. If we let τ string be our dependent (y) variable, and let α be our independent (x) variable, identify the shape of graph we should get from this equation. Where would we find τ friction on the graph? ● τ friction would be the y-intercept 7. Where would we find moment of inertia, I , on the graph? 8. Since I equals the inertia of everything that is spinning, the inertia of an object on the turntable is I object = I – I table .

Physics with Vernier 5 Computer / Labstream – Rotational Inertia