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Rotational Inertia Experiment Physics 4A – Chabot College Scott Hildreth Part A: Measuring Rotational Inertia and Using the Parallel Axis Theorem. Using the rotating support and some accessory masses, we’ll investigate the rotational inertia of selected geometric shapes, and explore the relationship between torque, angular acceleration, and linear acceleration. 1. The rotational inertia experiment uses a drum wrapped with string, which will be pulled by a descending or “driving” mass to create a torque that turns the drum. Measure the diameter of the drum of the rotating support and record (along with your uncertainty!) You’ll need this measurement to determine the radius of the drum (R* in the picture), and with that, the torque that rotates the apparatus. 2. Connect the computer to the Labpro, and connect the photogate and smart pulley to the Lapro DIG/SONIC 1 port. Open up the “Atwoods Machine” file in “Physics with Computers”; note that a graph of velocity vs. time will be displayed. The velocity here will be the *linear* velocity of the descending drive mass, m, as it pulls the string around the pulley, and off of the rotating drum. 3. Determine the masses of the accessories: (a) flat disk; and (b) the two small solid cylinders. Record these with your uncertainties. Note that they may be larger in mass than the scale’s capability to measure, so you’ll have to use additional masses on the opposite scale pan to achieve a balance. How will this process affect your uncertainty in measurement of the masses? How many significant figures are reasonable? What are the percentage uncertainties in your measurements? 4. Using the pulley arrangement, add a small amount of mass to the descending side so that the drum begins to turn at a constant rate. You might assess whether the drum is turning by looking at the descending mass – it should move at constant speed linearly if the drum support is rotating at constant angular speed. Record this mass mo– it establishes the frictional resistance of the drum and pulley, and will have to be subtracted from other measured descending masses in the experiment to determine actual torques. 5. Attach a 50-g weight hanger to the string, and add the frictional mass mo to the hanger. The descending mass should now be m = (50+ mo) grams, and you can let it go after clicking the COLLECT button on the Computer to begin data collection. Once the drive mass hits the ground, stop the collection, and identify the portion on the velocity vs. time graph where the velocity was constantly and smoothly increasing. Analyze that portion, and determine the slope –the linear acceleration. 6. From the linear acceleration, find the rotational acceleration of the system f = a/R* Note that you use the drum radius here – not the radius of the plate or pulley. The string is providing a torque around the drum, and it is that radius we need to use to determine how fast the system is increasing its rotational velocity. 7. Calculate the experimental rotational inertia of the rotating support, based on the known values of r (drum), the descending mass, and the angular acceleration. Since the support will be used in the rest of the experiments, you’ll have to subtract this value from the measured rotational inertias you get for the other accessories. Since the descending mass is pulled downwards by gravity, and upwards by the tension T in the cable, its equation for motion (in the vertical “y” direction, with DOWN being positive) is: mg - T = ma (note the signs of these terms! Do they make sense??) Therefore, T= mg-ma = m (g-a) The tension on the cable pulls on the drum, creating an applied torque that turns the drum. That applied torque is equal to the force time the distance from the axle (and since the cable pulls at right angles away from the drum, the angle involved between the force and radius vector is 90 degrees, and the lever arm is equal to the radius): a = lever arm x Force = R* T The friction in the rotating support system creates another torque, f , directed backwards (retarding the drum’s acceleration). If you measured the mass mo correctly, and the descending mass was moving at a constant rate, it wasn’t accelerating, and Tfriction = mog so: f = R* mog But torques produce angular accelerations, so the NET torque results in the angular acceleration of the rotating support. and net = Ia - f I (experimental) = net/= (a - f )/ = [ (R* T) – (R* mo g) ] / = ( ) ) Here we know = a/R* from the measurements above, so the value for I can be decomposed into known and measured quantities – the mass of the descending hanger, the frictional mass, the radius of the drum, and the acceleration of the descending mass. As usual with an algebraic derivation, you should check to see if the units match. Rotational inertia should have units of mass x distance2 (kg m2). The right side of this equation has units of Distance x Force/ angular acceleration = meters x Netwons / rad/sec2 = m (kg m/sec2) sec2 = kg m2 (Correct) 8. Experiment (a) Now place the flat disk accessory on the rotating support, add more mass to the hanger so that it descends with constant speed (record this value as moa ). (Question! Why would this be more than the previous value of mo from the support alone?) Now add extra mass so that the hanger descends slowly to the floor. Record the distance fallen and time taken for (3) trials. 9. Experiment (b): Repeat with the disk plus two solid cylinders. For this experiment, place the two solid cylinders on top of the disk at the edges, but not too close so that they hit the pulleys or supports. Measure the distance “d” between the center of the support wheel and the center of the cylinders, needed for the parallel axis theorem! Again, first determine the small mass mob needed to make the system descend at a constant rate. 10. Calculate the measured rotational inertia of the support plus disk, and support plus disk + cylinders, and then subtract the rotational inertia of the support alone found earlier. Record the experimental inertia of the accessories alone. 11. Calculate their theoretical inertia using the masses and radii, according to the formulas in the textbook for similar geometric solids rotating about central axes. Compare and determine your percentage errors. Account for the difference(s) between your results and the theoretical values with a brief paragraph of analysis. For the experiment using two cylinders on top of the rotating disk, the rotational inertia will be the sum of the inertia of the disk they rest on, the rotation inertia of each cylinder, AND their extra inertia because they are rotating a distance “d” from the axle (= Md2): I (disk + cylinders) = I disk + 2 (I cylinder) + 2(Mcyl d2) You can read more about this use of the “Parallel Axis Theorem” in the textbook. Part B: Estimating Rotational Inertia You’ll use a similar apparatus to investigate the rotational inertia of a vertical flywheel. Your goal here will be to first estimate the rotational inertia of the flywheel geometrically, publish your estimate by providing that with your team members’ names submitted to the professor, and then to the experiment, comparing that estimate to the experimental value you find. The team with the closest published match between their estimate and the actual measured results from the experiment for their flywheel will not have to submit a lab report for this lab and will receive full marks for the lab report itself, IF you can complete the experiment and analysis within the class period. That team need only submit their data tables from the experiment; their completeness and clarity will be factored into the overall score for the experiment. All other teams will be expected to submit a complete write-up, including graphs and data analysis along with the data tables, in one week. You should create your own data table for this Part of the Lab – clearly labeling all measured quantities. The clarity and completeness of your data will be factored into your overall team lab score if you are not fortunate to have been the team with the closest match between experiment and theory. B1. First estimate the theoretical rotational inertia of the flywheel by measuring (and recording, with uncertainties) all appropriate variables. Be sure to record the descending mass m, which will be supplied by the professor. Draw and clearly label two free-body diagrams of the problem – including one for the descending mass (pulled down by gravity, and up by the tension in the paper tape), and one for the flywheel (rotating from the torque created by the tape’s tension pulling down on the edge of the wheel). Just as with the earlier Atwood’s machine problems with two masses hung from a pulley, or the two-mass system with one sliding up a slope while connected by a string over a pulley to another mass falling down, there are TWO variables that are in common with both the rotating flywheel and the descending mass. What are they? What two equations allow you to solve for for I? From the value of the linear acceleration of the descending mass, establish the rotational acceleration of the wheel, from that, the inertia of the flywheel. Determine the % uncertainty in your estimate from the uncertainties in the measured quantities. Make a clear sketch of the flywheel, with its dimensions, and clearly show how you arrived at the estimated value of its rotational inertia. Note that there are two flywheels – BLUE and GREEN. Be sure you identify which one you are using to measure and analyze, because you’ll have to use the same flywheel for the experiment later. Be careful with significant figures here! B2. Submit your team’s estimate for I(flywheel), with uncertainty, including the color of the wheel (BLUE or GREEN), and the names of your team members to me: Names: Date: Flywheel Color: Theoretical I = _____________ +/- ______________ (what units??) B3. Now do the experiment! To determine the experimental value of I, you’ll again use a measured descending mass “m” pulling on a paper tape that unrolls from the flywheel, and a spark timer (set to create a spark every 1/10th of a second) and thermal strip paper record which will give you dots on the paper as a record if its motion. Establish the acceleration of the descending mass by measuring how far the paper had moved in time, and fitting that data to a spreadsheet program with a quadratic equation to determine the linear acceleration a of the descending mass. (If possible, we’ll try to use the same Vernier apparatus to use the computer and logger pro software to collect this data as well. But you need to be able to use the supplied tape.) B4. Determine the percentage difference between your theoretical estimate and your experimental value. B5. Defend your results – if your experimental value was different from the theory, what happened to account for the difference? How close are the two values given the uncertainties you have in your measurements? Data Table for Part A Diameter of Support Drum: +/- Radius of Support Drum (R*): +/- Mass of Accessories: a) disk +/- c) individual cylinder Dimensions of Accessories: a) disk radius b) cylinder radius (Don’t forget units!) +/- +/+/- Distance from center of support to the center of the cylinders Rotating Support Alone mo (frictional mass) a) Support with Disk moa = Descending mass m Linear Acceleration From Vernier LoggerPro Graph Angular Acceleration Calculated I of system Net I (Measured) of accessories alone Subtract I support from the calculated values in a & b Theoretical I of Disk, or Disk + Cylinders Percentage Difference d = +/- b) Support with two cylinders mob =