# Moment of Inertia & Rotational Energy Physics Lab IX Objective

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1 Moment o Inertia & Rotational Energy Physics Lab IX Objective In this lab, the physical nature o the moment o inertia and the conservation law o mechanical energy involving rotational motion will be examined and tested experimentally. Equipment List The rotary motion kit (shown on the right); a LabPro unit; a smart pulley (SP); string; a mass holder and a set o masses; a plastic ruler and a meter stick; a digital caliper; ACCULAB VI-100 mass scale. Theoretical Background This lab extends the exploration o the Newtonian Mechanics to rotational motion. Moment o Inertia Analogous to the mass in translational motion, the moment o inertia, I, describes how diicult it is to change an object s rotational motion; speciically speaking, the angular velocity. I is deined as the ratio o the torque (τ ) to the angular acceleration (α ) and appears in Newton s second law o motion or rotational motion as ollows: τ = Iα For objects with simple geometrical shapes, it is possible to calculate their moments o inertia with the assistance o calculus. The table below summarizes the equations or computing I o objects o some common geometrical shapes. solid disk or cylinder 1 MR Thick ring with inner and outer radius R in and R out 1 + R in ) M(R out thin rod rotating about the center 1 1 ML thin rod rotating about one end 1 3 ML thin loop or point mass MR

2 Experimental Determination o the Moment o Inertia Fig. 1 shows a schematic o the experimental setup that you will use to experimentally determine the moment o inertia o the spinning platter. Considering the rotational part o the system (taking a disk as an example) and ignoring the rictional torque rom the axle, we have the ollowing equation rom Newton s second law o motion. τ = rt = Idiskα, (1) where I is the moment o inertia o the disk, r is the radius o the multi-step pulley on the rotary motion sensor and T is the tension on the string. Considering the hanging mass (m H ), the analysis rom the ree-body-diagram tells us that mh g T = mh a = mh ( rα) () Combining Eq.(1) and (), we have rm H = Idiskα, (3) rom which I disk can be determined as rmh Idisk = (4) α Fig. 1 Schematic o the system o the spinning disk and dropping weight. Spinning objects o dierent shapes can also be determined experimentally in the same way. Conservation o Mechanical Energy in Rotational Systems In an earlier lab, we have considered the mechanical energy in terms o the potential and kinetic energy in the linear kinematics. As noted beore, kinetic energy is the energy expressed through the motions o objects. Thereore, it is not surprising to recognize that a rotational system also has rotational kinetic energy associated with it. It is expressed in an analogous orm as the linear kinetic energy as ollows: 1 1 K = mv K Iω (5) trans rotate = ω is the angular speed in the unit o radian. It describes how ast an object spins. Now the conservation o mechanical energy can be generalized to the rotational systems as: I there are only conservative orces acting on the system, the total mechanical energy is conserved. E i = E, E = K + P Ki + Pi = K + P, K = Ktrans + K Using the terms in this experiment, rotate

3 m Hvi + Idiskω i + mh = mhv + Idiskω + mh gh (6) In this equation, m H is the mass o the hanging weight, and v its speed. I disk is the moment o inertia o the disk, and ω is the angular speed. h is the height o the hanging weight measured rom the ground. In the experiment, the hanging weight and the disk are released rom rest, and we measure the inal speeds as the hanging weight reaches the loor. So Equation 6 becomes 1 1 mh = mhv + Idiskω (7) It is also noticed that the linear motion o the hanging weight is related to the spinning rate o the disk through the equation, v = rω, where r is the radius o the multi-step pulley around which the string wraps. Using this equation, Equation 7 becomes 1 1 mh = mhr ω + Idiskω. (8) In this equation, I disk is the moment o inertia o the disk, and r is the radius o the multi-step pulley. Solving this equation or ω, mh ω, theo =. (9) I + m r disk H You will use this equation to calculate the theoretical values o the inal angular speeds. Quantities in Translational Motions Analogous Quantities in Rotational Motions M (mass) I (moment o inertia) v (velocity) ω (angular velocity) p = mv (linear momentum) L = Iω (angular momentum) 1 1 mv (linear kinetic energy) I ω (rotational kinetic energy) Experimental Procedure Setup and Use the LoggerPro program 1. Click the icon "Rotational Motion.cmbl" on the desktop. This will automatically load the LoggerPro program (Fig. ) with most o the coniguration done or you.. Following the steps (rom the tool bar at the top o the LoggerPro3 window) to connect to the LabPro interace or data collections. a. Experiment Connect Interace LabPro COM1 b. Experiment Set up Sensors Show All Interaces (Now you should see a pop-up windows or all the ports on LabPro.) c. For the "DIG/SONIC1" port, select "Rotary Motion". d. Now the gray arrow should turn green ( )and you are ready to take data. 3. To take data, click (collect). The program will take data or 10 second (deault). I your experiment needs longer time, click to change the setting or data collection.

4 You should see both the angular position and angular speed as unctions o time displayed in real-time in the graphs. I the data is not to your satisaction, simply repeat the experiment and take another data. 4. To save the data, ollow "File Export As CSV" to export your current data in "*.csv" ormat. Wisely choose the ilename to label the data. Since your data will be overwritten by new data, you should export the data BEFORE running next experiment. Fig. The screenshot o the LoggerPro3 interace or recording and analyzing the rotational motion data. Collect Data In the experiment, a hanging mass is attached to a string pulling the 3-step pulley (3SP) on the rotary motion sensor. The hanging mass will be released rom rest at a measured height. As the hanging mass alls, it pulls the string to spin the disk and causes the angular speed o the disk to increase. The angular motion o the disk is recorded by LabPro and the LoggerPro3 program. 1. Measure and record the ollowing quantities o the Aluminum disk: mass (M); diameter (D); calculated theoretical moment o inertia I theo = 1 MR = 1 8 MD.. Set up the rotary motion sensor and the disk as shown in Fig The direction and tilt angle o the SP should be adjusted so that the SP is aligned with the direction o the string and the string pulls the 3SP horizontally. 4. Attach the mass hanger to the string on the 3SP. The string should wrap around the largest "step". (You may choose the "middle" step i more appropriate or your setup. However, you need to use the corresponding radius or your calculations then.) The length o string must be long enough so that the hanging mass hits the loor beore the string is completely unwrapped. 5. Put a 10 g weight on the hanger and measure the total hanging mass, m H.

5 6. Rotate the disk until the hanging mass reaches a height o h i = 75 cm. Hold the disk at rest. 7. Click, then release the disk. You should see data plotting as the mass drops. The data must record the motion until ~1 second ater the hanging mass hits the loor. 8. Export the data in "*.csv" ormat. Find the angular acceleration (α) by inding the slope o the linearly rising portion o the ω(t) plot and the inal angular speed, ω,exp, using LoggerPro3 (discussed in "Data Analysis" below). Estimate the uncertainties o α and ω,exp. 9. Repeat step 5-8 with dierent weights (15g, 0g, 30g, and 40g) released rom the same initial height (h i = 75 cm). Remember to record the total hanging mass, m H. 10. (optional) Remove the disk rom the rotary motion sensor, and mount the hallow aluminum rod on the pulley. Attach the masses to the rod with the locking screws. Find the moment o inertial o the "rod+masses" system. Change the positions o the masses (moving them closer or arther rom the axis), and ind how the moment o inertia changes. 1. Use the swivel mount to attach the rotary motion sensor to a stainless steel rod. 3. Attach the hub and Al disk to the 3-step pulley using the spindle screw, as shown in the picture.. Insert the 3-step pulley to the axle o the rotary sensor. 4. Remove the rubber screw cap on the swivel mount. 5. Attach the smart pulley to the swivel mount. Loosen the thumb screw to unlock the swivel mount and adjust the position o the smart pulley. Make sure the pulley is at the proper height and aligned with the direction o the string. Tighten the screw when inished. Fig. 3 Procedure or setting up the rotational motion experiment. Data Analysis Determine the Moment o Inertia Perorm the ollowing analysis to determine the moment o inertia o the platter. 1. Ater taking data or each run, click the "Velocity" graph (this is the ω(t) graph) to select the graph, then click. A linear it over the whole data will appear with a text box containing all the itting parameters. Drag the brackets at the ends o the graph to

6 determine the speciic range o data that you want to it. Find the slope o the rising part in the "Velocity" plot or each run. The slope is the angular acceleration, α, produced by the hanging weight.. Calculate the value, rm H or each hanging mass, m H. 3. Calculate the I exp1 with Equation (4) or each hanging mass. 4. Plot rm H as a unction o α,(" rm H " in y-axis and "α"in x-axis). and determine the slope o the graph. According to Equation (3), the slope should also be the moment o inertia o the disk (I exp ). Use Excel to ind the uncertainty in the slope, δ(i exp ). (Ask the instructor or help i needed.) δ ( Iexp ) 5. Estimate the % uncertainty o I exp using 100%. I 6. Compare I exp1 and I exp with I theo. Conservation o Mechanical Energy Perorm the ollowing calculations or the same set o data. 7. Find the inal angular speed on each "Velocity" graph by clicking. A cursor will appear to allow you speciy which data point to look at. Move to the "kink" at the end o the rising section (this is when the hanging mass hits the loor). The "Velocity" (angular speed) o this point is the ω,exp. 8. Calculate the inal speed o the hanging mass rom the inal angular speed using v = rω., exp,exp 9. Use I theo as I disk. and calculate the theoretical inal angular speed, ω,theo, using Eqn. (9). 10. Calculate the % Dierence between ω,exp and ω,theo. 11. Calculate the initial potential energy o the system using, P = m i H gh i 1. Calculate the inal kinetic energy using, 1 1 K = Idiskω,exp + mhv,exp 13. Calculate the ratio K Pi. 14. Plot K (y-axis) as a unction o P i (x-axis) and determine the slope and y-intercept. exp

7 Selected Questions 1. The Equation (4) used or calculating I exp1 does not consider the internal torque generated by the riction on the axle. Would this rictional torque aect the values o I exp1? I yes, how? I no, why? Would this rictional torque aect the values o I exp obtained rom the slope o Plot 1? I yes, how? I no, why? (Hint: one o them is not aected by the presence o the riction i the riction is constant.). For the conservation o energy experiment, what eect does riction have on this conservation principle? Is this a source o random or systematic error? Why?

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