\fll'o find the rotational inertia of the ring and disk experimentally, a known torque is applie d to the - ring and disk, and the resulting angular acceleration, I1, is measured. See Fig. 3. r=Itt {3} where 1: is the torque cause d by the weight hanging from the thread wrapped around the three-step pulley of radius, r. *' r=rF {43' The tension in the string, F, is less than the weight, due to the downward acceleration. Applying Newton's \"'1- Second Law for the hanging mass, m, (see Figure 5 yields F= mfg a) (5,\": IigLJ'L': 3 Fr:_'.--r_r-F.::nd~,- F]'(:_r_,-rr.71-'n Ta? 1\" Finally, the line ar acceleration, a, is relate d to the angular acceleration, :1, by a=ra M 31 known torque is applie d to the three-step pulley on the Rotary Motion Sensor, causing a disk and ring to rotate. The re sulting angular acceleration is measured using the slope of a graph of angular velocity U31" sus time. The rotational inertia of the disk and ring combination is calculated from the torque and the angular acceleration. The procedure is repeated for the disk alone to find the rotational inertias of the ring and disk separately 16.4 EQUIPMENT SETUP 1. Use the large rod base and the 45cm ro d to support the Rotary Motion Sensor as shown in Figure 1. Plug the sensor into the interface. 2. Use calipers to measure the radius (r) of the medium size pulley on the clear three-step pulley. 3. Cut apiece ofthread ahout TEcmlong. Run the thread through the hole in the medium size pulley as shown in Figure 4. Tie alarge knot inside the pulley to keep the thread fro m pulling though. 4. Clamp the black pulley onto the Rotary Motion Sensor as shownin Figure 5. Note how the pulley is clampe d at an angle to match the tangent to the cle ar three-step pulley. It must also be adjusted vertically using the two thumhscrews, to match the height of the clear pulley keing used. 5. Connect a mass hanger to the end of the thread. Adjust the length so that it doesn't quite hit the table. 5. When you wind up the thre ad onto the three-step pulley, make sure the thre ad winds smoothly with no overlaps. Pilso, do not place too much thread on the pulley: You only want a single layer of thread, so that the radius stays constant. 'T. In PESCCI Capstone, create agraph of Pangular 'uhlocity 1.rs. Time. Set the sample rate to 20 Hz. l'nv-alup t'll ll':l a,\" Figure 5: Adjusting Angle of Clamp-on Pullev Figure 4: Attaching Threod to Pull-3',\" 165 PROCEDURE RING AND DISK l. Fasten the disk to the shaft of the Rotary Motion Sensor using the thu mbscrew. Position the ring on top using the two pins to key it to the disk. 2. Hang alzout 50 g onthe mass hanger, andlet it fall. With the mass ofthe hanger, this is the Total Hanging Mass I'u-'ITE. The pulley will start spinning. IClollect arun of angular velocityvs. time data. Record all values intable below. 3. Use alinear curve fit to find the angular azceleration (See figure Efor an example) 4. Piccounting for Friction: Put just enough mass on the thread to make it fall at a constant spee d. after you give it a starting push. It will only be about agram so you won't be able to use the mass hanger. This is called the friction mass, Mr, and is subtracte d from the Total H anging Mass, Mm to obtain the hanging mass, m, to use in further calculations: m= Mm M]. 5. Remove the ring, and repeat the above procedure for just the disk.Use about 15 g, and also find the friction mass. 5. Remove the disk and repeat for just the pulley. Use about lg, but don't bother with the friction mass, as it will be too small. 115.6 PROCEDURE PDINT MASSES We will now find the rotational inertia of point masses. This will be done with two point masses attached to the black rodfrom the Rotation Acce ssory, once with the two masses close to the ends of the rod, and again with the two masses close to the center of the rod. 1. Using a meter stick, position the brass massesonthe black rod as accurately as possible so that their centers of mass are at the same distance from the center of rotation. Inthe first case, keep the two masses as far apart as youcanonthe rod (Fig TI. r'kttazhthe black rodto the clear pulley {see Fig. '1'"). 2. Now repeat the procedure to find the Rotational Inertia as was done earlier for the disk. 3. Repeat the procedure with the two masses near the center of the rod (see Figure B). 4. The moment of inertia for a point mass is: E MR' 5. Consider how you would account for the effect of the rod, and include it's effect in your calculations. Figure 6: Plot of angular velocity versus time 6 --135 x 01014 r - 1000 10 Time 17 Jonon the heel Figure 7: Masses at Far position Figure 8: Masses at Near Position 16.7 PRECAUTIONS: 1. Make sure the pulley is horizontal. 2. Don't force the three-step pulley in the shaft. When properly aligned, it will slide in easily. 3. Don't use too much or too little hanging mass. We want an angular speed that is not too high or too small.4. The hanging mass should not be swinging like apendulum while it is going do wm. 5. Make sure that the string is going to both pulleys properly 168 IC-16 ROTATIONAL INERTIA REPORT FORM DATA FOR RING AND DISK Ring: mass ME: Inner radius, R1 Outer Radius, Re: Disk: mass Mr Radius, R: Three-step pulley: mass Radius on which thread is wrapped, r: Outer radius of 3-step pulley: Symbol, Run #1 Pulley+disk trin Run # 2 Run # 3 units Pulley+ disk Pulley Total Hanging Mass (including hanger) E Angular acceleration (from graph) radis Linear acceleration a m/ $2 (em. 6) Friction mass (incl. hanger, if use d) E Hanging mass to use (MTE - MY E Tension in string T N (e q. 5) Torque (eqn. 4) N.m Moment of inertia I kg.m MOMENTS OF INERTIA Experimental Theoretical % error I for Ring kg .m I for disk kg.m I for pulley kg.mDATA FOR ROD AND POINT MASSES Rod: mass. Length: Point masses Mass-1: Distance from center: Far: Near: Mass-2: Distance from center: Far: Near: Radius of pulley used, r: Run # 2 Symbol, Run # 1 Rod+masses Run # 3 units Rod+ masses at Ne ar Rod alone at Far position position Total Hanging Mass MTE Angular acceleration rads (from graph, Linear acceleration (eqn. 6) Friction mass Hanging mass to use Mi MTH - MF E Tension in string (e qn. T N Torque (e qn. 4) T. N.m Moment of inertia I kg.m? MOMENTS OF INERTIA Experimental Theoretical % error I for Rod kg .m= I for Masse s kg.m (large radius)