\fwhere Mis the mass and Rais the radius of the disk. The rotational inertia of the ring about an axis through its Of and parallel to the symmetry axis of the ring is IRoy = 42 M(RY + R/) (3) where Mi is the mass of the ring, R, and R, are the inner and outer radii of the ring. When the ring is dropped on the disc, it will usually not be exactly centered on the disc. The axis of rotation will still be the center of the disc. If the rotation axis of the ring is displaced by a distance Dfrom its Of the rotational inertia of the ring can be calculated from the parallel axis theorem and we have In = # M(R/ + R)) + MD' (4) Where, ( see figure 3): D = ((X, -XD)|2 (5) The final rotational inertia will be the sum of the rotational inertia of the disk plus the ring. The rotational kinetic energy of a rotating object is given by KE = Wlo' (6) Ring & Disk Axis of Axis of Rotation Rotation 45 cm Rod Thick Ring Disk Figure 2: Rotational Axis for Ring and Disk Figure 1: Conservation of Angular MomentumR1 Ra X2. Figure 3: Ring, Disc, and possible position of ring on disc after being dropped. 174 EQUIPMENT SETUP 1 Use the large rod base and the 45cm rod to support the Rotary Motion Sensor as shown in Figure 1. Plug the sensor into the interface 2 Measure the mass and radii for the disk and ring. 3 Attach the disk to the clear three-step pulley on the Rotary Motion Sensor using the thumb-screw 4 Place alevel on the disk and level the system using the adjustable feet on the bass . 5 In PASCO Capstone, set the sample rate for the Rotary Motion Sensor to 20 HE. Create agraph of Angular Velocity vs. Time. 175 PROCEDURE 1 Hold the ring with the pins up, so the ring is centered on the disk and 2 to 3 mm above it. Dropping from too high causes a large vertical force on the bearing which produces a spike in the frictional drag and results in a torque which decreases the angular momentum. However, it is also critical that your fingers are clear of the ring when it strikes the spinning disk 2 Spin the disk to give it a positive speed of about 30-40 radisec. Start collecting data and after about two seconds, drop the ring onto the spinning disk. Continue to collect data for a few seconds more. Fig 4 shows a screen shot of angular momentum versus time from a few seconds before to a few seconds after dropping the ring. 3 It is difficult to end up with the ring centered on the disk . Let the disc stop rotating. Me asure the minimum distance between the ring and the edge of the disk . Measure the maximum distance directly on the opposite side. The distance D that the ring is off-center is half of the difference between these two measurements 4 Use the Coordinates tool to measure the rotational speed of the last data point just before the collision, and the first data point just after the collision. These are the initial and final angular velocities, w; and of\fDISC. Mass My Radius Ra : Moment of Inertia (e gn. 2) Is : RING: Mass Me Inner Radius Ry Outer Radius Ry :_ Moment of Inertia about its axis (eqn. 3) Incar : Run 1 Run 2 Run 3 Angular speed just before dropping Ring Angular speed just after dropping Ring Minimum distance from Ring to edge of dis Maximum distance from ring to edge of dis: Ring Offeet D Moment of inertiaof the offeet ring IR Final Moment of Inertia Angular Momentum before dropping ring Angular Momentum after dropping ring Percent error in loss or gan of angular momentum Kinetic Energy before dropping king Kinetic Energy After dropping ring Percent loss or gan in kinetic Energy17.11 SAMPLE DATA DISC. Mass My : 120.16g Radius Ra : 4.662cm Moment of Inertia (eqn. 2) Is : 1305 9.em RING; Mass ME. 468.6 6 Inner Radius R1: 2.58 cm Outer Radius Ry. 3.67 cm Moment of Inertia about its axis legn. 3 Ism : 4715 g.em? Run 1 Run 2 Run 3 Angular speed just before 23.614 dropping Ring [rad/s) Angular speed just after 4.760 dropping Fing [rad/s) Minimum distance from Ring 0.5 to edge of disc (o ) Maximum distance from ring 1.3 to edge of disc (o ) Ring Offset [om ) D 0.4 Moment of inertia of the 4790 offset ring (g.cm Find Moment of hatia If 609 6 Angular Momentum before 30840 dropping ring (g.com /$) Angular Momentum after If my 29020 dropping ring (g.em?/$) Percent arorinloss or gain 5.9% of angular momentum Kinetic Energy before 3641 00 dropping Ring (g. om/:) Kinetic Energy After 69060 dropping ring (g.cm /$) Percent loss or gain in kinetic EnergySAMPLE CALCULATIONS Initial moment of inertia Case 1 Moment of inertia of the offset ring Final moment of inertia Angular Momentum before dropping ring Angular Momentum after dropping ring Percent error in loss of angular momentum Kinetic Energy before dropping Ring Kinetic Energy after dropping ring Percent loss in kinetic energy 7.350 8. 4.760 radis 25 3.0 35 4.5 50 5.5 60 65 70 75 4.5 190 195 120