Moment of Inertia, and conservation of energy - off to the races!!! We will work together to do the calculations to determine which of these items will win a race as they roll downhill together. Our objects all start from the same height (h = meters). After we do the math, we'll start the race! Mass of small metal cylindrical ring 91.6 g outer diameter 5.1 cm Inner diameter 3.8 cm Mass of black small solid cylinder 109 g diameter 5.1 cm Mass of large metal cylindrical ring 219 g outer diameter 10.0 cm Inner diameter 8.9 cm Mass of large black solid cylinder 525 g diameter 10.0 cm Mass of solid white ball 120 g diameter 5.1 cm Axis Axis Annular cylinder R Hoop about cylinder axis or ring) about cylinder axis 1 = MR2 1 = (RI+ RI) Axis Axis Solid cylinder Solid cylinder (or disk) about (or disk) about cylinder axis IRL central diameter 1 = MR 1 = MRS . MIZ Axis Axis Thin rod about Thin rod about axis through axis through one center _ to end J. to length length 1 = MI 1 = MP Axis Axis Solid sphere Thin 2R about any 2p spherical shell diameter about any diameter 1 = 2MR? 1 = 2MR Axis Axis Slab about Hoop about L axis through any diameter center 1 = MR2 (= M(a2 + 62)Moment of Inertia, and conservation of energy - off to the races! 1! Now do the math, finding the moment of inertia for ea ch, and then conserving energy for each one {calculate initial gravitational potential energy, and nal rotational kinetic energy + nal translational kinetic energy]. Use conservation of energy to nd the speed y for each item, then figure out how much time it will take for each to reach the bottom. Each person at the table should calculate for a different object, and print your name next to your mathematical prediction. You may assume no slipping, and you may assume that each object rolls without slipping. Small metal cylindrical ring Calculate the moment of inertia Calculate the initial energy (gravitational potential energy} Conserving energy, what should the total final energy be? The final energy will not have gravitational potential energy, but instead will have kinetic energy [sum of translational and rotational kinetic energies}. Using this sum, and relationships between translational and rotational velocity, solve for the translational speed v at the bottom of the ramp. Moment of Inertia, and conservation of energy - off to the races! 1! Now do the math, finding the moment of inertia for ea ch, and then conserving energy for each one {calculate initial gravitational potential energy, and nal rotational kinetic energy + nal translational kinetic energy]. Use conservation of energy to nd the speed v for each item, then figure out how much time it will take for each to reach the bottom. Each person at the table should calculate for a different object, and print your name next to your mathematical prediction. You may assume no slipping, and you may assume that each object rolls without slipping. black small solid cylinder Calculate the moment of inertia Calculate the initial energy (gravitational potential energy} Conserving energy, what should the total final energy be? The final energy will not have gravitational potential energy, but instead will have kinetic energy {sum of translational and rotational kinetic energies}. Using this sum, and relationships between translational and rotational velocity, solve for the translational speed v at the bottom of the ramp. Moment of Inertia, and conservation of energy - off to the races! 1! Now do the math, finding the moment of inertia for ea ch, and then conserving energy for each one {calculate initial gravitational potential energy, and nal rotational kinetic energy + nal translational kinetic energy]. Use conservation of energy to nd the speed v for each item, then gure out how much time it will take for each to reach the bottom. Each person at the table should calculate for a different object, and print your name next to your mathematical prediction. You may assume no slipping, and you may assume that each object rolls without slipping. large metal cylindrical ring Calculate the moment of inertia Calculate the initial energy (gravitational potential energy} Conserving energy, what should the total nal energy be? The final energy will not have gravitational potential energy, but instead will have kinetic energy [sum of translational and rotational kinetic energies}. Using this sum, and relationships between translational and rotational velocity, solve for the translational speed v at the bottom of the ramp. Moment of Inertia, and conservation of energy - off to the races! 1! Now do the math, finding the moment of inertia for ea ch, and then conserving energy for each one [calculate initial gravitational potential energy, and nal rotational kinetic energy + nal translational kinetic energy]. Use conservation of energy to nd the speed v for each item, then figure out how much time it will take for each to reach the bottom. Each person at the table should calculate for a different object, and print your name next to your mathematical prediction. You may assume no slipping, and you may assume that each object rolls without slipping. large black solid cylinder Calculate the moment of inertia Calculate the initial energy (gravitational potential energy} Conserving energy, what should the total nal energy be? The final energy will not have gravitational potential energy, but instead will have kinetic energy [sum of translational and rotational kinetic energies}. Using this sum, and relationships between translational and rotational velocity, solve for the translational speed v at the bottom of the ramp. Moment of Inertia, and conservation of energy - off to the races! 1! Now do the math, finding the moment of inertia for ea ch, and then conserving energy for each one [calculate initial gravitational potential energy, and nal rotational kinetic energy + nal translational kinetic energy]. Use conservation of energy to nd the speed v for each item, then figure out how much time it will take for each to reach the bottom. Each person at the table should calculate for a different object, and print your name next to your mathematical prediction. You may assume no slipping, and you may assume that each object rolls without slipping. solid white ball Calculate the moment of inertia Calculate the initial energy (gravitational potential energy} Conserving energy, what should the total final energy be? The final energy will not have gravitational potential energy, but instead will have kinetic energy [sum of translational and rotational kinetic energies}. Using this sum, and relationships between translational and rotational velocity, solve for the translational speed v at the bottom of the ramp. Moment of Inertia, and conservation of energy - off to the races! 1! Record your calculated values, and your predicted order, in the table below. Moment of inertia Initial Final energy Final speed v Order of nish energy (sum) Small metal cylindrical ring Black small solid cylinder Large metal cylindrical ring Large black solid cylinder Solid white ball