Measuring the Moment of Inertia of a Flywheel
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Measuring the Moment of Inertia of a Flywheel
Objective ========= Measure the angular velocity of a flywheel and use conservation of energy to calculate its moment of inertia. Apparatus ========= Flywheel String Slotted mass on hanger Stopwatch Vernier caliper Metre ruler Theory ====== The rotational kinetic energy can be defined by the equation K=1/2 I Ï‰2. Where I is the moment of inertia of the body about the axis of rotation. In this experiment, the flywheel rotates freely about a horizontal axis. The radius of the axle of the flywheel can be measured with a caliper. As m falls, its gravitational potential energy is transferred into translational kinetic energy of m, rotational kinetic energy of the flywheel and work done by friction. As the flywheel completes N further turns, its original rotational kinetic energy is transferred into friction loss. Assume the flywheel decelerates uniformly. Thus, the moment of inertia of the flywheel can be determined. Procedure ========= 1. The flywheel was set as shown with the axle of the flywheel horizontal. A polystyrene tile was placed on the floor to avoid the impact of the mass on the floor. 2. The vernier caliper was used to measure the diameter d of the axle. The mean of two perpendicular measurements was taken. 3. The hanger with appropriate amount of slotted mass was put on the tile. Use the balance to measure the total mass m. 4. Sufficient length of string was attached to the hanger so that the free end wraps once round the axle of the flywheel. 5. The mass was winded up to an appropriate height. 6. Verified that the string fell off the axle when the mass hit the ground. A label was put on the curved surface of the flywheel. The mass was winded up again. 7. The height h of the mass was measured. The height h was recorded. The number of revolutions n1 that the flywheel made was calculated as the mass was wound up. 8. The mass was released and at the same time the stopwatch was started. 9. As soon as the mass hit the ground, timing was stopped and the number of revolution n2 that the flywheel performed was counted before it came to rest. 10. The mass was winded up again and steps 7 to 9 were repeated for at least 3 times. The mean values of the falling time t and n2 were obtained. Precautions =========== 1. The mass and the height from which the mass falls should be chosen so that the falling time is long enough for measurement to be taken accurately. The mass and the height should not be changed throughout the experiment once they have been chosen. 2. The first few turns of the string should overlap the others. 3. The mass should be wound up to the same height in all trials. 4. When using the stopwatch, the hand should be held straightly to minimum the reaction time error. 5. Do not stand too close to the polystyrene tile when releasing the mass. 6. When choosing the appropriate amount of the slotted mass, a smaller amount (e.g. 1 slotted mass) should be chose to try first. 7. The later turns of the string should not overlap the others. Result and calculation ====================== Mass m = 0.2 kg Height h = 0.80 m Axle diameter d = 0.0522 m â†’ radius r = 0.0261 m (1) (2) (3) (4) (5) (6) (7) (8) mean Time t/s 1.75 1.85 1.84 1.84 1.81 1.78 1.82 1.78 1.81 No. of revs n1 5 5 5 5 5 5 5 5 5 No. of revs n2 25 27 23 26 26 24 25 23 25 Suppose v is the final velocity of the mass when it reaches the floor and Ï‰ is the angular velocity of the flywheel at this instant. Then v = 2 Ã— average velocity during fall = 2h / t = 2 Ã— 0.8 Ã· 1.81 = 0.88 m s1 Ï‰ = v / r = 0.88 Ã· 0.0261 = 33.9 rad s1 From the conservation of energy, Decrease in Increase in Work done gravitational potential = kinetic energy + against energy of falling mass of mass and flywheel friction i.e. mgh = 1/2 mv2 + 1/2 IÏ‰2 + n1W where W is the work done against friction per revolution. Since the kinetic energy acquired by the flywheel ( 1/2 IÏ‰2) is dissipated in n2 revolutions, n2W = 1/2 IÏ‰2 W = IÏ‰2 / 2n2 Substituting, mgh = 1/2 mv2 + 1/2 IÏ‰2 + n1W mgh = 1/2 mv2 + 1/2 IÏ‰2 + n1IÏ‰2 / 2n2 I = mr2 ( n2 / n1 + n2 ) ( gt2 / 2h Â– 1) =0.2 Ã— (0.0261)2 ( 25 / (25 + 5) ) ( 10 Ã— (1.81)2 / 2 Ã— 0.8 Â– 1 ) =2.21 Ã— 103 kg m2 Conclusion ========== The moment of inertia of the flywheel is measured and found to be 2.21 Ã— 103 kg m2. Discussion ========== Errors and improvement 1. The reaction times error. This can be improved by straighten the hand when taking the time. 2. The number of revolution n2 that the flywheel performed cannot be accurately obtained. This can be improved by counting the number of revolution by two students instant of one and to repeat the experiment more times. 3. Unsteady hands. When the hand released the mass, force may be push to the mass. To improve this, student should release the mass slowly and softly. Reference ========= Physic Beyond 2000 How to Cite this Page
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