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Download Moment of Inertia - Ryerson Department of Physics
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D4-1 MOMENT OF INERTIA OBJECTIVE The purpose of this experiment is to verify Newton's Second Law for rotational motion; Γ = Iα, and to measure the moment of inertia of a disc and axle. METHOD A heavy flywheel is set in rotation by a mass attached to a string wrapped around the axle of the flywheel. The force exerted by the falling mass is related to the torque, Γ, and the rate of change of angular velocity of the wheel, that is, the angular acceleration, α. The moment of inertia, I, is the constant of proportionality between these two variables and depends on the mass and effective radius of the rotating object. The above law can be verified by using various masses and measuring the resulting acceleration for each as a function of the net accelerating torque. THEORY The magnitude of the torque is determined as the product of the applied force and its moment arm. In this case the applied force is the tension, T, in the string and the moment arm is the radius of the axle, r. In order to find the tension consider the forces on the mass m in the diagram. The net force (down) on m is F = W - T. Where W is the weight (mg) of the mass, m. Using the letter a to represent the downward acceleration of this mass we get the following equation: F = mg - T = ma When solved for T we get: T = mg - ma The net torque acting on the disc and axle is thus given by: Γ = mr(g - a) (1) It is clear that, in addition to other things, we must also measure the acceleration, a, in order to determine the torque. It is actually easier to measure α, the angular acceleration, directly and determine the linear acceleration using: a=αr (2) D4-2 The angular acceleration can be calculated by determining the time, t, required for the disc to rotate through a given angular displacement, starting from rest. For the experimental set up we have here it will be convenient to allow the disc to accelerate through exactly eight (8) revolutions. The average angular velocity, ωAV, for the eight turns (in radians per second) will then be given by: θ 8(2π ) 16 π = ω AV = = t t t Since the angular acceleration is constant, the final angular velocity, ωf, will be twice the average, that is, ω f = 2 ω AV . The angular acceleration will then be given by: α= _ω ω f 2 ω AV 32π = = = 2 t t t t (3) The moment of inertia for a solid cylinder rotating about its axis is given by: I = (½) MR2 (4) where R and M are respectively the radius and mass of the cylinder. When calculating the total moment of inertia remember the object in rotation consists of the disc and the axle each of which is a cylinder. The moments of each part must be calculated separately and added, do not add the masses. (Be careful to count the part of the axle inside the disc only once.) PROCEDURE 1. Measure all dimensions of the disc and axle (Note that the diameter of the disc is stamped on its perimeter to 5 figures in mm). Take all other pertinent measurements so that you can determine the moment of inertia of the disc and of the axle. 2. Wind the string around the axle eight turns, such that, when the wheel has turned exactly eight turns unwinding it the string will fall off. Make sure the string hangs freely - isn’t touching wood. Suspend a reasonable amount of mass, m, on the string and measure the time it takes after releasing the mass from rest until the string falls off (8 turns). Make two trials and average. Record data for both trials in your data table. 3. Repeat step 2 for five different masses. 4. For one of the masses used, measure the time taken from the moment the string falls off until the disc comes to rest. This may take several minutes. 5. Use equation (3) to determine the angular acceleration corresponding to each value of m used. Use equation (2) to determine the linear acceleration for each value of m. Use D4-3 equation (1) to determine the torque applied to the disc and axle for each value of m. Tabulate the results of all these calculations. 6. Plot a graph of torque against corresponding values of angular acceleration. Determine the slope of a best straight line representing these plotted points. Note that Lotus 1-2-3 may be useful for this part. What is the significance of this slope? 7. Calculate the total moment of inertia of the disc and axle using equation (4) and your measurements from procedure 1 above. In order to do this you will need to know the density of the flywheel which is 7.87×103kg/m3 (the density of iron). Compare this value to that obtained from the slope of the graph. 8. Extrapolate the straight line of the graph to intersect the torque axis. The point of intersection should correspond to a positive torque and of course a corresponding zero angular acceleration. This non-zero torque can be interpreted as the torque required to overcome the friction in the system. Determine this frictional torque from your graph. 9. In step 4 of the procedure it was the frictional torque that caused the disc to slow down and finally to stop. Use the equation Γ = Iα to determine the frictional torque. Use the calculated value for I (step 7). The angular deceleration, α, can be determined by first calculating the angular velocity at the instant the string fell off the axle using ω f = 2 ω AV . Then use α = Δω/t to determine the angular deceleration. Compare the frictional torque found here to that found in step 8. PRELAB QUESTIONS D4-4 1. 2. 3. 4. [1] Define moment of inertia? [1] What is the mathematical formula for the moment of inertia of a solid cylinder? [1] What causes the wheel to rotate? Explain. [1] Does the mass suspended at the end of the string fall with an acceleration equal to the acceleration due to gravity? Explain. 5. [1] How do you relate the angular acceleration of the wheel with the linear acceleration of the falling mass? REPORT WRITING INSTRUCTIONS Read the procedure on previous two pages and prepare a tables to help you record all quantities needed for this experiment. I. Procedure: 1. Take the pertinent measurements to determine the moment of inertia of the disc-axle directly from its dimensions (cf. figure below) Schematic diagram of the disc and axle 2. Determine the moment of inertia from the data recorded from the motion of the wheel. 3. Determine the torque due to friction in the bearings. II. Conclusions: 1. How do the two values obtained for the moment of inertia compare? Give the percentage deviation. Which value do you think is more reliable? Justify your answer. 2. Compare the two values obtained for the frictional torque. Give the percentage deviation. Which do you consider more reliable? Justify your answer.