PC1221 Fundamentals of Physics I Inertia Wheel


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1 PC1221 Fundamentals of Physics I Inertia Wheel 1 Purpose Determination of the angular acceleration of the inertial wheel as a function of the applied torque Determination of the moment of inertia I of the inertia wheel from the slope of the data for applied torque versus angular acceleration Theoretical calculation of the moment of inertia of the inertia wheel Comparison of the experimental and theoretical values for the moment of inertia of the inertia wheel 2 Equipment Inertia wheel (attached with pulley and light string) Meter stick, mass holder and mass sets Stopwatch Balance 3 Theory For linear motion, Newton s second law F net = ma describes the relationship between the net applied force F net, the mass of an object and its acceleration a. The force is the cause of the acceleration and the mass is a measure of the tendency of the object to resist a change in its linear translational motion. For rotational motion of some object about a fixed axis, an equivalent description for the relationship between the net applied torque τ net, the moment of inertia and the angular acceleration α of the object is given by τ net = Iα (1) Thus, the torque is the angular acceleration α and the moment of inertia I is a measure of the tendency of the body to resist a change in its rotational motion. Page 1 of 5
2 Inertia Wheel Page 2 of 5 The moment of inertia I of a rigid body depends on the mass of the body and the way in which the mass is distributed relative to the axis of rotation. Each of the elements of the mass that make up the entire mass of the body contribute in the same way to the total moment of inertia of the body. If m i ri 2 represents the contribution to the moment of inertia from an element of mass m i located at a distance r i from some particular axis, then the total moment of inertia of the body is given by N I = m i ri 2 (2) i=1 where N is arbitrarily chosen but it must be large enough so that each element of mass approximates a point mass. For most objects, the calculation of the total moment of inertia is best done by using calculus. This has been done for many symmetrically shaped objects. In particular, for a uniform solid disc of radius R and total mass M, the moment of inertia about its axis of symmetry can be shown to be I = 1 2 MR2 (3) Note that the moment of inertia of a body is different for different axes of rotation. Figure 1: Setup for the inertia wheel The basic setup of this experiment is shown in Figure 1. The inertia wheel, which consists of a large uniform disc and a small pulley attached to its center, is mounted on the wall. The disc will be caused to rotate about its central axis by application of a torque caused by the weight of a mass m on a string wrapped around the pulley. The mass m moves linearly downward as the disc rotates about its fixed axis. The two forces acting on the mass m are the weight mg and the tension T in the string. Applying Newton s second law to the linear motion of the mass gives mg T = ma T = m(g a) (4)
3 Inertia Wheel Page 3 of 5 The inertia wheel of moment of inertia I rotates from the torque applied by the tension in the string T on the pulley. The definition of torque is the product of the an applied force through a perpendicular lever arm. For the torque due to the tension T, this gives where r 2 is the outer radius of the pulley. τ = T r 2 (5) When the mass m is released from rest, it accelerates for a distance h and then strikes the floor as shown in Figure 1. It takes some time t to move this distance h. The acceleration a can be determined from the kinematic relationship h = 1 2 at2 a = 2h t 2 (6) Since a point on the outer radius r 2 of the pulley must travel the same linear distance that mass m travels as it moves the distance h, the angular displacement, velocity and acceleration are related to its linear displacement, velocity and acceleration by the ratio of the outer radius of the pulley. In particular, the linear and angular acceleration are related by α = a r 2 (7) where r 2, the lever arm of the torque, is equal to the outer radius of the pulley. The relationships described above will be used in this laboratory in the following way. A series of different masses will be placed on the string and the time it takes for each mass to accelerate the distance h will be determined. Equation (6) will then be used to determine the acceleration for each m. Next, equation (7) will be used to determine the angular acceleration α for each mass m. Equation (4) will then be used to determine the tension T and using T in equation (5), the torque τ will be found. According to equation (1), these data for τ versus α should be linear with the moment of inertia I as the slope. A linear least squares fit to the data will determine I. The moment of inertia of the inertia wheel can be determined theoretically by treating the system as a uniform disc. It is an approximation, but a rather good one, to ignore the effect of the pulley on the calculation of the moment of inertia of the inertia wheel. A better approximation is to calculate the contributions from each part separately and add them together I = 1 2 MR M ( ) p r r2 2 (8) where M and M p are the masses of the disc and the pulley respectively, R the radius of the disc, and r 1 and r 2 the inner and outer radius of the pulley respectively.
4 Inertia Wheel Page 4 of 5 4 Experimental Procedure 1. Record the LABEL of the inertia wheel which you carry out in your laboratory work. 2. Record the values of M, M p, R, r 1 and r 2 in the Data Table 1. These values are pasted on the wall, or else it may be available from your laboratory demonstrators. 3. The inertia wheel should have already mounted on the wall in a permanent position. The apparatus should have a light string long enough to reach the floor with one end fixed to the edge of the pulley. The other end of this string should be attached to the mass holder. 4. The apparatus has a small tripping platform on which the mass holder on the end of the string rests before it is released. Measure the distance from the top of this platform to the floor and record it in the Data Table 1 as h. 5. Wrap the string around the pulley until the mass holder is on the platform which is to be tripped to release the mass. 6. A set of masses is supplied with the apparatus. Put one of the masses on the mass holder. Measure the total mass of the mass set with the mass holder m and record this value as m in the Data Table Release the tripping platform and simultaneously start a timer. Stop the timer when the mass holder strikes the floor. Take extreme care to start the timer exactly when the mass is released. Make sure that the mass holder is just resting on the platform with no slack in the string when it is released. Record the value of the time t in the Data Table 2. Using this value of m, repeat this process FOUR more times for a total of FIVE trials. 8. Repeat the procedures in 67 for FIVE different combinations of masses. Perform FIVE trials with each value of m. Record all the values of m and t in the Data Table 2.
5 Inertia Wheel Page 5 of 5 5 Data Analysis D1. Using the values of the masses of the disc and pulley respectively M and M p, radius of the disc R, and, inner and outer radius of the pulley r 1 and r 2, calculate the theoretical value of I the moment of inertia of the inertia wheel. State your best theoretical value of the moment of inertia of the inertia wheel. Hint: The uncertainty of the theoretical moment of inertia of the inertia wheel can be calculated by R 4 I = 4 M 2 + M 2 R 2 R (r2 1 + r2) 2 2 Mp 2 + Mp 2 (r1 r r2 r 2 2) 2 where s are the corresponding uncertainties. D2. Enter your data in the Data Table 2 into the Excel spreadsheet. Calculate the mean t for the repeated trials of the time for each mass in the spreadsheet. D3. Using the values of t in equation (6), calculate the acceleration a for each value of m in the spreadsheet. D4. Using the values of acceleration determined above in equation (7), calculate the angular acceleration α for each value of m in the spreadsheet. D5. Using the values of acceleration determined above in equation (4), calculate the tension in the string T for each value of m in the spreadsheet. D6. Using these values of T in equation (5), calculate the value of the torque τ for each value of m in the spreadsheet. D7. Use the LINEST() function available in the Excel to perform a linear least squares fit to the data, with the torque τ as the yaxis and angular acceleration α as the xaxis. Determine the slope and intercept with the corresponding uncertainties of the least squares fit. State the best experimental moment of inertia of the inertia wheel. D8. Plot a graph of the torque τ against the angular acceleration α in the spreadsheet. Also show on the graph the straight line that was obtained by the linear least squares fit to the data. D9. Use percentage discrepancy to compare the experimental value to the theoretical value. What does this imply about the accuracy of your results? Hint: The percentage error is defined as Percentage discrepancy = Experimental value Known value Known value 100% D10. How is the frictional torque acting on the inertial wheel τ f determined from your data? State the best experimental value of the frictional torque acting on the inertia wheel.
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