Uhapter iz Rotational Motion II: Moment of Inertia (Composite Systems) In the previous lab we examined the concepts mo tion rotating about a xed axis of a platform which we approximated as a rod as well as with a thin disk platform. In this lab we continue to explore with rotational motion around a xed axis of other ob jects and combinations of them. The same concepts (torque, angular velocity and acceleration, moment of inertia) and equations are involved, so keep the manual for Lab 11 open as well. 1 2.1 Background Almost all the necessary theory for this experiment has already been discussed in Lab Manual 11. Please refer to it. Parallel Axes Theorem Parallel axis theorem (for moment of inertia). If you are not familiar with the concept of center of mass yet, here is a quick intuitive explanation: center of mass of a system (object) is the average of the posi tions of all masses inside the system (object). If an object is placed within a uniform gravity eld (like that of the Earth near its surface), then center of gravity of the object happens to be the same point as center of mass. For a uniform symmetric object, center of mass is the centroid of the obj ect. E.g. cen ter of mass of a uniform flat disk is its center. Now. for the parallel axis theorem itself: suppose you hap pen to know the moment of inertia IQ\" of an object of mass m relative to an axis that passes through its center of mass. Suppose you also x the object to ro tate around an axis, that is parallel to original center of mass axis, but displaced by distance at from the center of mass. Then the moment of inertia of the object relative to this new axis is: I = Io\" + and?2 (12.1) 12 .2 Experiment In this lab we will attempt to verify that the moment of inertia (rotational inertia) for a system of objects rotating about the same xedaxis will be the sum of the constituent moments of inertia plus the addi tional terms as predicted by the parallel axis theorem. This will be accomplished by experimentally measur ing the moment of inertia of two or more solids and comparing your results with the moments of inertia expected from the developed theory. Recall that we are using the extension of Newton's Laws to rotation as discussed in the rst rotational motion manual. specically the dynamics interpreta tion of the denition of moment of inertia: 2 Rotational Motion 11: Moment of Inertia (Uomposite Systems) I = . 12.2 a ( J Recall that we are approximating the aluminum bar platform as a rod. For a uniform, thin rod of mass mp and length L the moment of inertia is given by I = 1'an2- ,, 12 (12.3) For a uniform disk of mass, fu'd and radius, R, the moment of inertia is given by deZ I = . d 2 (12.4) From the disk, one may punch a hole through the center like a donut. For such a hollow cylinder or ring of inner radius R1, outer radius R2, mass 111,, with uniform mass density, the moment of inertia is given by MW? + R3) If: 2 (12.5) Note that the height of the cylinder is not needed for these derivations. 12.2.1 Equipment 0 Rotary motion sensor 0 Rigid aluminum platform 0 Rigid thin disk 0 Rigid thin ring 0 Lightweight thread 0 Mass hanger and assorted masses 0 Scale 0 Bubble level a Vernier calipers 0 Personal computer with PASCO Capstone soft ware 12.2.2 Using PASCO Capstone Primary documentation for the PASCO Capstone software, the PASCO 550 Universal Interface, and the sensor and actuator attachments used for this exper iment are all available through the PASCO Capstone help menu. Sensor You already know how to start a new exper iment and select the required sensor. The sen sor for this experiment is Rotary fotion Sensor which will be set to 1, the rst small yellow cir cle you see on the computer for the PASCO 550 Universal Interface. Graph Option From the right panel labeled Dr's plays drag the Graph icon into the middle panel. On the yaxis click on Seteet Measurement and choose .4 again?\" Speed(md/s). The xaxis will au tomatically be set to Timea) CAUTION: Make sure to keep hair and clothing away from the rotating platform when it is in motion! 12.2.3 Procedure Refer to Lab manual 11 for how to accelerate the sys tem with the hanging masses and finding values for linear acceleration a of the hanging mass Mr , tension force, T, torque, T and moment of inertia, I. Experiment 1: MOI Disk and Ring Repeat the procedure for Experiment 2 in Lab man ual 11 except that you add the ring onto the disk. De termine the combined moment of inertia of the disk and ring from the plot of torque, 7' vs angular acceler ation, (1. Compute the moment of inertia of the ring only by subtracting the moment of inertia of the disk only (obtained in Lab 11) from that of the combined system. Compare this experimental value for the moment of inertia of the ring to the value predicted from theory. Make sure you measure the two radii for the ring and the mass of the ring for your calculations and comparisons. Experiment 2: MOI Platform with point masses Keep the values of hanging mass, It! and reel radius R xed. Replace the ring and disk with the alu minum bar platform that we are approximating as a rod for calculations. Add the two extra masses, m to the platform. Fix them at the same distance, it" away from the center. Vary distance, r for at least 4 dif ferent points and measure the corresponding angular acceleration, (1. Calculate linear acceleration a of the hanging mass, M, tension, T and torque, T as before. Then calculate the total moment of inertia, I using equation (12.2). Recall that the masses, m on the platform are acting as point masses. Compare these values to the predicted total mo ment of inertia of the platform \"mp plus two attached masses, m given by the following equation: I = Ip + 2mr2 (12.6) Plot torque, T vs angular acceleration, a. From the slope of this graph, determine the experimental value of the moment of inertia, Id of the thin disk. 12.3 Specic Report Expecta- tions The following subsections list the requirements for meeting expectations as indicated in the rubric. Please remember that if any section isn't mentioned that all of the general section requirements are ex pected. 12.3.1 Introduction Introduce moment of inertia, torque, angular accel- eration, and angular velocity quickly and the intro- duce the expectations for the moment of inertia of composite systems. Then describe the objective (s) of this lab. There is no need to include hypotheses or predictions. 12.3.2 Procedure Briefly describe your experimental setup. 12.3.3 Results Neatly include data tables, graphs and calculations for Experiments 1 and 2, with clear labels. 12.3.4 Discussion Comment on whether or not your results agree with the parallel axis theorem for dynamics of rotational motion as it applies to moment of inertia. How well do the experimental values for moments of inertia compare to the predictions from theory. Speculate as to what are the sources of discrepancies (again \"human error\" doesn't count)