PHY 221 Experiment 8 Moment of Inertia and Angular Acceleration Reference: Review Chapter 10 in University Physics / by OpenStax. Pre-Lab Quiz on Canvas: Review the sections, lab outline, and necessary calculations before beginning. In your report, begin with the title, date, your name, your partners' names, and the scientific purpose. Background/Introduction: Torque, also called moment or moment of force, is the tendency of a force to induce rotation of an object about some axis, fulcrum, or pivot point. If a force is used to start or stop something spinning, a torque is generated. Just as a force is a "push" or a "pull", torque can be thought of as a "twist". For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque (turning force) that loosens or tightens the nut or bolt. Torque is a vector quantity defined as the cross product of the lever-arm vector and the force vector: T = F XF [1] Figure 1: Arbitrary radius and force and the resulting torque. where r is the position vector between the point where the torque is being measured and where the force is being applied as in the Figure 1. The SI unit of torque is a Newton-meter, which is also a way of expressing a Joule (the unit for energy). However, torque is not energy. To avoid confusion, the unit Nom is used instead of joules, J. The distinction arises because energy is a scalar quantity, whereas torque is a vector. The directions we ascribe to angular velocities, angular accelerations, and torques, by convention, describe the axis of rotation. There is actually no physical motion in the direction of the angular velocity vector-in fact, all of the motion is in the plane perpendicular to this vector. Likewise, there is no physical acceleration in the direction of the angular acceleration vector-again, all of the acceleration is in the plane perpendicular to this vector. Finally, no physical forces act in the direction of the torque vector-all of the forces act in the plane perpendicular to this vector. The rotational version of Newton's second law is: Er = la [2] That is, the net torque is equal to the moment of inertia multiplied by the angular acceleration. The moment of inertia, /, depends on the masses and their distance from the axis; namely, 1 = Emit. [3] Formulae for the moments of inertia for many different arrangements of mass are tabulated in your textbook (Figure 10.20, p. 497). The angular acceleration is related to the linear tangential acceleration by: atan = a XT [4] We will use the same apparatus we used in the centripetal acceleration lab, but with a different crossbar. This crossbar is a threaded rod, with wing nuts that clamp masses at different radii. A string is attached to, and wrapped around the vertical shaft. The other end of the string passes over a pulley, and is attached to a mass that can fall. As the falling mass descends, the string will unwrap from the shaft, exerting a torque on the shaft, and the shaft will rotate. PHY 221 Exp. 8 Editted by J. Hermann, M. Watry, T. Tarbuck, and V. PlausThe moment of inertia for the system (consisting of the shaft, crossbar, masses and wing nuts) can be found using conservation of energy. The potential energy change of the falling mass (m) is the sum of the kinetic energy changes of the falling mass and the rotating system: mgh = =mv2 +=Iw2 [5] Here g is the acceleration due to gravity, h is the height the mass falls through, v is the final velocity of the falling mass, I is the moment of inertia of the entire rotating system, and @ is the final angular velocity of the rotating system. The angular velocity (@) of the system can be related to LE the linear velocity (v) of the falling mass by noting that the distance traveled by the falling mass is the same as the distance traveled by a point on the circumference of the shaft of radius r. Hence w=wr, where r is the radius of the rotating shaft. Equation 5 reduces to Equation 6 after some algebra Figure 2: Spinning apparatus experimental set-up. and after substituting for the final velocity v, with the expression from kinematics that gives the final velocity caused by a constant acceleration as a function of the height and time. I = mr2 (95 - 1) [6] Equation 6 gives the moment of inertia of the entire system that is rotating. To determine the moment of inertia of the two masses alone without any supports (,'), the experiment is done with (/w) and without the masses (/.). This is similar to determining the mass of water in a beaker by measuring the beaker's mass with and without water. Im = Itot - Io [7] The last equation follows, provided the hanging mass falls through the same height in each case. This provides an experimental method of determining the moment of inertia of masses m ' at a distance r' from the axis of rotation. With today's experiment, we can compare the mathematical definition of moment of inertia with the experimental calculation in Equation 7 and use the moment of inertia to compare the predicted tangential acceleration using Equations 1,2, and 3, with the experimental linear acceleration. Equipment: Mass hanger & assorted masses, spinning apparatus with threaded crossbar, slotted 100 g masses and wing nuts, stopwatch. Note: Please note that the notation in this outline uses a prime ( ) not for derivatives, but to denote "different mass", which is a common notation in physics. The primed notation is used for the crossbar masses and radii.Procedure: 1. Place the threaded crossbar symmetrically through the top of the main post and secure it. 2. Use calipers to measure the radius of the main post, 7, and its uncertainty, or. Determine a length of string that will reach to the floor from the post and wrap it around the shaft evenly. 3. Place the end of the string over the pulley and hang a total of 100-150 g from the string. Record the mass, m, and its uncertainty, om. Record the height of the mass from the floor, h, and its uncertainty, oh. 5. Measure the time it takes for the hanging mass to drop to the floor, 7. Repeat the experiment enough times that you have confidence in your data for the average time, forg, and its uncertainty of (standard deviation of the mean). 6. The crossbar assembly has four butterfly nuts, two slotted masses, and the bar itself, attached horizontally at the top of the main post. Choose two 100-gram slotted masses and record each mass together with its butterfly nuts, my ' and my', and their uncertainties, omy ' and om?" 7. Attach them symmetrically, close to the ends of the crossbar. Measure the distance from the center of the post to the center of each mass, ', and its uncertainty, or'. a. The math is simpler if they are at exactly the same distance, but don't spend a lot of time on this. Just make sure that they're similar enough that the system doesn't wobble. 8. Diagram your set-up, labeling all items and measurements. 9. Repeat 3-5, re-measuring all new or changed measurements. 10. Check the equations in the introduction and be sure you have measured all the required parameters for this experiment. 1 1. Repeat 7, 9 with another location of the slotted masses. 12. Repeat 7, 9 with a third location of the slotted masses. Calculations: Calculate the experimental moment of inertia of each of the systems, and then just the crossbar masses (Eqn 6 and 7). o Be careful with your uncertainty calculation. Predict the theoretical moment of inertia of the two crossbar masses my ' and my' (Eqn 3) Calculate the experimental linear acceleration of m from time and distance using kinematics . . . Predict the moment of inertia of the system (Eqn 3 and 6) Predict the tangential acceleration from applied torque and moment of inertia (Eqn 1 and 2) o Show your algebraic work is finding this equation. Hint: 7 / mg, so it might be helpful to draw a FBD or otherwise explain your work. Discussion: Did the two methods of calculating the moment of inertia of the two masses agree? Why/why not? Did the predicted tangential acceleration agree with the linear acceleration calculated from kinematics? Why/why not? Discuss any assumptions that were made about the apparatus or your method, whether they should be valid, whether they actually are valid, and how breaking these assumptions would affect the agreement/disagreement of your results.F experime ntal inertia of experimental predicted ol, pred. theoretica systthe ol, exp. inertia of ol, exp. inertia of system inertia of ol, theo. ems system system (s) masses mass (s) systems (S) mass mass (s) 1 0.000595 0.000012 IN/A N/A N/A N/A N/A N/A 2 0.00585 0.00009 0.00525 0.00009 0.00651 0.00004 0.00591 |0.00004 3 0.00367 0.00007 0.00307 0.00011 0.00404 0.00003 0.00345 |0.00003 4 0.00179 0.00003 0.00119 0.00007 0.00187 0.00002 0.00128 |0.00002 da, linear systekinematic linear da, kinematic linear predicted linear acceleration ms acceleration acceleration (m/s^2) acceleration (m/s^2) (m/s^2) 1 0.0857 0.0009 N/A N/A 2 |0.00879 0.00007 0.0079 0.0018 3 0.01401 0.00012 0.01271 0.00014 4 0.0287 0.0003 0.0274 0.0003