1. Thoroughly discuss possible systematic errors and explain how they would affect the value (e.g., smaller or bigger than the theoretical value) or the experiment.

MOMENT OF INERTIA OBJECT: To determine the moment of inertia I in the simple case of a (total) mass M on a cross bar at radius R; to verify the dependence of I on both M and R; to do a full analysis of random and other errors which may affect the experiment; to discuss other possible systematic errors. EDUCATIONAL PURPOSE: To learn about moment of inertia. SUGGESTED READING: Halliday & Resnick FUNDAMENTALS OF PHYSICS, 12th Edition, See. 10.5 REQUIRED EQUIPMENT: Rotational motion apparatus Hooked mass set, slotted mass set Stopwatch, vernier calipers Triple beam balance Metre stick, cord, etc. THEORY: Uniformly accelerated angular motion is completely analogous to uniformly accelerated linear motion. One can set up a table of equivalencies: LINEAR ANGULAR displacement Ax (m) displacement A6 (rad) velocity v (In/s) velocity w= v/r (rad/s) acceleration a (I'D/52) acceleration a: = air [rad/secz) force F (N) torque 'r = Fr sin6 (N 'm) mass m (kg) moment of inertia I = E mr2 (kg-m2)* momentum p = mv (kg-m/s) momentum L = I'm (kg-mfs)* kin. energy Ek = 2mm)2 (J) kinetic energy Ek = 2 14:02 (J)* Newton II F = ma Newton II r= Io: * [There are some restrictions on the equations marked with a *.] Moment of inertia is the angular quantity analogous to the linear quantity mass. One way of comprehending mass is: the greater an object's mass, the more force is required to accelerate it. Similarly, the greater a rotating object's moment of inertia, the more torque is required to accelerate it. Note that the moment of inertia is dependent not only by its mass but also on how the mass is distributed. We will nd the moment of inertia of a body as follows: A (total) mass M, whose moment of inertia we want to nd, is attached to a vertical shaft at radius r by means of a "moment of inertia" rod (refer to the diagram in the procedure). The vertical shaft is mounted on bearings which make it (and the mass m) free to rotate when a torque is applied to it. The torque is supplied by a falling mass m which connected to the vertical shaft by means of a cord, one end of which is wrapped around the circumference of the shaft and the other end of which is attached to the falling mass. If the hanging mass m starts from rest and falls through a distance h in time t, potential energy of the amount mgh will be converted to kinetic energy. This kinetic energy consists of the translational energy of the falling mass % mvz, plus the rotational energy of the rotating system, V2 I (1)2 where v = the nal speed of the falling mass, I = the moment of inertia of the rotating system, a) = the final angular velocity of the rotating system. Due to the conservation of energy mghzyzmv2+y21w2 (1) Because the force of gravity is constant, the mass m will be uniformly accelerated, so from the kinematics equation Ax = vt + h=vo2vt wherevo=0 V! 2 h h = , = 2 so 2 v t ( ) The linear velocity v, the angular velocity 0), and the radius of the vertical shaft are related by v = r (a (3) Combining equations ( 1), (2), and (3), and solving for I gives 2 I = 2 g 1 4 mr [ 2 h J t ) Equation (4) gives the moment of inertia of the entire rotating system. We want the moment of inertia of just M, the load attached to the horizontal rod. To nd this, the following theory will be utilized: The moment of inertia of the entire rotating system can be considered to be simply the sum of the two parts, Io + I1, where L3 is the moment of inertia of the vertical shaft with the horizontal rod attached, and 11 = 1140'?2 is the moment of inertia of the (total) load M attached at radius R from the rod. PROCEDURE : 1. Check that the (threaded) moment of inertia rod is centered and secured to the top of the vertical sha (tighten the locking screw if necessary). Use the wing nuts to secure a 100 g slotted mass to each side of the rod 12.0 cm from the center of the rod (as measured from the center of the slotted mass). Secure one end of a cord to the screw in the vertical shaft and wind it evenly around the shaft (overlapping turns change the radius by up to 10% giving corresponding errors in your results). Pass the other end of the cord over the pulley and attach a 100 g mass to it. Ensure that there is enough cord so that when the mass is released for just below the pulley, it will just reach the oor below. 2. Release the 100 g mass and let it fall downward causing the vertical shaft to rotate. Measure and record the following: h = the distance the accelerating mass falls t= the time for the mass to fall the distance h m = the value of the falling mass r = the radius of the vertical shaft (thickness of the cord?) R = the distance of the slotted mass from the axis of rotation M = the value of the slotted masses plus wing-nuts (BOTH sides) Estimate the uncertainty in each of your measurements. 3. Repeat the experiment several times to get a good average value of t. 4. Remove the slotted masses and wing nuts from the moment of inertia arm. Again wind the cord, release the accelerating mass and measure the time t over several trials. 5. Repeat steps 2-4 for various values of mass M/2 (suggested values 50, 100, 150, 200 g each side and more if you feel necessary) for a fixed value of R = 12.0 cm. 6. Repeat steps 2-4 for various value of the radius R, for a fixed value of M (say 2 X 100 g + 4 wing nuts). Suggestion: for reasons described below, increase the number of trials at the larger radii R. ANALYSIS: For each set of data: 1. Use equation (4) to calculate the experimental moment of inertia I of the rotating system with the slotted mass and wing nuts, and the moment of inertia Ia without the slotted masses and wing nuts. Estimate the typical uncertainty due to measurements for the calculated moment of inertia. 2. Plot a graph of I (experimental) vs. M for a constant R. Find the slope of the line of best t and compare with the theoretical value of R2. If you are using the spreadsheet to calculate the least squares t, compare the average y deviation with the error estimate determined above. 3. Plot a graph of I (experimental) vs. R for a constant M, and another for] vs. R2. For the second graph, determine the slope of the line of best t and compare with the theoretical value of M (the total rotating mass = 2 slotted masses and 4 wing nuts). 4. Calculate the theoretical ID = moment of inertia with M = 0. (Weigh the threaded rod, estimate the mass of the center pole and use the stande formulas from your book, Section 11-7.) How does this value compare with that determined experimentally above? Can you explain any discrepancy