M11 Rotational Inertia I. OBJECTIVE Test some aspects of rotational dynamics, specifically • the relationship between torque and angular acceleration • the rotational inertia of different objects. II. EQUIPMENT PASCO rotational

dynamics apparatus 9279, mass holder for the rotational dynamics apparatus, 2 x 5 g, 2 x 10 g masses, meterstick, stopwatch. III. INTRODUCTION In Figure M11-1 is shown a flat rigid object that is fixed at point A in the plane of the page. The object is free to rotate about an axis that passes through point A per- pendicular to the plane of the page. The object has a reference line marked on it that is initially in line with the x axis at time t = 0. At some later time t > 0, the object has rotated into a new position. We measure the rotation with the angle in radians. A rotation through an angle of 360 degrees (one complete revolution) is equal to a rotation of 2 radians. The angle of rotation is shown as the angle between the x axis and the reference line for the figure at time t > 0. Note that points B and C on the reference line, their initial and final positions connected by dotted lines in the figure, will mark out a circle centered on A if the object rotates through one complete revolu- tion. Just as every point on a rigid body that undergoes translational motion moves the same distance when the body moves, every point on a rigid body that is rotating moves through the same angle when the body rotates./n58 y-axis Reference Line INIMINININ. 1>0 Reference Line x-axis Figure M11-1. A rigid body rotating about an axis perpendicular to the page that passes through point A. Table M11-1 lists a series of quantities that are used to describe linear motion along- side their analogues in rotational motion. Displacement x in linear motion corresponds to the rotation angle measured in radians. Velocity v in linear motion corresponds to angular velocity in rotational motion. Note that the symbol presented here as the angular velocity is different than the used in Experiment M9: Simple Harmonic Motion. Acceleration a corresponds to angular acceleration a. The mass m of an object corresponds to its moment of inertia I. Force F corresponds to torque T. Both types of motion also have their own expression for associated kinetic energy./nQuantity Displacement Velocity Acceleration Mass Linear Motion Units Force cm cm S cm S g g.cm s² Kinetic Energy g cm Equation x Ax At a= Au At - m F = ma K = my² 2 Quantity Displacement radians Angular Velocity Rotational Motion Angular Acceleration Moment of Inertia Torque Rotational Kinetic Energy Units Equation 0 Figure M11-2. A rectangular bar. rad S rad g.cm² g cm g cm² (1))= 40 At = A At See figures α = T = la K = 10² An object that has a certain mass will always have that mass, without regard to its shape or orientation. In contrast, an object's moment of inertia depends not only on the mass of the object and how that mass is distributed, but it also depends on the axis about which the moment of inertia is calculated. The details of these calculations are not important for this experiment but they may be found in your lecture textbook. The equations for calculating the moment of inertia for the shapes required for this experiment are given below. In all of these equations, m is the mass of the object in question, and the body is rotating around the indicated axis. The mass of these objects is distributed uniformly./n60 A uniform rectangular bar of mass m with the dimensions a and I as shown in Figure M11-2 has a moment of inertia about an axis perpendicular to the plane shown that passes through the center of the bar given by: 1 = 12(a²+1²³) Figure M11-3. A solid disk. A solid uniform disk as shown in Figure M11-3 has a moment of inertia about an axis perpendicular to the plane shown that passes through the center of the disk given by: mr (M11-2) A hollow uniform cylinder as shown in Figure M11-4 has a moment of inertia about an axis perpendicular to the plane shown that passes through the center of the cylinder given by: 1 = 2 (r² +r²) (M11-1) " Figure M11-4. A hollow cylinder. (M11-3)/n64 V. PROCEDURE 1. Use the disk-step pulley system of mass 991 grams and radius 12.7 cm. Measure and record the mass of the mass holder to the nearest 0.1 gram. Wind the string around the 1.50 cm (smallest) step of the step pulley. Route the string over the slotted pul- ley. Suspend the mass holder and 10 grams of additional mass from the end of the string as shown in Figure M11-6. Make certain that the string is not rubbing on the table or the photogate. Turn on the counter/timer. Press the "Select Measurement" button (button #1) until the word "Accel:" appears on the display. Press the "Select Mode" button (button #2) until "Accel: Ang Pulley" appears on the display. The counter/timer has been designed to work with this particular slotted pulley and a photogate to determine the angular acceleration of the slotted pulley. Allow m, = mulier + made to drop from rest. When the mass has fallen about 10 centi- meters, press the "Start/Stop" button (button #3). An "*" will appear under "Accel: Ang Pulley" while the counter/timer is calculating the angular acceleration of the slotted pulley. When a number replaces the "*", this is the angular acceleration of the slotted pulley in rad/s. When you have measured the angular acceleration of the slotted pulley, rewind the string using care to ensure it is wound around the correct step pulley. Repeat your measurement of the angular acceleration of the slotted pul- ley, ay, 6 more times. Place the following masses on the mass holder and repeat the measurement de- scribed above: 15, 20, 25, and 30 g. 2. Mount the rod (mass 691 grams) on top of the steel disc. Using the same masses as in V-1, measure ad 7 times for each mass. Measure and record the dimensions of the rod. 3. Mount the hollow cylinder (mass 701 grams) on top of the steel disc. Using the same masses as in V-1, measure a 7 times for each mass. Measure and record the dimensions of the hollow cylinder. VI. FOR THE REPORT 1. Tabulate the data of Section V-1. For each measured value of ad, calculate the acceleration a of m.. At a minimum, your table should include columns for m, m, your ape measurements, and calculated values of a. Using Excel, plot a vs. m,. Using LINEST in Excel, calculate the slope, s, the intercept, b, and the uncertainty in those quantities o, and ₁. From these, calculate In and Op.co Assume the uncer- tainty in r is o, = 0.02 cm. Report your answer as Ipp± 0. The expected value/nthe experimental value of the rotational inertia for the rod. Report I ±018.cop Calculate the value of I, using Eq. M11-1. You may assume that the uncertainties in the dimensions of the rod and the mass of the rod to be negligible. Do the calculated value and the experimentally determined value agree? 3. Repeat the analysis of Sections VI-1 and VI-2 for the data of Section V-3. Determine the experimental value of the rotational inertia for the hollow cylinder. Report Ic ±0kc.co Calculate the value of le using Eq. M11-3. You may assume that the uncertainties in the dimensions of the cylinder and the mass of the cylinder to be negligible. Do the calculated value and the experimentally determined value agree? VII. QUESTIONS 1. Derive the expression a = m in Section V-1. mgr ID.+mr² for the acceleration a = ar of the falling mass 2. One can never completely eliminate the effects of friction in an experiment. What effect, if any, will friction between the rotating steel disk and the stationary part of the apparatus have on your experimentally determined value of In? As a simplifying assumption, take the torque on the pulley to be equal to mgr. 3. Calculate the rotational kinetic energy of the step pulley/disk when the falling mass of 15 grams has fallen 90 cm using the data from Section V-1. You may assume a frictionless system. 4. In Section V-2 and V-3, you used the fact that the moment of inertia of the step pul- ley/disk and rod system (or step pulley/disk and cylinder system) is simply the sum of the individual moments of inertia. Suppose the experiment were repeated and this time you added a disk of diameter d = 25.4 cm and mass m = 894 grams to the step pulley/disk system. You now repeat the experimental procedure using the same series of masses that were used in V-1, V-2, and V-3. What is the slope of the plot of a vs m, for the combined pulley disk and added disk system? 5. Show that the following equation is dimensionally consistent with the rotational kinetic energy of a rotating body. E= - la ² 2