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Measurement of Moment of Inertia INTRODUCTION The property of a body by which it resists acceleration is called the inertial mass m. The rotational analogue to inertial mass is the moment of inertia I and it is the property of a body by which the body resists angular acceleration. Newton’s second law of motion F= ma for linear motion has a rotational analogue which is = I where is the torque and is the angular acceleration. OBJECTIVES In this experiment, you will Collect angular acceleration data for objects subjected to a torque. Determine an expression for the torque applied to a rotating system. Determine the relationship between torque and angular acceleration. MATERIALS Vernier data-collection interface Logger Pro or LabQuest App ring stand lightweight mass hanger string Vernier Rotary Motion Sensor Vernier Rotary Motion Accessory kit balance drilled or slotted masses THEORY If a torque is applied to a body which is constrained to rotate about a fixed point then the body will undergo and angular acceleration given by = I where I is the moment of inertia about the axis through the fixed point. For rotation about the axis of cylindrical symmetry the moment of inertia of a disk is Idisc = ½ M R2 . For the mass hanging over the pulley: mg _ T = ma where a = r. Substituting for a in the first equation gives the tension T = mg _ mr = m(g _ r). The torque on the disk is: = rT and substituting in the value of the tension from the line above gives = rm(g _ r). PROCEDURE 1. Connect the Ultra Pulley to the Swivel Mount and then to the Rotary Motion Sensor, as shown in Figure 1. Attach the sensor to a ring stand and position it so that a weight tied to the edge of the large pulley on the sensor and hanging over the Ultra Pulley can hang freely without touching the floor. 2. Find the mass of one of the solid aluminum disks, then attach it to the 3-step pulley on the sensor. Record the radius of the pulley ( measure the radius of the drum = r ). 3. Open a new file in the data-collection program. The default data-collection rate is fine, but reduce the length of the experiment to five seconds. 4. Place the lightest mass available on the hanger, then wind the string onto the largest pulley on the Rotary Motion Sensor. 5. Start data collection, then release the hanging weight. Catch the hanger when the string has completely unwound. 6. To determine the angular acceleration of the disk, perform a linear fit on the appropriate portion of the angular velocity vs. time graph. Record this value in your lab notebook, along with the mass of the hanger and weight for each value you use. 7. Repeat Steps 4–6, increasing the mass of the hanging weight, until you have at least five different values of angular acceleration. MOMENT OF INERTIA OF A DISK 1- Using the small (approximately 10 gram) masses, measure the angular acceleration for five different masses. Calculate the torque for each run and plot torque versus angular acceleration. The slope of the graph will be the moment of inertia of the system. 2- Write the equation that represents the relationship between the net torque, , acting on the disk and their angular acceleration, . 3- Calculate the moment of inertia of the aluminum disk and compare it to the theoretical value Idisc = ½ M R2 NOTE: As you are likely to have found before, the slope of a graph is usually some function of physical parameters of the system. For example, in the Newton’s Second Law experiment, the slope of the graph of net force vs. acceleration is the mass of the object accelerated by the force. The greater the mass of the object, the larger was the force required to produce a given acceleration. In effect, the mass is a measure of the resistance to the change in the motion of the object. Physicists call this resistance to change in motion inertia. In this experiment, the slope is also a measure of the resistance of the object to undergo an acceleration; in the case of rotational motion it is known as the moment of inertia. 4- From the previous paragraph, you might suspect that the slope is a function of the mass. What evidence do you have that supports this hypothesis?