Rotational Inertia

William M Jones

Annekathrin Friedrichs

Amanda Rainer

Emily Stone

Abstract

            Lab 14, Rotational Inertia, uses a rotating table with a cylindrical ring and disk to compare the theoretical moment of inertia with the experimental moment of inertia for the two masses and shapes.  A rotating table was connected to a hanging mass through the use of a system of two pulleys and a string of negligible mass.  The hanging mass was released from rest.  A smart pulley collected the acceleration and velocity data as the mass was accelerated toward the ground by gravity.  This data was collected using the Logger Pro program.  The data from Logger Pro was analyzed and compared to the theoretical moment of inertia for the two shapes using the given moment of inertia formulas.

            The rotational inertia lab verifies the physics used to derive the moment of inertia for various shapes including the ring and disk used in the experiment.  The shapes, disk and ring, were Annular Cylinders with a moment of inertia of .  The experimental rotational moments of inertia for the two cylinders were within the range of uncertainty.

 

Procedure

 

The lab group followed the experimental procedure outlined in the lab manual Physics 4A Laboratory Manual to conduct the experiment.  A metal rod was clamped to the lab table which had two pulleys clamped to the rod.  One of the pulleys was a smart pulley with a ULI motion detector attached to it.  The ULI interphase was connected to the computer to collect the acceleration data of the hanging mass in the Logger Pro software program.  A sketch of the setup is shown in Figure 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1: Experiment Setup Sketch

 

A 200 gram mass was hung from a height of 1.56 meters.  The mass was allowed to fall and timed to the point where it struck the floor.  The average acceleration was determined using the relationship , where a is the unknown.  This value was compared to the acceleration value given in the Logger Pro program and was within 16% of the Logger Pro value.  This indicates that the program was determining the acceleration value with more accuracy then measuring the height and attempting to use a stopwatch to time the fall.  The Logger Pro program was used to compile the data for the remainder of the experiment.

Three to four trials were completed for each of three experiments.  The first experiment was conducted to find the moment of inertia, I, for the rotating table.  The second experiment found the moment of inertia, I, for the rotating table and disk.  The third experiment found the moment of inertia, I, for the rotating table and ring.  The frictional torque was found as part of each of the experiments.

To determine the frictional torque a mass of the size less than what would take to turn the table was hung from the top pulley.  This mass was increased until the table started to rotate.  The Logger Pro program recorded the acceleration value.  This part of the experiment was completed with the disk and the ring placed on the rotating table.  The frictional torque was calculated by .

The mass of the disk is 3.64 kg while the mass of the ring is 4.27 kg.  The outer radius of the ring is 0.1265 m with an inner radius of 0.1115 m.  The disk has an inner radius of 0.009 m and an outer radius of 0.1265 m.

 

 

Analysis

 

            Step 1 of the analysis asked for the average acceleration for each configuration.  The three configurations are the rotating table, the rotating table with the disk shaped Annular Cylinder and the rotating table with the ring shaped Annular Cylinder.  The acceleration for each configuration was determined by taking the slope of the velocity vs. time shown in the graph below.  The average acceleration of the rotating table, the rotating table with disk and the rotating table with the ring is shown below the graph.  The absolute average deviation and average acceleration of all the trials were calculated using the Excel spreadsheet.

 

 

 

Figure 2:  Rotating Table/Ring Trial 2

 

 

 


 

 

 

 


Acceleration Tables

 


 


                        Acceleration Table

           

            As shown in the tables the average acceleration of the table with a 0.200 kg hanging mass is 0.435 m/s2 with a deviation of 0.097 m/s2.  The acceleration values of the ring and disk were determined using a 0.500 kg mass.  The average acceleration of the table and disk was 0.088 m/s2 with a deviation of 0.026 m/s2.  The ring and table had an average acceleration of 0.043 m/s2 with a deviation of 0.013 m/s2.

            The moment of inertia, I, was calculated using the following formula derived from the relationship .  The derivation is shown below.

 

 

 

                                               

                       

 

            The frictional torque of the system for each configuration was determined by hanging a mass from the top pulley at a height of approximately 156 cm until the table started to rotate.  The graphs show the acceleration value for each of the hanging masses.  The mass needed to start the table rotating without the disk or ring was 0.0073989 kg or approximately 0.0074 kg.  The table and disk required a hanging mass of 0.0378955 kg or approximately 0.0379 kg.  The ring combination needed 0.0449102 kg or 0.0449 kg to rotate the table.  The respective accelerations of the table and combinations are shown in the three graphs below.

 

 

Determining Rotating Table Frictional Torque

 

 

Determining Rotating Table and Disk Frictional Torque

 

 

Determining Rotating Table and Ring Frictional Torque

 

 

In the derivation above, the torque on the pulley is shown to equal the tension times the radius of the table shaft, which is also equal to the moment of inertia times the angular acceleration or: .  The first derivation requested in the analysis section # 2 is:

 

 

 

 

 

 

            Using the equation derived above, the rotational inertia for each of the three configurations was calculated.

 

                                                                       

 

 

 

 

 

 

 

 

 

 

 

 

 

            The theoretical values for the disk and ring were calculated using the formula for Annular Cylinders: .  The question is raised as to whether or not there is a difference in the calculations if the formula for a solid disk or cylinder is used instead of the formula for a hollow cylinder.  The answer to this question lies in the radius calculation for either equation.  The radius R2 is  and the radius (R12 + R22) is .  The difference is in the 10-5 decimal place, which should not significantly affect the rotational inertia calculation for the disk.  The theoretical rotational inertia for the disk and ring is shown below.

                                                                       

 

 

 

 

 

 

 

 

 


            The theoretical values for the rotational inertia are considerably smaller than the experimental values resulting from the experiment.  The frictional torque for each of the three configurations was determined as shown previously in this report.  Taking into consideration the mass of the pulley determined in lab 8 as 2.1 grams + or  1.29 grams the rotational inertia for the disk and ring was recalculated using the formula derived for the inertia including the frictional torque and pulley mass.

The rotational inertia for the disk/table combination was 0.1169 kgm2 and 0.1074 kgm2 for the disk.  The ring/table value was 0.2384 kgm2 and 0.2289 kgm2 for the ring without the table.  Note: The experimental values from the experiment are considerably higher than the theoretical values for the rotational inertia.

            The differences from the experimental and theoretical values for each configuration are shown here:

 

 

 

 

 

            The uncertainty from the experiment comes from several areas.  The first is the pulley mass that was calculated from a previous experiment and was plus or minus 1.29 grams.  This is a  mp of 0.61.  Throughout the year it has been noted that the masses used in the experiments are not the exact mass on the label.  A 200 gram mass may show to be 199 grams on the scale.  This could be an error in the scale or in the mass during production.  Worst case is the 200 gram mass is off by 1 gram inducing another error into the experiment.  For a 200 gram mass and 500 gram mass this gives an uncertainty of 0.005 and 0.002 respectively.  The shaft measurement of the rotating table was within 1 mm resulting in an uncertainty of 0.02.  This value is squared in the equations doubling the uncertainty to 0.04.  The frictional torque has an uncertainty from the fact that the value is calculated from a mass that accelerated the table’s rotation, although the acceleration was minuet.  It is uncertain as to the proper way to calculate this uncertainty and is not shown here.  The uncertainty for the acceleration of the disk and ring are shown in the tables above as 0.0262 for the disk and 0.0133 for the ring.  The uncertainty for the experimental rotational inertia of the disk is 0.0728 and 0.1523 for the ring.

            There is an uncertainty that is associated with the theoretical rotational inertia values shown above.  This uncertainty comes from the measurement of the ring and disk diameter.  The measurement is within 0.1 cm on each side of the diameter making the radius measurement off by as much as 0.2 cm.  Since both the inner radius and outer radius are squared in the formula this doubles the uncertainty values for the ring and disk.  For the ring the uncertainty is 0.0158 + 0.0179 for an uncertainty of 0.0334 and an uncertainty for the disk of 0.0158 + 0.11 = 0.126.

            The total uncertainty for the ring, experimental plus theoretical, is 0.0334 + 0.1523 = 0.1857.    The total uncertainty for the disk, experimental plus theoretical, is 0.126 + 0.0728 = 0.1988. 

The difference between the experimental rotational inertia value and theoretical value of the disk is 0.0781.  This value is less than the uncertainty for the experiment of 0.1988 and is within the range of uncertainty.  The difference between the experimental rotational inertia value and theoretical value of the ring is 0.1682.  This value is less than the uncertainty for the experiment of 0.1857 and is within the range of uncertainty.