11 May 2015: Moment of Inertia and Frictional Torque

Purpose:

The purpose of this experiment was to experimentally determine the moment of inertia and frictional torque of a large metal disk rotating about a shaft in its center.

Apparatus:

Figure 1: Apparatus of this experiment

The apparatus for this experiment consisted of a large metal disk that rotates about a metal shaft that goes through its center. This The disk and shaft are held up by metal stands attached to a metal plate. A large caliper was also needed to accurately measure the diameter of the large metal disk, and a small caliper was used to measure to the diameter of the central shaft. In addition, a camera was needed to capture the spinning of the disk, and  a computer with LoggerPro was used to collect the video using movie capture and to obtain data from it. Lastly, a 500 gram cart and a 1 meter track were needed to verify the accuracy of the calculated moment of inertia and frictional torque of the system.

Procedure (Part 1):

The dimensions of the large metal disk and the central shaft (the system) were measured using the calipers. Using the dimensions, the moment of inertia of the system was calculated. The derivation is displayed in figure 2 below:

Figure 2: Derivation of the moment of inertia using the measured dimensions

The mass of the system was provided as a stamp onto the side of the rotating metal disk. 

Discussion (Part 1):

Since the system is composed of three different cylinders of different diameters and lengths (the large metal disk, and the two cylinders of the central shaft on each side), the moment of inertia of the system was found by using the equation for the moment of inertia of a cylinder: 

1.

The moments of inertia of the three "cylinders" in this system were calculated separately, then added together. Because this system was composed of the same metal or metal alloy, the mass of each of the three cylinders of the system was obtained by finding how much of the total volume the cylinder consists of, then multiplying it by the total mass to get the mass of that cylinder.

2.

The dimensions measured of each cylinder were used to find each volume of the cylinder, and the total volume of the system was found by summing the volumes of all the three cylinders. The large metal disk, whose volume was pi((0.1000275 m)^2)(.0157 m) = 4.96 x 10^-4 cubic meters, was found to contain around (4.96 x 10^-4 m)/(5.75 x 10^-4 m) = 86.3% of the volume of the whole system, and so its mass was found to be (0.863)(4.808 kg) = 4.15 kg. Using the same approach, the mass of the cylinder of the central shaft that contained the length of 5.00 cm was found to be 0.324 kg (6.74 % of the volume/mass), and the mass of the other cylinder was found to be 0.332 kg (6.96 % of the volume/mass).

Summing the moments of inertia of all three cylinders, the moment of inertia of the system was found to be 0.0209 kg squared meters.

Procedure (Part 2):

Once the moment of inertia of the system was obtained, the frictional torque between the large disk and the central shaft of the system was obtained. The data needed to obtain the frictional torque was taken from a video capture that was performed. The video capture consisted of the recording of the rotating large disk decelerating by the frictional torque in the central shaft. A point on the rim of the disk was used for the analysis of the video capture to obtain the change in the angle that the disk rotates during a set time period. This point was marked by a large dot on a piece of tape attached to the rim of the disk. Once the video capture of the rotating disk was analyzed by plotting points of the position of the dot on the rim using LoggerPro, the data obtained consisted of the x and y positions of the dot as well as the velocity of the dot in the x and y position.

Discussion (Part 2):

Theta, the angle that the dot on the rim moved through during each time period, was found by taking the inverse tangent of the quotient of the y position and the x position, as shown below:

Figure 3: Finding frictional torque in the system

The purple dot is the dot on the rim that was monitored during the video capture. Once theta at each time period was found, a graph of theta versus time was obtained, shown below:

Figure 4: Graph of theta vs time

A quadratic fit was performed for both slopes on the graph. The reason that the graph had two different slope is that each slope counts for one rotation, and two rotations were recorded in the video capture. A quadratic fit was performed because the slope of the graph is related to a rotational kinematics equation, shown below:

3. 

Using the average value of A, the angular deceleration of the disk was found as follows:

Figure 5: Derivation for the angular deceleration of the rotating metal disk

Knowing that torque is the product of inertia and angular acceleration, the frictional torque was obtained using the derivation below:

Figure 8: Derivation of frictional torque

Procedure (Part 3):

To determine the accuracy of the derived frictional torque and angular deceleration resulting from the frictional torque, a 500 gram cart was attached to one of the cylinders of the central shaft using a long string, and a meter metal track was placed 60 degrees above the horizontal from the ground. Calculations, shown below, were performed in order to determine a theoretical time it takes for the cart to roll down the meter track from rest. Then, three trials were performed where the cart, initially at rest, was rolled down the track while attached to the system; the time it took for the cart to travel down the track was measured. It took 6.9 seconds for the cart to reach the bottom of the track for the first and third trial, and it took 6.8 seconds for the second trial.

 Figure 9: Derivation of the theoretical time needed for the cart to travel down the track from rest

The above calculation is performed when the angle was at a theoretical 40 degrees.

Discussion (Part 3):

 First, the major external forces acting on the cart were found. These forces include its weight, the tension from the string, and the normal force. Summing forces in the direction parallel to the track, an expression for tension was found in terms of theta, the carts weight, and acceleration (eq. 1). Note that the acceleration in this experiment is different from the angular deceleration caused by the frictional torque.

Then, the net torque on the system was recorded. The torques acting on the system were the frictional torques and the torque resulting from the tension in the string, which are going in opposite directions. Then, the net torque was equaled to the inertia of the system and the angular acceleration, using that relationship. Then, the torque from the string was put in terms of the tension and the radius it is pulling at from the center, and the angular acceleration was replaced with a/r. Next, the tension of the string was solved for in terms of acceleration, inertia, frictional torque, and radius (eq. 2).

Next, equations 1 and 2 were set equal to each other, and the acceleration was solved for in terms of inertia, radius, frictional torque, theta, and the mass of the cart. Then, using the kinematic equation used for the slope of the theta versus time graph in part 2, the theoretical time for the cart to roll down the track was found.  The calculation shown above for acceleration was done for a theoretical angle of 40 degrees. Using the true angle of 60 degrees, the following calculation is performed:

4.

The acceleration of the cart was found to be 0.0409 m/s/s, and the time it would take for the cart to roll down the track was 6.99 seconds.

As can be seen, the theoretical time and actual time was very close. The difference was 0.09 seconds (1.3%) between trials 1 and 3, and the difference was 0.19 seconds (2.7%) between trial 2.

Conclusion:

The sources of uncertainty in this experiment were very small, making the difference between the theoretical time and actual time it took for the cart to roll down the track from rest very small. The major source of uncertainty that caused the actual time to be slightly less than the theoretical was the slight slipping of the string from around the central shaft at around the end of the trial, where the cart was almost about to get to the bottom of the track. To reduce this uncertainty/error, the central shaft at the edges where the disk does not come into contact can be made a little rougher. Other than that error, all the other uncertainties in this experiment were so small that their error had very little to no effect on the experimental results.