The purpose of this experiment was to determine the components of applying a known torque on a rotational object that influence the angular acceleration of that object. Using this information, the experimental moment of inertia of the rotating object can be found.
Apparatus:
Figure 1: Apparatus of this experiment
The apparatus of this experiment consisted of a rotational stand, where metal disks are placed on the stand for rotation. Only one disk or multiple disks can be rotated together on the stand through the use of running air through a hose clamp; when the hose clamp is open, the bottom and top disks rotate independently of each other. However, when the hose clamp is closed, both the bottom and top disk will rotate together. In order to allow the disk(s) to rotate with the air on, a drop pin had to be placed in the center of the disks where the rotational stand is. Air is also used to decrease greatly any friction between the disks. Next to the rotational stand is a rotational sensor. On the sides of the disks there are 200 marks, which the rotational sensor reads and uses to measure the change in the angle of the disks (theta) and the angular velocity of the disks (omega). The rotational sensor reads 200 counts each rotation of the disks. On the end of the apparatus is a nearly frictionless pulley, where a hanging weight's string is placed to allow for the smallest energy loss possible in the system.
A computer with LoggerPro software was also used to obtain and store the data measured by the rotational sensor. An analytical balance was also used to measure the mass of the disks and the torque pulleys (the torque pulleys are placed between the disks and drop pins). Lastly, to measure the diameters of the disks, the following caliper was used:
Figure 2: Large caliper used to measure the diameters of the disks
A smaller caliper was used to measure the diameters of the torque pulleys.
Procedure (Part 1):
The diameters and masses of all the disks and pulleys used in this experiment were measured. The data is shown below:
- diameter and mass of top steel disk: d = 12.630 cm ; m = 1360 g
- diameter and mass of bottom steel disk: d = 12,630 cm ; m = 1348 g
- diameter and mass of top aluminum disk: d = 12.630 cm ; m = 465 g
- diameter and mass of smaller torque pulley: d = 2.81 cm ; m = 10.0 g
- diameter and mass of larger torque pulley: d = 5.36 cm ; m = 36.3 g
Then, LoggerPro was set up and the rotational sensor was hooked up to the computer. Once the equipment was all set up, the air was turned on to the apparatus with the hose clamp open so that the bottom and top disk rotated independently of each other. Then, once the string was wrapped around the torque pulley and the hanging mass was at its highest point, the data measurements was started and the mass and disks were released. Six experiments were performed, with one component changed in each experiment. Experiments 1, 2 and 3 involved changing the weight of the hanging mass, experiments 1 and 4 involved changing the radius of the torque pulley, and experiments 4,5 and 6 involved changing the mass of the rotating disks. Using the angular velocity graphs obtained from the experiments, the angular accelerations of the rotating disks as the mass was going upward and as the mass was going downward were found by looking at the slopes of the angular velocity graphs. The angular velocity graphs obtained as well as data obtained for each experiment is found below:
Figure 3: Angular velocity graph of experiment 1
α up = 1.080 rad/s/s
α down = 1.189 rad/s/s
Figure 4: Angular velocity graph of experiment 2
α up = 2.366 rad/s/s
α down = 2.194 rad/s/s
Figure 5: Angular velocity graph of experiment 3
α up = 4.069 rad/s/s
α down = 3.051 rad/s/s
Figure 6: Angular velocity graph of experiment 4
α up = 2.312 rad/s/s
α down = 2.080 rad/s/s
Figure 7: Angular velocity graph of experiment 5
α up = 6.938 rad/s/s
α down = 6.238 rad/s/s
Figure 8: Angular velocity graph of experiment 6
α up = 1.157 rad/s/s
α down = 1.056 rad/s/s
Figure 9: Data obtained for experiments 1 through 6
To ensure that the angular acceleration obtained by the rotational sensor is the actual acceleration value, the acceleration of the hanging mass was obtained using the velocity graph obtained from a motion detector through its slope. This test was performed for experiment 5. The velocity graph is displayed below:
Figure 10: Velocity graph of the hanging mass in experiment 5
a = 0.1652 m/s/s
Discussion (Part 1):
It was expected that the angular acceleration when the hanging mass was accelerating upward was larger than the angular acceleration when the hanging mass was accelerating downward. This is because the net torque on the rotating disk is larger when the hanging mass is accelerating upward than when accelerating downward. When the hanging mass is accelerating upward, the torque applied on the rotating mass by the tension in the string of the hanging mass is in the same direction (clockwise) as the torque from friction (since both are slowing down the rotating disk), resulting in a larger net torque. A larger net torque results in a larger angular acceleration (torque = moment of inertia x angular acceleration) since moment of inertia is constant in this experiment. On the other hand, when the mass is accelerating downward, the torque from the hanging mass is in the opposite direction of the torque from friction; the torque from the mass is speeding up the rotating disk while the torque from friction is slowing it down. Therefore, a smaller net torque is obtained. Friction in the system is seen from the graphs; the slopes are not perfectly straight lines in that there is some small zagging up and down in the graphs. Therefore, in order to help cancel the effect of friction on the system, the average angular acceleration for each experiment was taken by taking the average of the upward and downward angular acceleration.
Looking at experiment 5, in order to determine if the acceleration measured is the true acceleration, the relationship between angular acceleration and linear acceleration can be used to find the "experimental" radius, and see if that radius agrees with the measured radius of the pulley. Doing so,
1. 
The radius of the large torque pulley used for that experiment was 2.68 cm. The values are fairly close, proving that the acceleration data gathered is the true acceleration of the rotating mass. The reason for the small difference between the two radii is uncertainties in measurements as well as the friction in the system.
Taking a look at experiments 1 through 3, it can be seen that the hanging mass does affect the rotational acceleration of the disks. when the mass is nearly doubled (from 24.6 g to 49.6 g), the angular acceleration nearly doubles (from 1.135 rad/s/s to 2.280 rad/s/s), and when it triples (from 24.6 g to 74.6 g), the angular acceleration nearly triples (from 1.135 rad/s/s to 3.560 rad/s/s) as well. Therefore, the hanging mass is directly proportional to the angular acceleration of the rotating disks.
Looking at experiments 1 and 4, it can be seen that the radius of the torque pulley also has an effect on the angular acceleration. When the radius of the pulley was nearly doubled (from 1.40 cm to 2.68 cm), the angular acceleration nearly doubled (from 1.135 rad/s/s to 2.196 rad/s/s). Therefore, the radius of the torque pulley is proportional to the angular acceleration of the rotating disks.
Lastly, looking at experiments 4 through 6, it can also be seen that the mass of the rotating disks affects the angular acceleration. when the mass was made about three times lighter (from 1360 g to 465 g), the angular acceleration was nearly tripled (from 2.196 rad/s/s to 6.588 rad/s/s). In addition, when the mass was nearly doubled (from 1360 g to 2708 g), the angular acceleration was about twice as slow (from 2.196 rad/s/s to 1.107 rad/s/s). Therefore, the rotating mass is inversely proportional to the angular acceleration.
Procedure (Part 2):
Using the data obtained from part 1, the moments of inertia of the disks or disk combinations in each experiment were found. The experimental values for moment of inertia were found by using the derived equation shown below:
2. 
For comparison of the experimental moment of inertia to see its accuracy, the actual moment of inertia of the rotating mass was found using the following formula:
3.
The calculations are shown below:
Figure 11: Experimental and actual moments of inertia of exps 1/2
Figure 12: Experimental and actual moments of inertia of exps 3/4
Figure 13: Experimental and actual moments of inertia of exps 5/6
Discussion (Part 2):
Looking at experiment 1, the percent error between the experimental value of moment of inertia (0.00298 kg m^2) and the actual value (0.00271 kg m^2) is 9.96%. Looking at experiment 2, the percent error between the experimental value of moment of inertia (0.00299 kg m^2) and the actual value (0.00272 kg m^2) is 9.93%. Looking at experiment 3, the percent error between the experimental value of moment of inertia (0.00287 kg m^2) and the actual value (0.00271 kg m^2) is 5.90%. Looking at experiment 4, the percent error between the experimental value of moment of inertia (0.00292 kg m^2) and the actual value (0.00271 kg m^2) is 7.75%. Looking at experiment 5, the percent error between the experimental value of moment of inertia (0.000964 kg m^2) and the actual value (0.000927 kg m^2) is 3.99%. Lastly, looking at experiment 6, the percent error between the experimental value of moment of inertia (0.00583 kg m^2) and the actual value (0.00540 kg m^2) is 7.96%.
Although the percent error varies between the experiments, the percent error was less than 10%, indicating that the values are accurate and that the error is due to uncertainty. There is more percent error in some experiments than others, showing that there is more uncertainty in some experiments than others. These uncertainties are found in measurements of dimensions (different equipment used to measure dimensions of different parts of the apparatus) as well as varying friction (due to the use of different disks or disk combinations and the use of different torque pulleys).
Conclusion:
There are some sources of uncertainty/error in this experiment. These uncertainties/errors include:
- uncertainty in the measurement of the diameters of the disks and torque pulleys
- uncertainty in the measurement of the mass of the disks and torque pulleys
- the use of different equipment with different disks and torque pulleys depending on the size and mass (using different analytical balances and calipers between the larger disks and the smaller disks and torque pulleys)
- uncertainty in the measurement of the angular acceleration of the rotating mass by the rotational sensor
- uncertainty in the measurement of the linear acceleration of the hanging mass by the motion detector
- friction between the disks, and different friction between the disk and the rotational stand
In order to lower the uncertainty, better equipment would be needed. In addition, higher experimental values would also be needed to lower the effect of error and uncertainty on the results.
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