The goal of this activity was to explore how torque changes with different hanging masses. A pulley system consisting of a disk, ring, and hanging bolt was used to find angular acceleration at varying masses in two different systems. From the angular acceleration, the torque of the system could then be calculated. Theoretical values of torque were calculated from Newton’s Second Law of rotational motion equation after deriving a value for the moment of inertia for both the ring and disk. The experimental and theoretical values were compared on both plots of the systems. Our group is somewhat confident in our results, as our experimental values, when compared to the theoretical, were close in some respects, but not others.
Introduction
In this experiment, the goal was to change the mass of the weight on the hanging bolt in order to see its effect on the torque of the system. Torque is a measure of the force acting on an object to make it rotate. The pulley system was used along with Pasco data studio to measure the angular acceleration at different masses. From the angular acceleration, torque can be found by also using the moment of inertia. The moment of inertia is defined as the distribution of mass with respect to an object’s axis of rotation. Moment of inertia obeys the law of superposition, the law in which the total value X of a system equals the sum of all the values of X in the system. For our purposes the following equation was used.
of the pulley is negligible)
In order to calculate the theoretical torque at different hanging masses, the moment of inertia of the system must be known. Because the moment of inertia of the pulley was considered negligible, we needed to find the moment of inertia of the disk and the ring. The following equations were used to determine the moment of inertia of the disk and ring.
(
The model for both systems were able to be created by the equation of torque using Newton’s Second law. Using Newton’s second law allowed us to determine an equation between torque, angular acceleration, and moment of inertia. The derivation is below.
Another equation was able to be derived from Newton’s Second Law using known and measurable parameters.
F = Tension
m = hanging mass
r = radius of the pulley
There are some assumptions in the model. The model assumes that there is no friction, which is not true because if friction were included, then the calculated torque would be different. The independent parameter in our model is the angular acceleration while the dependent parameter is the torque. The model is applicable to the goal because the value of torque varies with different masses. Therefore, our model reflects the change in torque when altering the hanging mass.
The interaction between the forces is explained in the force diagram below
The model has its limitations. The biggest limitation was that the force of friction and air resistance were considered negligible. Friction affects the acceleration and torque up to 15%. Furthermore, there were many other sources of error such as the string falling out of the pulley, timing mishaps in recording, and human error. There also could have been errors in the recording device. The sensor could have picked up on extraneous movement and altered the angular acceleration recorded.
Procedure
In this experiment a three pulley apparatus was used to measure angular acceleration. Independent design decisions were made in order to increase the quality of the experiment. We decided to turn the string counterclockwise in order to receive positive values of angular acceleration as opposed to negative. The angular acceleration was set to be recorded in radians per second squared instead of degrees per second squared. We also decided to wrap the string around the smaller circle because when we used the larger one, the string would fall off when the hanging mass was dropped. If there was an error in the conducting of a trial, it was redone. The individual that held the hanging mass counted down from three then dropped the mass. At the same moment, another group member began the recording program to find the angular velocity. There were five trials of five different masses that were conducted. The masses ranged from 0.025 kg to 0.045 kg. We chose to do 5 trials at every mass in order to increase the accuracy of our data. After the first set of five different masses had concluded, the same procedure was used with the additional ring resting on top of the disk.
Data
Mass (kg)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Average
STDEV
0.025
27.4
27.3
27.8
27.9
27.7
27.62
0.258843582
0.03
32.9
32.5
32.2
32.5
32.6
32.54
0.250998008
0.035
37.3
36.8
36.8
36.9
36
36.76
0.472228758
0.04
41.3
41
41.1
40.9
41
41.06
0.151657509
0.045
45.6
45.2
45.8
46
45.7
45.66
0.296647939
Mass (kg)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Average
STDEV
0.025
0.01009197
0.01026683
0.01006941
0.01006377
0.01013145
0.010124686
7.49496E-05
0.03
0.01173808
0.01176516
0.01178546
0.01176516
0.01175839
0.01176245
1.51956E-05
0.035
0.01334697
0.01338645
0.01338645
0.01337855
0.01344963
0.01338961
3.33553E-05
0.04
0.01489268
0.01491975
0.01491073
0.01492878
0.01491975
0.014914338
1.22412E-05
0.045
0.01631768
0.01635829
0.01629737
0.01627706
0.01630752
0.016311584
2.69409E-05
Mass (kg)
Trial 1 α
Trial 2 α
Trial 3 α
Trial 4 α
Trial 5 α
Average α
STDEV
0.025
6.09
6.03
6.07
6.08
6
6.054
0.037815341
0.03
7.08
7.13
7.24
7.11
7.15
7.142
0.060580525
0.035
8.31
8.25
8.22
8.13
8.11
8.204
0.083546394
0.04
9.2
9.2
9.24
9.26
9.3
9.24
0.042426407
0.045
10.4
10.3
10.5
10.3
10.4
10.38
0.083666003
Mass (kg)
Trial 1 τ
Trial 2 τ
Trial 3 τ
Trial 4 τ
Trial 5 τ
Average τ
STDEV
0.025
0.011293986
0.01129737
0.011295114
0.01129455
0.011299063
0.011296017
2.13302E-06
0.03
0.013485773
0.013482388
0.013474943
0.013483742
0.013481034
0.013481576
4.10054E-06
0.035
0.01563627
0.015641008
0.015643377
0.015650484
0.015652063
0.01564464
6.59755E-06
0.04
0.0177897
0.0177897
0.01778609
0.017784285
0.017780675
0.01778609
3.82898E-06
0.045
0.019891575
0.019901728
0.019881422
0.019901728
0.019891575
0.019893606
8.49471E-06
The standard deviation of both data sets are very low. The error bars are nearly unremarkable in the first plot and cannot be made out in the second because of this. The error bars were made by calculating the standard error of the data. The x error bars represented the standard error of the angular acceleration, while the y error bars represented the standard error of the calculated torque values. The standard deviation could have been lowered even more if we conducted more trials.
Analysis
The slope of the two plots represents the moment of inertia. The slope of the theoretical plot for the disk system was 0.000135 while the experimental was 0.0003. For the disk and ring system, the theoretical moment of inertia was .0006 while the experimental was 0.0002. The moments of inertia for the theoretical and experimental data were relatively close. The slope was determined by taking a trend line for both the plots. Both systems differ in the slope because of the law of superposition. The moment of inertia of just the disk system was lower than the disk and ring system. This is because in the disk and ring system, both moments of inertia must be added together according to the law of superposition.
While the moments of inertia between the experimental and theoretical were similar, the calculated torque values were different. This could be due to the fact that friction was neglected in the model to calculate torque. Furthermore, there could have been some human error with the recording and there could have been issues with the recording software.
Increasing the hanging mass increased the torque in the system. As seen in the data, when the mass increased, the torque did as well. The torque increased with every one of the five weight increments in the experiment. Therefore, it can be seen that increasing the hanging mass, increased the torque in the system. This is to be expected because the derived equation for torque is the following.
In the equation, when mass increases, so does the torque. Furthermore, in our data when the mass increased, the angular acceleration did as well. If the angular acceleration goes up, the torque does too.
Conclusion
The goal of the experiment was to see how changing the hanging mass affected the torque of the two systems. We were able to see in our data that an increased hanging mass led to an increased torque. The quality of the data is satisfactory because the calculated values of torque fall nicely along the lines of regression in both plots. However, our confidence in our results is less than fair because while the moments of inertia were similar, the calculated values of torque were different. This difference can be accounted for by the lack of friction in the model and other possible errors.
We learned that increasing the hanging mass increased the torque in the system. This was because increasing the mass increased the tension.
In the illustrated equation for force (tension), the mass and acceleration determine the tension force. If the mass increases, then the tension will as well, which also increases the torque.
This experiment could be improved on by including friction in the model. Our calculated torque values would have been more on par with the theoretical values if, friction was not omitted, there was less human error, and if other outside forces acting on the system were included. The experiment also could have been improved if we used more variations of masses and did more trials.