Angular force is measured in a quantity torque, represented with the symbol τ. Torque is said to be applied when a force is exerted on a rigid body pivoted around an axis. Rotational motion can be described with the equation Στ = I α, where counterclockwise rotation means torque is positive and clockwise motion means torque is negative. I is rotational inertia or object resistance to rotation, and α is angular acceleration. The rotational inertia of a rigid object (a system of particles) is defined as : I = Σm_{i}r_{i}^{2}, where m_{i} is the mass of the ith particle and r_{i} is the radius for the ith object from the rotation axis.
Once the mass distribution around a point of rotation is known, rotational inertia can be calculated. Theoretically, the rotational inertia of a disk rotating through its center of mass is I = 1/2MR^{2} and one rotating around its diameter has a rotational inertia of 1/4 MR^{2}. The theoretical rotational inertia of a ring is 1/2M(R_{1} + R_{2})^{2}, where M is the mass of the ring, R_{1} is the inner radius of the ring and R_{2} is the outer radius of the ring.
To experimentally determine rotational inertia, we used the equation I_{total} = τ/ α where torque is caused by the weight hanging from the string, wrapped around the 3 step pulley of the rotational apparatus. We also used the equation τ = rT, where r is the radius of the step pulley and T is the tension when the apparatus is rotating. In addition to both these equations, the equation a = r α was also used where a is the linear acceleration of the string.
Applying Newton’s Second Law for the hanging mass, the equation for tension can also be derived as T = m(ga). Once the linear acceleration of the hanging mass was experimentally determined, the tension (T), torque (τ = rT) and angular acceleration (α = a/r) were obtained to determine the total rotational inertia (I_{total} = τ/ α).
Part I. Theoretically Determine Rotational Inertia of Disk and Ring
Part II. Experimental Determination of Rotational Inertia of Disk (through center of mass) and Ring
4.1 Experimental Data
Table1: Theoretical Rotational Inertia
Object 
Mass (kg) 
Radius (m) 
Rotational Inertia (kg m^{2}) 
Disk 
M = 1.4248 
0.1185 
Through center of mass (I = 1/2MR^{2}) = 0.01 
Through diameter (I = ¼ MR^{2}) = 0.005 

Ring 
M = 1.4293 
R(inner): 0.054 R(outer):0.063 
I = 1/2M(R_{1}+R_{2})^{2} = 0.0098 
Table2
Step pulley radius: r = 0.0175 m
Case 
Run 
Hanging Mass, m (kg) 
Linear Acceleration (m/s^{2}) 
Tension (N) T = m(ga) 
Torque(Nm) τ = rT 
Angular Acceleration (rad/s) α = a/r 
Total Rotational Inertia (kg*m^{2}) I_{total} = τ/ α 
Disk^{1} 
1 
0.35 
0.0579 
3.41 
0.0597 
3.309 
0.018 
2 
0.35 
0.0576 
3.41 
0.0597 
3.291 
0.018 

3 
0.35 
0.0577 
3.41 
0.0597 
3.297 
0.018 

Steppulley and shaft 
1 
0.05 
3.62 
0.309 
0.005 
206.9 
2.4 x 10^{5} 
2 
0.05 
3.47 
0.317 
0.005 
198.3 
2.5 x 10^{5} 

3 
0.05 
3.52 
0.314 
0.005 
201.1 
2.5 x 10^{5} 

Disk^{1} + Ring 
1 
0.35 
0.0372 
3.42 
0.0599 
2.126 
0.028 
2 
0.35 
0.0384 
3.42 
0.0599 
2.194 
0.027 

3 
0.35 
0.0376 
3.42 
0.0599 
2.149 
0.028 

Disk^{2} 
1 
0.35 
0.109 
3.40 
0.0595 
6.229 
0.0096 
2 
0.35 
0.109 
3.39 
0.0593 
6.229 
0.0095 

3 
0.35 
0.107 
3.39 
0.0593 
6.114 
0.0097 
Disk^{1} = Disk through center of mass, Disk^{2 } = Disk through diameter
Table 3
Case 
Rotational Inertia (kg*m^{2}) 
% difference 

Theoretical 
Experimental 

Disk through center of mass 
0.01 
0.018 
80% 
Disk through diameter 
0.005 
0.0096 
92% 
Ring 
0.0098 
0.01 
2.04% 
4.2 Calculations
The first 2 columns of data table 3 were given or measured in experiment. The following 4 columns were calculated with the equations in the column head titles. Sample calculations are as followed:
(All for trial 1 run 1)
Tension (N), T = m(ga)
T = (0.35 kg)(9.8 m/s^{2}0.0579 m/s^{2}) = 3.41 N
Torque(Nm), τ = rT
τ = (0.0175 m)(3.41 N) = 0.0597 Nm
Angular Acceleration (rad/s), α = a/r
α = (0.0579 m/s^{2})/(0.0175 m) = 3.309 rad/s^{2}
Total Rotational Inertia (kg*m^{2}), I_{total} = τ/ α
I_{total} = τ/ α = 0.0597 Nm/3.309 rad/s^{2 }= 0.018 kg*m^{2}
4.3 Error Analysis
There was a lot of error in this experiment. Most of the error was probably caused by performing the trials with weights that were too heavy, potentially affecting the results when unintended. A sample error calculation is shown below:
Sample error calculation for Table III, Disk through center of mass
E_{abs} = observed/experimental value – theoretical value
E_{abs} = 0.018 kg*m^{2} – 0.01 kg*m^{2} = 0.008 kg*m^{2}
0.018 kg*m^{2} ± 0.008 kg*m^{2}
E_{rel} = E_{abs }/theoretical value
E_{rel} = 0.008 kg*m^{2} / 0.01 kg*m^{2}
E_{rel} = 80%
The theoretical rotational inertia for selected objects were not close at all to the experimental results. In all likelihood, performing the trials with weights that were too heavy could have affected the results negatively, spurring a greater linear acceleration which might have created increased tension, angular acceleration, and rotational inertia for the trials with 350 g weights. Had we performed the trials with lighter weights, the results probably would have been a lot more accurate and closer to the theoretical. It is notable that the calculation for the rotational inertia of the ring (calculated by subtracting the experimental disk’s rotational inertia from the combined rotational inertia of the disk and the ring) had very little error, leading me to believe the faster trials really were to blame for the irregular results. Since the same weights were used for both the disk and disk and ring trials, it is possible the initial difference in speed would have been canceled out after the disk’s rotational inertia was subtracted, leaving a rotational inertia that was the difference, the difference being a more accurate calculation of inertia for the ring. Were the experiment to be performed again, I would suggest slowing down the trials with lighter weights and closely monitoring the system to make sure any intended increase in acceleration does not affect results erratically.
Besides that, sources of error could have included errors of reading the velocity measurements on the screen, error in manually recording and measuring data, error in setting up the experiment, using incorrect weight amounts from ones intended, using the wrong items for the trials (disk, ring) or placing them in the wrong orientations (upright or horizontal), failing to check for possible defects in equipment like the weight hanger, chips in rotational disks and the base, among other things. There could have also been mechanical error in the electronic equipment, like malfunctioning programs or broken parts etc. Systematic error could have occurred because supposing the massing device was not zeroed properly, the base was not fully leveled, or the 10 spoke pulley was tilted a little askew from the line of motion, results could have been skewed up or down, depending. Random errors could have occurred as well. It was easy for the string on the pulley to wind around the pulley in unintended ways sometimes, and some trials may have been performed without noticing this error. For some trials, the mass hanger was not always brought up to the exact same position above the ground. For these reasons, there could have been some unpredictable error with results.
There was a little possible error incurred by ignoring the rotational inertias of the pulley. For a hoop rotating around a cylindrical axis, as was the case here, the rotational inertia is equal to MR^{2}. If the mass of the wheel was estimated to be around 10 grams, and the radius around 3 cm, the rotational inertia of the wheel (omitting spokes) would be around 9 millionth kg m^{2}.
As this lab is concluded, I now know how to experimentally determine the rotational inertia of a rotating body by measuring angular acceleration and applying the equation Στ = I α.I also know how to calculate rotational inertias for different objects. The results were not optimal since the trials were performed with too many weights, but they were close enough that I could see how running slower trials could affect my results. Since I performed the theoretical calculations and understand how they work, I have come through with an understanding about how to accurately perform the lab, were it to be done again and of the underlying concepts the lab tries to teach. Next time, I would be sure to slow down trials and observe results more closely.
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