# Lab 127 Torque and Rotational Inertia

**Lab 127: Torque and Rotational Inertia**__Introduction__

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(g-a). 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).

__Experimental Procedure__

__Part I. Theoretically Determine Rotational Inertia of Disk and Ring__

- The rotational apparatus was set up with an “A” base, a mass hanger strung over the step pulley and onto a wheel and over its side. On top of the step pulley is a center shaft where various items can be placed for the trials, depending. There are 3 items that can be used, a ring, a disk, or just simply the step pulley itself without any weights. The base of this rotational apparatus is leveled.
- The computer is logged on and the USB cable of the 850 Universal Interface is connected to a USB port on the computer. The AC adapter power cord is connected to an electrical outlet underneath the lab table. The power push button on the left front corner of the interface is pressed. The green LED indicator lights up. The phone jack connector from the photogate sensor is plugged into “Digital Inputs Port 1” on the Universal Interface. The lab file “Lab 127 Torque and Rotational Inertia for 111A” file is opened.
- The masses of the disk and the ring were measured with a scale.
- The radius of the disk, the inner and outer radii of the ring are measured.
- The theoretical value for the rotational inertia of the disk about the center of mass and about the diameter are calculated and recorded in Data Table 1.

__Part II. Experimental Determination of Rotational Inertia of Disk (through center of mass) and Ring__

- Determination of the rotational Inertia of disk rotating through its center of mass
- The rotational disk was removed from the setup and the radius of the step pulley was measured with a ruler. The radius r was recorded in Data Table II.
- The disk was placed back on the center shaft with the side of the disk with circular grooves facing up.
- The mass hanger of 50 grams was attached to the end of the thread.
- The thread was wound around the pulley until the top of the 50 g mass hanger was close to the 10-spke pulley. The disk was held stationary.
- “Record” was clicked on the program interface and the disk released. After 2 rounds of trip, the computer “STOP” button was pressed.
- The slope on the linear speed v. time graph was determined and recorded. The slope was equal to the linear acceleration of the mass hanger. This value is recorded in Data Table II.
- The above steps were repeated 2 more times and the values we had averaged.
- The rotational inertia measured was the total rotational inertia not only for the disks, but for all other rotating parts like the step pulley, shaft, and 10 spoke pulley. The above steps were then repeated without the disks and hanging 10 g mass. The results were recorded in Data Table 2.
- The rotational inertia of the disk rotating through its center of mass was compared to the theoretical result.
- Determination of the rotational inertia of both disk and ring
- The disk was placed back on the center shaft with the grooved side up.
- The ring was placed on top of the disk, settled into the groove.
- The same steps as in other trials were repeated again.
- The rotational inertia of the ring only was experimentally determined and compared to the theoretically calculated result.
- Determination of the rotational inertia of disk rotating on an axis through its diameter
- The disk and ring from Part B were removed from the apparatus. The disk was placed back on the apparatus with the diameter of the disk as the rotational axis.
- Steps in previous trials were followed again.
- The same steps as in other trials were repeated again.
- The experimental rotational inertia of the disk was compared to the theoretical result.
__Results (Data and Conclusion)__

__4.1 Experimental Data__

__Table1: Theoretical Rotational Inertia__

Object |
Mass (kg) |
Radius (m) |
Rotational Inertia (kg m |

Disk |
M = 1.4248 |
0.1185 |
Through center of mass (I = 1/2MR 0.01 |

Through diameter (I = ¼ MR | |||

Ring |
M = 1.4293 |
R(inner): 0.054 R(outer):0.063 |
I = 1/2M(R 0.0098 |

__Table2__

Step pulley radius: r = 0.0175 m

Case |
Run |
Hanging Mass, m (kg) |
Linear Acceleration (m/s |
Tension (N) T = m(g-a) |
Torque(Nm) τ = rT |
Angular Acceleration (rad/s) α = a/r |
Total Rotational Inertia (kg*m I |

Disk |
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 | |

Step-pulley and shaft |
1 |
0.05 |
3.62 |
0.309 |
0.005 |
206.9 |
2.4 x 10 |

2 |
0.05 |
3.47 |
0.317 |
0.005 |
198.3 |
2.5 x 10 | |

3 |
0.05 |
3.52 |
0.314 |
0.005 |
201.1 |
2.5 x 10 | |

Disk |
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 |
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 |
% 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(g-a)

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%

__Discussion__

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}.

__Conclusion__

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.