Sample Lab Report

AP  Physics,  Fall  2013  

The Law of Conservation of Angular Momentum Data collected 4/12/13 – 4/13/13

Jill McLean, Ron Elliott, Jennifer Campbell

Any object or system that moves around an axis has angular momentum. One way to change the system without changing the total angular momentum is to lightly drop a non-rotating object onto the original system. The resulting system will rotate at a slower rotational rate, but the angular momentum can be shown to remain the same. Theory The angular momentum of a rotating object can be found if the rotational inertia of the object and its angular velocity are known, using the relation L = Iw. The calculation of I requires the selection of a formula based on the object’s shape. In this investigation, all shapes are disks with a hole in the middle therefore the formula I = 1/2M(R1

2 + R22) will be used.

A simple comparison of initial (before the drop) and final (after the drop) values for L will demonstrate conservation to a point, but a graph of Lfinal vs. Linitial will provide the general trend and provide a visual for the conservation of angular momentum. Ideally, the slope of this graph will be 1, representing 100% conservation. A value greater than 1 might indicate that the second object had an initial rotation, while a slope less than 1 might suggest losses due to an outside force such as friction.

Data Three metal disks were used within this experiment . In order to calculate values of rotational inertia, dimensions and mass were measured for each disk. Values of radius were measured using a ruler to find diameter first and then divide by two. A triple beam balance was used to measure mass. The calculation of values of I is shown in more detail within the analysis.

Note that disks 1 and 2 have very similar properties, including their rotational inertias. After mounting a Vernier rotational motion sensor with the axle pointing upward, disk 1 (aluminum, small hole) was attached. The disk was given a small angular velocity and then the second aluminum disk (disk 2) was dropped carefully onto the first. It was important to simply drop disk 2, without any initial angular speed. From a graph of w vs. t produced by the LabQuest, we found the value of w before and after disk 2 was dropped.  

Graphs of w vs. t were used to extract values of angular velocity before and after dropping. A sketch of a typical LabQuest screen is shown here. The stylus was touched to particular locations on the graph, marked with circles, and the value of w was recorded.  

disk 2

disk 1

rotational motion sensor

Disk 2 dropped onto Disk 1. Multiple trials were used so that a graph could be produced.  

The experiment was repeated using a much heavier disk (disk 3) instead of disk 2.

Most data seems to follow a general pattern that when winitial is greater, wfinal is also greater. However, trials 1 and 6 in the second set deviate. One of these will likely be ignored on the final graph. Since Disk 2 had nearly the same rotational inertia as Disk 1, the final I is about double the initial I. Therefore it seems logical that the value of w cuts in half during the first set of trials.

Analysis As mentioned in the theory section, the formula I = 1/2M(R1

2 + R22) was used to

calculate values of rotational inertia. For example, using data for disk 1:

I = 1/2M(R12 + R2

2) = I = 1/2(0.1064 kg)((0.044 m)2 + (0.0025 m)2) = 1.03 x 10-4 kg·m2

To calculate values of angular momentum, L = Iw is used. Using data from trial 1 of data set 1:

L = Iw = (1.03 x 10-4 kg·m2)(40.336 rad/s) = 4.15 x 10-3 kgm2/s

Note: When calculating Lfinal, the combined I for both disks must be used. For example, using data from trial 1 of data set 1:

L = Iw = ((1.03 +1.02) x 10-4 kg·m2)(21.506 rad/s) = 4.41 x 10-3 kgm2/s

Disk 2 dropped onto Disk 1  

The slope of the line is very close to 1, as expected.  

Disk 3 dropped onto Disk 1  

The slope of this line is not very close to 1. As mentioned earlier, two of the trials do not agree with each other in terms of a logical pattern.  

A second version of this graph was produced with the elimination of trials 1 and 6. The results are much more logical, with a slope much closer to 1.  

Summary When no outside force or torque acts on a rotating system, we expect that the system’s angular momentum will remain constant, even if the system is suddenly expanded to include additional objects. Therefore, when we dropped a non-rotating disk onto a rotating one, we expected the angular momentum to remain the same. After eliminating two data points, it does appear that angular momentum is conserved. Looking at the first set of trials, the final value of L was 95% of the initial. For the second set, the final was over 99% of the initial. Ideally these would have been 100%, but results appear to support the law of conservation of angular momentum. The fact that better results were found for the second set of trials suggests that the experimenters understood better at the end how to drop the disk within providing any initial rotation. The second set also involved a heavier dropped disk so its inertia would have made it easier to keep it from rotating as it was being dropped. Reproducing the first set of trials may by itself improve the quality of the data. However, the way the w values were chosen from a graph must not be ignored. The exact locations along the graph involve a judgment by the experimenter and therefore invite an inconsistent source of error. Using the Vernier’s max/min capabilities may have been a better method. In conclusion, the data showed to a high degree of consistency that angular momentum is conserved in this type of system.