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(R12 + 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. disk 2
disk 1
rotational
motion sensor
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 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(R12 + R22) was used to
calculate values of rotational inertia. For example, using data for disk 1:
I = 1/2M(R12 + R22) = 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.