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The ‘spinning disk touches stationary disk’ problem revisited:
an experimental approach
Mário S.M.N.F. Gomes1, Pablo Martín-Ramos1,2, Pedro S. Pereira da Silva1,
Manuela Ramos Silva1*
1 CFisUC, Department of Physics, FCTUC, Universidade de Coimbra, P-3004-516 Coimbra,
Portugal. Phone: +351239410648. Email: manuela@uc.pt.
2 EPS, Universidad de Zaragoza, Carretera de Cuarte s/n, 22071 Huesca, Spain.
Abstract
A popular Newtonian Mechanics problem, featured in textbooks, Physics Olympiads and
forums alike, concerns two disks with different radii and moment of inertia that rotate differently
and that touch each other. Most students struggle to calculate the final angular velocity of the
disks, erroneously attempting to use different conservation laws. In this paper we propose a simple
experiment that should help Physics teachers explain this challenging exercise in an engaging
way for the students. By using a smartphone/tablet and video analysis tools, the angular velocity
of both disks can be easily tracked as a function of time, clearly showing the three stages of the
interaction (before the collision, with only one disk rotating; the collision of the disks with
slippage; and the rotation of the two disks in harmony, without frictional forces in between).
Processing and plotting of the data in a spreadsheet immediately shows which quantities are
conserved and which are not. Several extensions to the core experiment are also suggested.
Keywords: disks; friction; Newtonian Mechanics; smartphone; video analysis.
1. Introduction
Most engineering degrees start with a Newtonian Physics course, and such introductory
Mechanics course aims to provide students with an understanding of the fundamental concepts of
Physics and their interrelationships. Concepts like force, torque, or angular momentum and their
relationships (including conservation laws) are first introduced when studying a particle, then a
rigid body, and finally a system of rigid bodies. An introductory course should also teach students
to diagnose which of the conservation laws to apply in a given situation, if any. The problem with
two spinning discs that touch (and the calculation of their final velocities), frequently found in
textbooks as a challenging problem, establishes a fertile situation for a thorough discussion of
what conservation laws to apply. Most students find the problem very difficult and it is common
to find in web platforms (i.e., Physics Stack Exchange, PhysicsForum, …) several enquiries about
the correct solution. Actually, this problem has been featured in three video posts by Walter H.G.
Lewin, former Physics Professor at MIT, in YouTube (1-3).
In this paper we show how the problem can be tackled experimentally, in an easy way, using
video analysis. In recent years, video analysis has emerged as a very useful tool for slowing down
(4-11) or quickening (12, 13) movements that otherwise would be difficult to follow with the
naked eye. Video analysis allows the student to monitor the rotation of the two disks, to retrieve
hundreds of (t, θ1, θ2) measurements, so as to easily identify linear or accelerated movements, and
to probe what physical quantities are actually being conserved during the interaction of the disks.
2. Theoretical background
2.1. Problem formulation
A common phrasing of the problem (Scheme 1) is:
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‘A disk of radius R and moment of inertia I1 rotates with angular velocity ω1. A second
disk, of radius r and moment of inertia I2 is at rest. The axes of the two disks are parallel.
The disks are moved together so that they touch. After some initial slipping, the two disks
rotate together. Find the final rate of rotation of the smaller disk’.
Scheme 1. Nomenclature used in the problem formulation.
In this situation, the following forces act upon the system (depicted in Scheme 2): the
gravitational forces (
and
), vertical, pointing to the centre of Earth; friction forces (
and
), in the horizontal plane, tangent to the disks, opposing the rotational movement, equal in
magnitude; normal reaction forces with a vertical component (cancelling the gravitational force,
and
) and an horizontal component, exerted by the axles into the disk, cancelling the
frictional force (
and
).
Scheme 2. Forces and moments involved.
2.2. Popular misconceptions
Those students who think of the problem as a collision of two bodies try to use the conservation
of momentum first (often an extremely powerful tool in solving problems): in fact, no net external
force acts on the system of the two disks, and therefore the total linear momentum of the system
cannot change. Nonetheless, this reasoning soon gets to a dead end: the centre of mass of either
disks, or of the system, is not moving, before or after the collision, and therefore the value for the
rotation of the small disk cannot be retrieved by using of this law.
Other students try to approach the problem by using the conservation of energy, but –for a
system of bodies– the law states: the work done by all forces, internal and external, applied to a
system of bodies in a certain time interval equals the variation of kinetic energy of the system,
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during the same interval. Since there is work done by the frictional forces, the kinetic energy is
not conserved.
When the two aforementioned laws fail, students turn their attention to the conservation of the
angular momentum of the system, which can be applied in the case of a falling disk into a rotating
disk, because in that situation no external net torque acts on the system along a specific (the
rotation) axis, and the angular momentum of the system remains constant. In the present situation,
however, for any of the two axis that we could choose, there is a non-zero torque from the
horizontal
components.
2.3. Correct solution
Since aforementioned conservation laws fail, the problem has to be solved by rotation
dynamics equations, applied to both disks, and bearing in mind that the two frictional forces,
and
, are a third-law force pair.
Therefore, for each disk, the torque exerted by the frictional force will cause an angular
acceleration
[1]
[2]
Because the frictional forces are a third law pair, they are equal in moduli and using the
definition of angular acceleration, one gets
⟹
0 [3]
0⟹
constant [4]
that is,
is a conserved quantity.
If so, it will remain constant before and after the collision, being and the angular
velocity of disk 1 and disk 2, respectively, before the collision and
and
, the velocities of
disks 1and 2 after the collision.
[5]
When the slippage is over, the frictional forces become zero, and
0 6
⟹
7
⟹
8
3. Experimental
A complete rotational system ME-8950A (Pasco, Roseville, CA, USA) was adapted for the
experimental procedure presented herein. Furniture wheels may be used instead as a cheap
alternative, as discussed in the extensions to the activity (see section 5).
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Figure 1. Left: the two disks used for the experiment (the screw was removed from the smaller disk); right:
experimental setup.
The 22.8 cm diameter disk (dark grey, Figure 1) and the 7 cm diameter small black disk
(intended to be a 900 g counterweight) were mounted on two cast iron bases using the rotating
axles included in the kit. The large disk was spun at a certain angular speed and then the small
disk –initially at rest– was approached till they touched. Small pieces of white tape were stuck on
their periphery so as to enable rotation tracking through video analysis.
A tablet’s camera was used to record the experiment (a Lenovo Tab A10, 5 MPx rear camera)
from a zenith position (see Figure 2). The video file was then processed using Tracker –a free
open source video analysis and modelling tool–, and the data obtained was exported to a .csv file,
so as to complete the data processing and plotting steps in a MS Excel spreadsheet. OpenOffice
Calc may be used instead, but the former may be more ubiquitous and easily understandable (14).
Figure 2. Experiment recording with a tablet.
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4. Results and discussion
Several trials were performed changing the initial velocity of the big disk, disk 1, or changing
its moment of inertia. Figure 3 shows the results of the video analysis of one of such trials, in
which a point on each of the disks periphery was followed with time. Their angular position times
their radius is plotted in Figure 3(a) and its derivative in Figure 3(b). The first region of the graph
corresponds to the interval before the collision, with only disk 1 rotating; the second interval (with
a light grey background) corresponds to the collision of the disks with slippage; and in the third
region the disks rotate in harmony, without frictional forces in between.
123456789
-150
-100
-50
0
50
100
123456789
0.0
0.5
1.0
1.5
2.0
angle (rad)
Time (s)
(a) (b)
Angular velocity radius
(radꞏs
-1
ꞏm)
Time (s)
ꞏR
|ꞏr
Figure 3. (a) Angle as a function of time for the two disks; (b) angular velocity×radius plot. 1 out of 3
points has been plotted for clarity reasons.
The rotation kinetic energy of each of the disks and their sum are depicted in Figure 4(a). As
predicted above, the energy decreased while the slippage occurred. For the angular momentum
(see Figure 4(b)), there was a decrease for the big disk and an increase (in module) for the small
disk. A careless sum of their values (as the one that students would attempt) would also show a
decrease while friction occurs.
123456789
0.0
0.5
1.0
1.5
123456789
0.00
0.05
0.10
0.15
Energy (J)
Time (s)
E
1
E
2
E
system
(a) (b)
Angular momentum
(kgꞏm
2
ꞏs
-1
)
Time (s)
L
1
|L
2
|
L
system
Figure 4. (a) Energy and (b) angular momentum plots, for each of the disks and their sum. 1 out of 4 points
is plotted for clarity reasons.
Figure 5 represents
,
||, and
as a function of time. One can see
that while the former two vary linearly during the slippage interval, the later remains constant
throughout the interaction.
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123456789
0.00
0.25
0.50
0.75
1.00
1.25
1.50
(I/radius)
Time (s)
(I
1
/R)ꞏ
(I
2
/r)ꞏ|
(I
1
/R)ꞏ
(I
2
/R
2
)ꞏ
Figure 5. (I/radius)ꞏω for each of the disks and conserved quantity. 1 out of 3 points is plotted for clarity
reasons.
To make calculations from the experimental data retrieved from the video analysis, the value
of the moment of inertia for disk 1 was taken from the PASCO manual ( 0.0091 kgꞏm2) and
that of the counterweight (disk 2) was experimentally determined by coiling a string around it and
letting a weight, tied to the other end of the string, fall. The moment of inertia was found to be
6.310
kgꞏm2.
By fitting the curves in Figure 3(a) with a second order polynomial function in the slippage
stage, one can retrieve the value of the angular acceleration, and –from that– the modulus of the
frictional force. For the run shown above, the retrieved values were α1=2.513 radꞏs-2 and
α2=10.829 radꞏs-2, and the calculated frictional forces were 0.20 N and 0.195 N, respectively. This
is a convenient way for the students to confirm the goodness of their data.
Likewise, fitting of the curves in the third interval with a linear fit yields the values of
ω1=7.515 radꞏs-1 and ω2=24.866 radꞏs-1. By multiplying by their respective radii, one gets the
value of the velocity of a point at the periphery (0.86 mꞏs-1 in both cases), confirming that the
video analysis approach is accurate enough.
5. Alternative procedures and suggested extensions to the proposed experiment
5.1. Different speeds
If the experiment is repeated spinning disk 1 at different speeds, one can probe the relationship
in equation 8 by plotting the observed final angular velocity of disk 2 and that calculated from the
initial angular velocity of disk 1. Such plot is shown in Figure 6, together with its linear fit. The
slope of such fit, as expected, was close to 1 (0.93, with R2=0.9993).
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0.0
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8
[(I
1
ꞏrꞏR)/(R
2
ꞏI
2
+r
2
ꞏI
1
)]ꞏ
1
(radꞏs
-1
)
2
'| (radꞏs
-1
)
Figure 6. Plot of |
| as a function of ω1, using equation 8.
5.2. Estimation of the frictional effect of the axles
Since the axles are not completely frictionless, another possible extension would consist in
tracking the movement of the disks until they completely stop. Since the friction is small, so that
the video to be processed is not very long, it is convenient to spin the disk at a small/medium
initial speed. A fit of the curve θ(t) after the slippage interval until the disk stops with a second
order polynomial yielded for our setting a value of -0.482 radꞏs-2 angular acceleration.
5.3. Using the small disk to move big one
In a classroom, some of the groups may also try spinning the smaller disk instead of the larger
one, and should reach similar conclusions.
5.4. Low-cost option with furniture wheels
If the laboratory is not equipped with nice disks and their almost friction-free rotation axles, it
is possible to use a cheap alternative. A set of 5 different wheels would cost less than 20 euros,
and they stand up by their own while their axles remain horizontal. The initial rotation and the
collisions are easy to perform, as well as the video analysis of a point in the periphery. As noted
above, the moment of inertia may be determined by a simple experiment in which a string is rolled
up around the disk, and a weight at its free end is left to fall. Figure 7 shows one of these setups
and the corresponding results. The major advice, if this setup is to be used, is to look for smooth
rotations at the DIY shop.
3456
0.0
0.5
1.0
1.5
Angular velocity radius
(radꞏs
-1
ꞏm)
Time (s)
Big wheel
Small wheel
Figure 7. Left: alternative experimental setup; right: angular velocity × radius plot for the furniture wheels.
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5.5. Tracking in the smartphone/tablet with VidAnalysis or Vernier Video Physics apps
If the teacher prefers to conduct the video analysis directly in the students’
smartphones/tablets, VidAnalysis (15) free app can be installed in Android devices. This intuitive
and easy-to-use app requires the setting of the x-y axes, a length scale and the tracking of the
position of the marker through screen touching, frame by frame. The app can export the data to a
.csv, in a similar fashion to how Tracker does. Vernier Video Physics (16) paid app would offer
similar features for iOS devices.
It should be clarified that a tablet would be the preferred option if any of these apps is to be
used instead of Tracker, given that the larger size of the screen –as compared to a smartphone–
would improve the precision when touching with the finger on the tape in the different frames.
5.6. Using the in-built smartphone’s accelerometer instead of video tracking
If the teacher prefers to avoid video analysis, it should also be possible to use the in-built
accelerometer of the smartphone and apps such as Physics Toolbox Gyroscope (17) to obtain the
angular velocity as in references (18-21).
6. Summary and conclusions
Video tracking is a powerful tool for analysis the movements difficult to follow with the naked
eye. By applying it to a challenging Newtonian Mechanics problem, it can help the students
retrieve real data and, by its subsequent processing and plotting, gain insight on the actual physical
parameters involved. Several possible extensions, including a low-cost alternative to commercial
equipment, are also suggested, making it adaptable to students of different ages.
7. Acknowledgments
CFisUC gratefully acknowledges funding from FCT Portugal through grant
UID/FIS/04564/2016. P.M-R would like to thank Santander Universidades for its financial
support through the “Becas Iberoamérica Jóvenes Profesores e Investigadores” scholarship
program.
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