Measuring the Moment of Inertia of a Bicycle Wheel
Measuring the Moment of Inertia of a Bicycle Wheel
Alexandre Dub, (# 110234667)
Paris Hubbard Davis, (# 110222984)
Philippe Roy, (# 110235920)
Guillaume Rivest, (# 110227286)
McGill University, February 3, 2002
ABSTRACT
FIGURE 1:Measuring the Moment of Inertia of a Bicycle Wheel.
Illustration of the basic setup for the experiment. A more detailed
scheme is presented in the third section, Apparatus and
Procedure.Moment of inertia is a quantity which varies as the axis
rotation varies, or, more clearly, as the distance from the axis
varies. Our goals were two: 1) experimentally measure the moment of
inertia and then check this result with a theoretical measure, and
2) play with the bicycle wheel. The former was conducted with
weights exerting torques about the rim for a revolution of the
wheel. The resulting period was measured, and via conservation of
energy, moment of inertia was derived. The latter consisted of a
summation: the weight and distance from the centre of the various
parts of the wheel: hub, spokes, nipples, and rim. The conservation
of energy method and a good approximation of the theoretical value
were consistent within error.I. INTRODUCTION
II. THEORY
The experimental derivation of moment of inertia was conducted
by equating energy at two different times. Our first approximation
is that g is constant. We set the zero of potential energy to be at
the floor, and energy is given by:
(1)1The period of the wheels half revolution ( radians) is
measured, allowing us to derive the angular velocity as
follows:
= / t(2)
Of course, initially velocity and angular velocity are zero.
Next, we consider the energy just after the falling mass has
touched the ground. We approximate that the period measured for the
subsequent torque-free revolution is equal to this quantity (i.e.
frictional loss is minimal over a revolution).
(3)
Next, we assume that energy is conserved (i.e. frictional loss
is minimal), and therefore
(4)
The velocity of the falling mass is related to the angular
velocity of the wheel. From rotational dynamics, we recall that the
velocity of any point on the rotating body (the thread at edge of
the rim, for example) is given by:
(5)2Thus we may calculate the velocity of the falling mass just
before impact. By measuring h, m, and one may calculate I, moment
of inertia.
Theoretically, more approximations are made. The summation is
performed as though the distance between the flanges was zero, i.e.
the wheel rotates in a plane. Each element of mass contribution was
calculated via the relationships
(6)
(7)
Deftly written as one in the form adopted by Kleppner and
Kolenkow, (I = mjj2). We approximate that the mass of the rim is
centered at a radius r where
(8)
We assume that the nipples are inside radius of the rim:
(9)
And that the spokes are rods of uniform density (i.e. rcm = L)
rotating in the plane of the rim:
where L is given by
(10)
FIGURE 2:Figure of Geometry. Illustration of some approximations
used.We then invoke the parallel axis theorem, which states
that
(11)
where h is the separation between the two axes. 3
The only final approximation is the hub, which proves to be
difficult. The shape of the hub is a cylinder with some erratic
behaviour including a taper in the middle and a bulge at the
flange. The design of the hub, however, allows some insight. The
hub flange is the area of mechanical stress on the hub, whereas the
hubshell itself is designed to be as light as possible. We
approximate, therefore, that the hub is a cylinder of radius
(12)
The theoretical result requires no further assumptions. One
other definition we find useful is the average, given by
(13)
III. APPARATUS AND PROCEDURE
FIGURE 3:Experimental Setup. Detailed scheme of the apparatus
used during the experimentation. The numbers are referred to in the
following explanatory paragraphs.
The whole point of the experiment was to find the moment of
inertia of our bicycle wheel. To do so, we went through and
experimental method and a theoretical one.
For the experimental part, we used the conservation of energy
principles to find the moment of Inertia of the wheel. First, we
wound a string(1) around the wheel and affixed a hanging mass(3) at
the end of it. Then we applied a torque on the wheel by putting the
mass(3) at a certain height h from the ground(4), which was half of
the circumference of the wheel, and let it go down. The
conservation of energy principle states that the total energy of a
system must be the same at every time t. We did not wish to measure
the heat energy affiliated with friction, because the effects of
friction were negligible (see appendix V). So, in our case, the
potential energy of the hanging mass(3), when the system was at
rest, was equal to the sum of the kinetic energy of the hanging
mass(3) when it touched the ground(4) and the kinetic energy of the
bicycle wheel when the mass(3) touched the ground(4). Since the
weight of the mass(3), the gravitational constant and the height
were known, we only needed the angular velocity of the wheel and
the linear velocity of the mass when it touched the ground(4).
Fortunately, these two were related by this simple equation: . So
we only had to find one of them, in this case the angular velocity
of the bicycle wheel, which was very easy to obtain. To do so, we
taped two sensors(10,11) of identical width on the wheel, separated
by half of circumference, such that they were detected by an
optical detector(9) and transmitted to an oscilloscope(5) (see
figure). To do so, we referred to the spokes of the wheel to have
the best precision. One of the sensors(10) was placed right before
the detector(9) when the mass(3) was at height h. Each time a
sensor(10,11) passed the detector(9), a peak was displayed on the
oscilloscope(5)(previously tuned to setup 3). With that, we only
had to measure, with the oscilloscope(5), the distance between the
first two peaks of torque-free revolution ( but not between the
first two peaks, because the wheel was still accelerating at this
time) to obtain the time for the wheel to do half of a revolution.
Using the time between the second and the third peaks, we found the
angular velocity of the wheel when the mass(3) touched the ground
by relating the number of radians travelled with the time, = / t.
With these values, we had all the necessary things to find the
experimental moment of inertia of our bicycle wheel using the
conservation of energy equation applied in our case:
(14)
Legend: m: weight of the hanging mass(3)
g: gravitational constant = 9.806431 m/s2h: height of the
hanging mass(3) when the system is at rest (h = r )
I: moment of Inertia of the bicycle wheel
: angular velocity of the wheel when the mass(3) touched the
ground(4)
v: final linear velocity of the mass(3)
To find the theoretical moment of inertia of the wheel, we had
to disassemble it. After that, we only had to measure the weight of
each part and to measure some dimensions (length, width,
circumference, etc). With this done, we were able to find the
moments of inertia of each of the spinning parts of the wheel and
finally add them to obtain the theoretical moment of inertia of the
entire wheel.
We have also done some measurements to determine the effects of
the friction on our main experiment (see appendix V).
IV. DATA AND ANALYSIS
All our measurements are tabulated in appendix I whereas the
calculations are detailed in appendix II (experimental) and in
appendix III (theoretical). The produced results are given in
appendix IV. The following tables contain our most important
results.
TABLE 1 :Moment of Inertia of a Bicycle Wheel. Experimental and
theoretical moments of inertia of a bicycle wheel. (%) refers to
the percentage of difference between the two first values. (I
final) is the weighted average of the two values.I expI theo%I
final
kg m2kg m2kg m2
0,050260,000160,05040,00040,280,050280,00015
GRAPH 1:Theoretical Moment of Inertia by Component. Distribution
by component, expressed as a percentage of the total theoretical
moment of inertia. Grey = Rim (83%). Green = Spokes (12%). Red =
Nipples (4.5%). Black = Hub (0.11%). The precise values for the
moment of inertia for each part are tabulated in table 8 & 9,
appendix IV.
ERROR ANALYSIS:
Throughout the experiment, our team encountered many possible
sources of error that could affect in some way our results.
Therefore, it is important to mention that all the uncertainties on
the produced values are systematic. Indeed, since we did not
perform enough measurements to consider a random analysis for the
errors especially for time averages in the experimental
calculations (see appendix II), we preferred to limit ourselves to
the systematic error analysis. Thus, except for certain particular
cases, the errors on the measurements were determined by the
smallest readable division of the instrument we used. One example
of a case when this is does not apply is drop height of the masses
attached. We could not only consider the uncertainty on the meter
stick with which we measured the height since this distance varied
with the starting point of the mass. Hence we came up with the
following value: h = (0.986 0.005) m (see table 3, appendix I).
Our theoretical derivation required approximations and,
inherently, error. Each of the four parts was assumed to by
symmetrically machined, so that the symmetry could constantly be
invoked to simplify the measurements taken. Next, we had to
approximate a radius for the rotating elements. We approximated
that some of the masses were point masses (spoke nipples) and that
others were rings (hub and rim) with their mass centered at a given
radius. Spokes were approximated to be uniform (the spokes in
question are not butted or tapered) rods, and, as discussed
previously, rotating in the plane of the rim. The biggest error on
the result was the approximation of the rim, which was the most
straightforward. The rim is of hollow construction, with most of
the weight added to reinforce the sidewalls (the prime area of
stress for a wheel). What allowed us greater freedom was the
minimal width of the rim; we knew that the weight of the rim was
centered somewhere in the space of .0237 m. The area of greater
uncertainty, approximation of the hub, contributed so little to the
final result that it was outweighed by error on the instruments. We
cannot say precisely what the error contribution of the
approximations amounted to without first determining the actual
value. We can say, however, that the assumptions and approximations
involved in our measurement were much smaller than the quantity
measured, and perhaps one order of magnitude (at worst) above the
error affiliated with our instruments. Moreover, the results
indicate that the error contributed by the approximations is nearly
equal to the error due to friction, which we assumed to be
negligible. V. DISCUSSION
The determination of the moment of inertia of a bicycle wheel by
two distinct methods experimental and theoretical, which have been
thoroughly discussed in section II led to surprisingly good
results, from a consistency point of view. Indeed, the two separate
results, tabulated in table 1, section IV, are off by a percentage
of 0.28 %, which is surprisingly close when we consider all the
possible sources of error involved and the numerous approximations
made throughout the whole experimentation. In fact, despite
closeness of the values, we must consider a wide domain of
uncertainty for the two calculated moment of inertia, larger than
the one that produce the strictly systematic error analysis see
error analysis, section IV. For example, we know that the friction
has affected our results which ads up to possible error sources and
that was not taken in account during the systematic error analysis
a detailed discussion in appendix V quantifies the impact of
friction on our results. Moreover, we did utilise frequent
approximations during the theoretical calculations in order to
simplify this extensive task which incidentally enlarges the
uncertainty domain. It is however somewhat difficult to quantify
this effect since it depends on how our approximations are in
accordance with reality see error analysis section IV.
It is also important that the results were obtained using
specific calculation methods as there were many possibilities for
calculating the same values, especially for the experimental part.
For instance, we used the theory on energy conservation when we
could have used theory on torque. We preferred the energy
conservation method since brief attempts using calculations with
torque proved to be particularly inconsistent for extensive
description of the procedure and results, refer to appendix
VII.
VI. CONCLUSION
VII. ACKNOWLEDGEMENTS
We would like to thank Prof. Charles Gale for teaching us the
mechanics necessary to undertake this experiment.
VIII. REFERENCES
1 KLEPPNER, Daniel & Robert J. Kolenkow. An Introduction to
Mechanics, McGraw-Hill, 1973, p.189,266.
2 Ibid, p.265
3 Ibid, p.252
4 Ibid, p.92-935 Ibid, p.95
APPENDIX I
MEASUREMENTS TABLES
TABLE 2:First (t1) and Second (t2) Half-Period. The half-period
is the time it takes for the wheel to complete half a revolution.
The exact value for the masses are given in table 3, appendix 1 .
Numbers on the top of the columns refer to the number of the
attempt.
Mass123
t1t2t1t2t1t2
g 0,001s 0,001s 0,001s 0,001s 0,001s 0,001s
104,4401,6004,3601,6004,2801,600
202,4801,1602,4401,1602,5201,160
501,3600,6801,3200,6901,3300,680
1001,0600,5501,0800,5501,0700,560
2000,8200,4200,8100,4300,8100,420
3000,7100,3700,7000,3700,7000,370
5000,6120,3160,5880,3160,5960,316
10000,5400,2720,5320,2760,5280,272
TABLE 3:Setup Specifications. Except for 0.3 Kg, all value
values (m) are from direct weighting. The radius (r) was derived
from the measured circumference (c). (h) refers to the drop height.
The gravitational acceleration (g) is cited from data used from the
PHYS 257 Katers pendulum lab.
mch
Kgmm
0,00980,00011,9730,0020,9860,005
0,01980,0001
0,06260,0001rg
0,10010,0001
0,19990,0001mm/s2
0,300000,000140,3140,0029,806431
0,49970,0001
1,00000,0001
TABLE 4:Wheel Specifications. (m) refers to a mass as (d) is the
corresponding distance.PartmRimdHubd
0,1gcmcm
wheel841,6Outer diameter63,40,2Flange diameter 5,380,01
rim448,2Width2,370,01Spoke end diameter 4,770,01
all spokes199,2Depth0,870,01Axle diameter0,890,01
1 spoke7,1Middle diameter2,840,01
all nipples26,5Spoke & NippledEnd diameter3,870,01
1 nipple0,9cmLength (Inside flange)7,160,01
hub197,8Spoke length27,70,2Length (Outside flange)7,820,01
hub w/out dust caps153,4Nipple length1,210,01
TABLE 5:Torque-Free Revolutions of the Wheel. (t) is the
duration of the corresponding half revolution ( R). Numbers on the
top of the columns refer to the number of the attempt.
R123
ttt
0,001s 0,001s 0,001s
10,2200,4600,950
20,2200,4700,920
30,2200,4700,970
40,2200,4800,950
50,2300,4801,010
60,2200,4901,000
70,2300,4901,060
80,2300,5001,040
90,2300,5101,130
100,2300,5101,090
110,2300,5101,180
120,2400,5301,150
130,5201,260
140,5401,220
APPENDIX II
SAMPLE CALCULATIONS : Experimental Part
Calculations relative to table 6, appendix IV:a) Average time
calculation (ex. 100 g , 2)
2 = =
= 0.55333333 sb) Deviation on average time calculation (ex. 100
g , 2)
= 0.0005773502692 s = 0.0006 s
So, 2 = (0.5533 0.0006) sc) Angular velocity () calculation (ex.
100 g)
= = 5.67758 rad/s
d) Squared deviation on calculation (ex. 100 g)
= = 0.0000379
So, = (5.678 0.006) rad/s
e) Linear velocity (v) calculation (ex. 100 g)
= 5.678 0.314 = 1.782892 m/s
f) Deviation on v calculation (ex. 100 g)
EMBED Equation.3 = 0.01151 m/s
So, v = (1.78 0.01) m/s
Calculations relative to table 7, appendix IV:g) Potential
energy (E) of the falling mass calculation (ex. 100 g)
= 0.1001 9.806431 0.986 = 0.968 J
h) Deviation on E (ex. 100 g)
= 0.0050030696 J
So, E = (0.968 0.005) J
i) Moment of inertia (I) calculation (ex. 100 g)
By energy conservation,
I = = 0.050179561
j) Squared deviation on I calculation (ex. 100 g)
=
So, I = (0.0502 0.0003) kg/m2k) Standard deviation on average I
calculation
=
l) Average I calculation
=
So , I = (0.05026 0.00016) kg/m2
APPENDIX III
SAMPLE CALCULATIONS : Theoretical Part
Calculations relevant to table 8, appendix IV: Our goal is
simple: compute I = mjj2 as accurately as possible
I. Spoke Nipples, Rim, and Hub:
a) radius calculation: combination of measured quantities (ex.
spoke nipples)
meters
b) radius error (ex. spoke nipples)
.002002498 mc) moment of inertia (ex. spoke nipples)
I = mjj2 = mnipplesrnipples2
d) moment of inertia error (ex. spoke nipples)
So, I nipples = (0.00228 0.00002) kg m2Calculations relevant to
table 9, appendix IV: II. Spokes
a) ideal length:
1. First, we determine the origin of the spoke by an
average:
2.
b) Moment of inertia of a rod through center of mass and
perpendicular to length:
,
c) Parallel Axis Theorem:
Again, errors add in quadrature, as before (see previous
examples). Therefore,
I total spokes = (0.00630 0.00014) kg m2APPENDIX IV
RESULTS TABLES
TABLE 6:Average Time and Velocities. (1) and (2) refer to the
average of the corresponding time values given in table 2, appendix
I . () is the average final angular velocity as (v) is the average
final linear velocity for a point on the rim. The exact value for
the masses are given in table 3, appendix I.Mass12v
gssrad/sm/s
104,36000,00061,60000,00061,96350,00070,6170,004
202,48000,00061,16000,00062,7080,0010,8500,005
501,33670,00060,68330,00064,5970,0041,440,01
1001,07000,00060,55330,00065,6780,0061,780,01
2000,81330,00060,42330,00067,420,012,3300,015
3000,70330,00060,37000,00068,490,012,670,02
5000,59870,00060,31600,00069,940,023,120,02
10000,53330,00060,27330,000611,490,023,610,02
TABLE 7:Energy and Moment of Inertia. (E) corresponds to the
potential energy of the drop mass as (I) is the moment of inertia
of the wheel for the attempt with the respective drop mass. The
experimental value is the final I value produced by the
calculations performed in appendix II. The exact value for the
masses are given in table 3, appendix I.MassEI
gJkg m2
100,0950,0010,04820,0006
200,19140,00140,05030,0004
500,6050,0030,05110,0003
1000,9680,0050,05020,0004
2001,930,010,05050,0005
3002,900,010,05090,0006
5004,830,020,04850,0008
10009,670,050,04780,0015
Experimental0,050260,00016
TABLE 8:Theoretical Moment of Inertia, part I. These are the
results of the first part of the calculations performed in appendix
III. (m) stands for mass as (r) is the corresponding radius.Spoke
NipplesHubRim
m (kg)0,02650,00010,15340,00010,44820,0001
r (m)0,2930,0020,01940,00010,3050,002
mr2 (kg m2)0,002280,000020,00005740,00000040,04170,0004
TABLE 9:Theoretical Moment of Inertia, part II. These are the
results of the second part of the calculations performed in
appendix III. The names for the values are explained in the same
appendix.Spokes
ideal length (m)0,2670,002
m (kg)0,19920,0001
I cm (kg m2)0,0011810,000009
distance (m)0,1600,002
md2 (kg m2)0,005110,00014
Itot-spokes (kg m2)0,006300,00014
APPENDIX V
A DISCUSSION ON FRICTION
While dealing with a quite complex and extensive apparatus as we
did for this project, it is most likely that our results were
affected by the effect of friction. Indeed, as the wheel spins,
many factors may slow it down such as the resistance of air, the
friction of the rope on the rim or just the friction due to the
bearings themselves. Thus, our team performed some measurements in
order to quantify the effect of friction on our results.
In order to estimate this effect, we let the wheel spin freely,
without any rope attached, and measured the half-period. Then, we
could compute the average velocity for about a dozen of
half-revolution. We carried out this task for three different
initial velocities (see table 5, appendix I). The results for the
angular velocity are given in table 11, appendix VI. Hence,
plotting of the average angular velocity as a function of the
half-revolution yields the following graph:
GRAPH 2:The Effect of Friction. The black set of points refers
to the data of attempt 1, the red set to the second attempt and the
green set to the third. Each corresponding line represents a linear
fit to the data (y = a + b*x) which does take in account the error
bars. The results of the fit, produced by the software Origin 7.0,
are tabulated in table 12 , appendix VI.
This graph clearly shows that friction affects the angular
velocity. However, we must notice that the slope on the different
fits being fairly small, variation on the velocity for a short
period will be equally small. Furthermore, since that for this
experimentation there was no thread attached to the wheel, the main
two sources of friction would have to be internal (bearings) and
external (air resistance).
The internal source of friction resides in the contact between
the components of the ball bearing and the hub itself. We might
recall that friction due contact between two surfaces is
independent of the velocity of the objects (F = N)4. Therefore,
this means that if only the friction due to the bearings would have
acted on the wheel, the three slopes of the linear fits would have
been equal.
As for the external source of friction, the so-called viscosity
of the air, we know that it is velocity dependent, obeying a
proportionality relationship (F=-cv)5. Thus, this implies that the
larger the velocity is, the larger the deceleration is. In other
words, the speed of the wheel must exponentially decrease as a
function of time. Results of the linear fits show that this
relation is effective in our case (see table 12, appendix VI).
Indeed, we may notice that if the three fits would have been a
single data set, it would be clear that the velocity decreases in
an exponential way. Later results will make more manifest this
assertion.
Hence, we conclude that the main source of friction on the wheel
is air resistance. This conclusion makes a lot of sense. The
hub/bearings area suffers tremendous stress while mounted on a
bicycle since it must support the weight of the bicycle frame and
the rider (N ( = F (). Therefore, it must be designed so that the
internal hub friction is minimized. So, from now on in this
discussion, we will only consider the friction induced by the
viscosity of the air.
It is possible to quantify the effect of friction on the wheel
using the proper plotted data. Indeed, we may deduce an average
value of c, the friction coefficient that is shape dependent.
F = -cv = ma c = - ma/v(15)
v = v0 e ct/m(16)
c = -(m/t) ln (v/v0)(17)
However, to perform this task, we must convert the
half-revolutions (x-axis) into time to compute the acceleration.
Moreover, since we are dealing with linear velocities and
acceleration, we have to convert all the angular data. To do so, we
consider the linear velocity of the rim (r = 0.314 m, see table 3,
appendix I).
The data seen in graph 3, is fitted with an exponential function
(v = v0 e ct/m). Still, it is even clearer that the deceleration is
proportional to the speed of the wheel. The results of the fit may
not be as satisfying as we would expect since the value of c varies
a lot (see table 4, appendix). In order to produce consistent
results, we should have measured the velocity of the wheel over a
larger period of time. Nevertheless, the measurements performed for
the calculation of the moment of inertia were themselves over a
short period of time. Therefore our results may not be consistent
with the theory but they are quite a good representation of the
experimental conditions.
An estimate of the air drag on the wheel can finally be
performed. After calculating an average value for c (0.0197 0.0002)
with the three values derived from graph 3, we computed a new final
linear and angular velocity, using v = v0 e ct/m, for each attempt
tabulated in table 6, appendix IV. Then, this procedure enable us
to calculate the new moment of inertia of the bicycle wheel
following the method presented in appendix II. The results appear
in table 10 this appendix.
GRAPH 3:Viscosity of the air and Friction. The colours are
representative of the same data sets as in graph . However the data
is fitted with v = v0e ct/m. The results of the exponential fit,
produced by the software Origin 7.0, are tabulated in table 14 ,
appendix VI.
TABLE 10:Extrapolation of a Friction-Free Moment of Inertia. (I)
is the first calculated value (experimental) of I whereas (I0) is
the friction-free one. (%) represents the percentage of difference
between the two values.II0%
kgm2kgm2
0,050260,000160,04750,00035,9
In brief, the air resistance did affect our results on the
experimental part as show the numbers in appendix VI. In fact, this
variation is not negligible at all since there is a percentage of
difference of 5,9% between the value initially produced and the one
that takes in account friction. Moreover, we may be puzzled to
notice the results of table 15 in appendix VI. Indeed, it seems
that the correction due to friction is inversely proportional to
the velocity of the wheel, which is the opposite of what we could
expect the percentage of difference is higher for lower velocities.
There is however a reasonable explanation to this fact. At lower
speeds, the air resistance acts for a longer time on the wheel
therefore producing a larger deceleration than at higher
velocities. Finally, as mentioned earlier the best way to quantify
the effect of friction is to carry out measurements on the largest
time period possible, so that the data can be fitted according to
theory.
APPENDIX VI
RESULTS TABLES: Friction Analysis
TABLE 11:The Effect of Friction on the Angular Velocity. () is
the average angular velocity for the corresponding half-period (R).
These data sets are plotted in graph 2, appendix V. Numbers on the
top of the columns refer to the number of the attempt.
R123
rad/srad/srad/s
0,514,280,066,8300,0153,3070,003
1,014,280,066,6840,0143,4150,004
1,514,280,066,6840,0143,2390,003
2,014,280,066,5450,0143,3070,003
2,513,660,066,5450,0143,1100,003
3,014,280,066,410,013,1420,003
3,513,660,066,410,012,9640,003
4,013,660,066,280,013,0210,003
4,513,660,066,160,012,7800,002
5,013,660,066,160,012,8820,003
5,513,660,066,160,012,6620,002
6,013,090,055,930,012,7320,002
6,56,040,012,4930,002
7,05,820,012,5750,002
TABLE 12:Results of the Linear Fits. (b) is the slope, the
change in angular speed expressed in rad/s per half a revolution
whereas (a) is the intercept. These values refer to graph 2,
appendix V.
Fitba
(rad/s)/(half-revolution)rad/s
Black-0,190,0114,500,04
Red-0,14210,00166,8640,007
Green-0,13590,00033,4710,002
TABLE 13:The Effect of Friction on Linear Velocity. (v) is the
linear velocity of a point on the rim at a time (t). These data
sets are plotted in graph 3, appendix V. Numbers on the top of the
columns refer to the number of the attempt.
123
tvtvtv
sm/ssm/ssm/s
0,2204,480,040,4602,1450,0140,9501,0380,007
0,4404,480,040,9302,0990,0141,8701,0720,007
0,6604,480,041,4002,0990,0142,8401,0170,007
0,8804,480,041,8802,0550,0143,7901,0380,007
1,1104,290,032,3602,0550,0144,8000,9770,006
1,3304,480,042,8502,010,015,8000,9870,006
1,5604,290,033,3402,010,016,8600,9310,006
1,7904,290,033,8401,970,017,9000,9490,006
2,0204,290,034,3501,930,019,0300,8730,006
2,2504,290,034,8601,930,0110,1200,9050,006
2,4804,290,035,3701,930,0111,3000,8360,005
2,7204,110,035,9001,860,0112,4500,8580,006
6,4201,900,0113,7100,7830,005
6,9601,830,0114,9300,8090,005
TABLE 14: Results of the Exponential Fits. (c) is the air drag
coefficient whereas (v0) is the intercept. These values refer to
graph 3, appendix V.Fitcv0
kg/sm/s
Black0,02370,00054,5370,004
Red0,01880,00032,1540,004
Green0,01830,00031,0950,004
TABLE 15:Extrapolation of a Friction-Free Velocity. (v) is the
final linear velocity and (v0) is the estimated real linear speed
of the wheel in a viscosity-free environment. (%) is the percentage
of difference between the two values. The exact values for the
masses are given in table 3, appendix I.Massvv0%
gm/sm/s
100,6170,0040,640,033,7
200,8500,0050,8740,0132,7
501,440,011,470,011,6
1001,780,011,810,011,3
2002,3300,0152,3530,0151,0
3002,670,022,690,020,86
5003,120,023,140,020,74
10003,610,023,630,020,64
APPENDIX VII
A DISCUSSION ON TORQUE ORIENTED CALCULATIONS
There are several methods leading to the determination of the
moment of inertia, since this intrinsic property of a body is
related to the multiple mechanical phenomena related to rotation.
Hence, our team initially had the idea to apply a constant torque
on the wheel, namely by attaching to it a dropping mass (as
detailed in section 3), and measure the change in angular velocity
to deduce the angular acceleration. As a matter of fact, that is
the reason why we made the drop height of the mass more or less
equal to a half circumference of the wheel. Indeed, this way we
knew that the torque would be applied for about half a revolution
of the wheel and we could measure this mass drop time on the
oscilloscope (see t1, table 2, appendix I). However, as we will
see, the calculations using this method lead to rather strange
results.
Ironically, the calculations for the experimental moment of
inertia using the torque method are much less complex. In fact, it
relies on the combination of two simple equations:
= / t = / t2 t = / t2 t1(18)
= I m g r = I (19)
Which, when combined together yields the following moment of
inertia formula:
I = m g r t1 t2 / (20)
Here, m refers to the mass of the drop mass, g is the
gravitational acceleration and r is the radius of the wheel. The
exact values of these variables are given in table 3, appendix I.
For t1 and t2, the corresponding values appear in table 2, appendix
I. Using theses formulas, we are now able to compute the moment of
inertia of the bicycle wheel with the torque method. The results
are as follows:
TABLE 16:Moment of Inertia Using the Torque Method. (I) is the
moment of inertia as computed by the energy conservation method
(see table 7, appendix IV). (%) is the percentage of difference
between the two values. The exact values for the masses are given
in table 3, appendix I.
MassII torque%
kgm2kgm2
100,04820,00060,06700,000839
200,05030,00040,05580,000511
500,05110,00030,05600,00049,7
1000,05020,00040,05810,000416
2000,05050,00050,06750,000434
3000,05090,00060,07650,000550
5000,04850,00080,09270,000691
10000,04780,00150,1430,001199
Final0,050260,000160,06680,000233
It is quite manifest from the results that our team encountered
a little consistency problem using the torque method calculations.
In fact, these incoherent results were the principal motivation for
us to perform the calculations using the theory on energy
conservation (see appendix II).
The reasons explaining our failure to produce consistent results
using the torque method calculations are few. We are not actually
sure of where lies the problem in this method as there is two
possibilities. First, the apparatus might not have been properly
set up in order to perform such an experiment. Although we adjusted
the drop height of the mass so that it matched exactly the
half-circumference of the wheel, it proved to be unhelpful in
producing some more logical results. However, we know that more
sensors on the rim (see figure 3, section 3) less distant from each
other would have helped us to measure more accurately the angular
acceleration of the wheel for the time the torque is applied. On
the other hand, we also have made several assumptions during our
computations that might have led to wrong results. In fact, we
neglected the tension of the string acting on the drop mass.
Moreover, as we calculated the angular acceleration , we only
considered the final angular velocity (derived from t2, see Theory)
over the time of change of speed, namely t1. Hence, the result of
this operation is the average over the appropriate time period
which is legitimate since we expect the acceleration of the drop
mass to be constant.
Therefore, it remains unclear why computations using the theory
on torque did not work out properly. However, we assume that the
explanation for this situation lies in either the calculations
themselves or in the experimental setup. In addition, it proved
that the procedure using the energy conservation theory was,
despite its heavier mathematics, more efficient.
GRAPH 4:Moment of Inertia: The Torque Method. Plot of the
results tabulated in table of this appendix. It is clear that (I)
is not constant for the bicycle wheel since it varies depending on
the drop mass (m). The linear fit, for illustrative purposes only,
is to demonstrate the fact that (I) appears to be proportional to
(m), the drop mass which is, of course, not consistent with
theory.
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0.0022796596
0.0062954512
0.0000574364
0.0417348254
Theoretical Moment of Inertia by Component
Sheet1
Theoretical Moment of Inertia
spoke nippleserrorunits
m0.02650.0001kgPartmRimdHubd
r0.29330.0020024984meter
mr^20.002280.00002kg/m2 0,1gcmcm
wheel841,6Outer diameter63,40,2Flange diameter5,380,01
spokes:rim448,2Width2,370,01Spoke end diameter4,770,01
ideal length:0.26668143920.0019902474meterall
spokes199,2Depth0,870,01Axle diameter0,890,01
m0.19920.0001kg1 spoke7,1Middle diameter2,840,01
Icm0.00118057520.0000088504kg/m2all nipples26,5Spoke &
NippledEnd diameter3,870,01
distance0.16024071960.0020902474separation of the two axes1
nipple0,9cmLength (Inside flange)7,160,01
md^20.0051148760.0001360089via parallel axis theormhub197,8Spoke
length27,70,2Length (Outside flange)7,820,01
Itot-spokes0.00629545120.0001362966kg/m2hub w/out dust
caps153,4Nipple length1,210,01
hub:
m0.15340.0001kgSpoke Nippleshub:Rim
r0.019350.0001meter
mr^20.00005743640.0000004214kg/m2m
(kg)0.02650.00010.15340.00010.44820.0001
r (m)0.2930.0020.01940.00010.3050.002
Rimmr2 (kg m2)0.002280.000020.00005740.00000040.04170.0004
m0.44820.0001kg
r0.305150.0020024984meterSpokes
mr^20.04173482540.0003874341kg/m2
ideal length (m)0.2670.002
0.8167761109m (kg)0.19920.0001
Sum0.0004113887Icm (kg m2)0.0011810.000009
Itot0.05040.0005477847kg/m2distance (m)0.1600.002
1.0875785013%md^2 (kg m2)0.005110.00014
Individual contributions:%d%Itot-spokes (kg
m2)0.006300.00014
Rim:0.11403495670.0076921649%
Nipples:4.52606413220.0004692042%
Spokes:12.49906612720.0027060486%
Rim:82.8608347840.0076921649%
Sheet2
Sheet3
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