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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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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