Physics IA rolling resistance

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PHYS 100

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Mechanical Engineering

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Dec 6, 2023

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Relationship between the air pressure of a bike and the Coef±icient of rolling resistance 1.0 Introduction Because I live in the Netherlands it is very common for people to use bikes as a way of transportation as bike paths are very common and new urban development is surrounded around making bike lanes safe and ef±icient to reduce CO 2 emissions. When I was younger I used to compete in biking marathons with my father, where I began to ±ind a passion for biking. I did competitions from age 15-16, however now I use my bike on a day-to-day basis to get to school which was approximately 4km each way, and then bike to Basketball practice which was 11km each way. I would constantly have to pump my tires as they would commonly go ±lat due to the age of my bike, therefore I began to notably see a difference in how much force I had to exert to bike faster when my tires were lower rather than when I would have recently pumped them up. During Bike marathons, my father would vary the pressure of our bikes depending on the terrain and weather. Consequently, I began to wonder what the most practical pressure is for biking the distances I do on a daily would be. Therefore throughout the report, I will be investigating the relationship between the change in the pressure of the bike tire and the coef±icient of rolling resistance (C sr ). 1.1 Research question How is the Coef±icient of rolling resistance (dimensionless), of a bike tire affected by the variation of its pressure (60psi, 80psi, 100psi, 120psi, 140psi 2% )? ± 2.0 Scienti±ic Context & theory Through this Lab report I will be investigating the effect that a change in the pressure of a bike tire has on the coef±icient of rolling resistance. The property inherent to the tire, based on factors affecting the resistance of the tire, force acting opposite to the motion of the cyclist is the coef±icient. Although pressure has an impact, factors such as surface characteristics for instance roughness, tire characteristics for example tire material/tread and the supported weight are likewise relevant factors which would in±luence the magnitude of resistive force, therefore re±lecting variations in the coef±icient. The equation of rolling resistance is expressed as, F=C sr N (Ryan Johnson). The friction experienced by the E-bike tire due to rolling on a surface is F ,

where the coef±icient property of the tire is C sr , and the weight of the E-bike body supporting the tire being equal to the Normal force N. The Normal force ( N ) can likewise be expressed as mg, m being the mass of the body and g being equal to the constant of Earth's gravitational acceleration (9.81 m/s 2 ). To make the coef±icient property of the tire ( C sr ) the subject, C sr = F/W , therefore expressing C sr as the force of rolling resistance on the tire per unit force of weight as W=mg. Consequently, for 0.5kg of weight (4.45N) to oppose the resistant force of rolling resistance and accelerate in a forward direction more than 0.5kg of force (4.45N) must be acquired. As there is an inverse relationship, The coef±icient of rolling resistance will inversely increase with the pressure of the tire (Greg Kopecky). From biking, it's known to me that an individual has to exert more force on the pedals when the tires have lower pressure, causing acceleration to be more dif±icult in the forward direction. This, therefore, demonstrates force opposite to velocity is larger. Intuitively, as the pressure of a tire decreases, and the surface area of the tire increases as it hits the ground, therefore more force is necessitated to move the tire. This occurs as the structural properties of a tire are affected by the pressure of the tire. As air is pumped into the tire, there's correspondingly an increase in air molecules, causing the volume to ultimately increase. The vacuity of the tire decreases, as the increasing number of molecules, occupy it. The pressure is increased as a result of a higher force exerted by the greater momentum change when molecules interact with the wall and collide at a more rapid rate. As pressure is equal to the perpendicular force per unit area (Aakanksha Gaur), As pressure decreases, the number of molecules would decrease, resulting in a lower density. Correspondingly fewer molecules will collide against the wall and the tire becomes less ±irm increasing the tire's contact with the surface, and increasing the frictional property (Kevin Shen). 3.1 Independent Variables The variation in the pressure of the E-Bike tire which I will investigate is (4.14 x 10 5 Pa,5.52 x 10 5 Pa,6.89 x 10 5 Pa,8.27 x 10 5 Pa, and 9.65 x 10 5 Pa). I shall be measuring the pressure of the bike tire through a pressure gauge in psi where 1 psi is equal to 6894.76Pa. The selected values were determined by the accuracy of the pump measurements. The measurement markings between 10 were indistinct and therefore values which have the highest accuracy were selected. As preliminary experiments demonstrated a signi±icant difference from 60psi to 140psi, it can be deduced that the selected range was suf±icient. In cogitation with section 2, the surrounding temperature of the tire might vary the accuracy of the E-bike when gauging the pressure within it if there's a sudden change in temperature. 3.2 Dependant variable

Due to the coef±icient of rolling resistance being dimensionless, initially the raw data will consist of measuring the deceleration (m/s 2 ) of the tire in relation to the varying pressure (Pa). The experimental set-up (section 4.2), was done with 5 trials to reduce the uncertainty through random error. The coef±icient will then be calculated through data processing ( section 5.1). 3.3 Controlled variable Controlled Variables Justi±ication why I chose this variable Method to control this variable Initial velocity of the bike tire (m/s) It's crucial to keep the initial velocity constant as through the experimental process the coef±icient will be discovered through measuring the acceleration of the tire. Acceleration is equal to the velocity over time therefore accurately measuring acceleration will require a constant initial velocity. I conceptualised that to accurately measure the velocity a tire is moving, and repeat this process with the exact identical constant initial velocity, I could use an electric bike (Sparta ultra M5B conversion kit on an X-treme bike frame). The electric bike has various settings where the speed can be controlled through a throttle which accelerates the bike's front tire at a constant force. I chose setting 1 which was 8km/h, which is 2.23m/s ± 0.1 Tire (Wanlihu), tire tread (mountain bike), and tire material (natural rubber) Controlling the tire used is essential to only measure the dependent variable varies when altering the pressure (Pa) within the bike tire. If the thickness of the tire is varied as a re±lection of the gauge pressure, the shape might alter and the tire might have more/less surface area touching the ground. Different tires would correspondingly have distinct designs which This will simply be controlled by using the exact same tire as a constant for all trials. The bike tire used is a 26-inch Wanlihu mountain bike tire made from natural rubber and has a mountain bike tread.

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affect the traction. Temperature ( °C ) Variations in temperature in±luence the pressure in a tire. This is as the gas particle's kinetic energy increases in correlation with an increase in temperature, causing them to move with greater energy and increasing their collisions with each other and the surface. However small changes in temperature should theoretically not signi±icantly skew the results. The air pressure of a bike is increased by 1.8% for every increase in degree celsius (Floadmin) To minimize the variation in temperature, the experimental procedure will be conducted in an ef±icient manner where all results can be collected within a 30min time period to reduce the probability of a change in weather. This will be monitored to be around 19 ± 1 °C Surface of the tire is tested on Changes in the surface would correspond to changes in its frictional properties of it. Rougher surfaces have higher friction because there are more irregularities, causing interlocking, therefore the force of kinetic friction against the bike tire will be greater. Conducting the experimental procedure will be done on a smooth stone. To control this variable by keeping the surface being used constantly at the same location Mass supported by the tire The mass of the bike frame supported by the tire can affect the resistance exerted onto the bike tire. The larger the mass in which the tire supports, the larger the kinetic frictional force between the wooden ±loor and the tire, due to the irregularities colliding against each other with a greater force. Controlling To control this variable, I will simply use the same E-bike and due to the bike having 2 tires the mass is distributed fairly evenly.

this will make sure the data collected is valid and reliable as the resistance force exerted on the tire remains constant. (Table 1 - controlled variables) 4.1 Equipment - Bike stand (28-inch double kickstand) - Air compressor (with pressure display in increments of 20) - Presta Valve - Stopwatch - Electric Bike ( Bosch ultra M5B conversion kit on X-treme bike frame) - Pressure Gauge 4.2 Experimental set up (Figure 1 - Method set up) 4.3 Experimental methods

1. First, set the bike to suspend the front motorised tire, by using the double kickstand. 2. Secondly, attach the Presta valve onto the air compressor, then de±late the tire. 3. Following that, when the front tire is de±lated, use the air compressor to in±late the tire to 60psi 4. Whereafter by using the pressure gauge, inspect that the tire is at 60psi for increased accuracy 5. Before conducting the experiment, ensure to conduct the experiment at a monitored temperature (19 °C ) and to execute the experiment on the same surface for each trial. 6. Henseforce set the bike to setting 1 (8km/h) by using the arrows above and below the display and stand by until the bike displays 8km/h and has reached the maximum Velocity. 7. Next, release the throttle and place the front tire on the ground (make sure not to push down onto the bike resulting in a greater force exerted down onto the wheel). 8. As soon as the wheel makes contact with the ground surface, use the stopwatch to record the time taken for the tire to reach 0m/s and come to a complete stop. 9. Subsequently repeat steps 3-7, 5 different times, to have multiple trials 10. Finally repeat steps 3-8 for the various pressures (80psi,100psi,120psi and 140psi) 4.4 Risk assessment To ensure the E-Bike is secure and won't potentially cause damage to itself or harm to the individual conducting the experiment through falling, when mounting the bike caution should be taken. Although there are no signi±icant risks when conducting this experiment, to some degree the individual will have involvement with sharp points as well as fast-moving components such as the bike chain, spokes and gears. Gloves are suggested to avoid cuts and ensure the individual's safety. Comprehensively, the experimental procedure has minor safety concerns and no ethical or environmental issues and therefore are not applicable to this investigation. 6.0 Raw Data Time taken to decelerate from 2.24m/s to 0.00m/s (s) ±0.01 Pressure of the Tire (psi) Trial 1 (s) ± 0.17 Trial 2 (s) ± 0.17 Trial 3 (s) ± 0.17 Trial 4 (s) ± 0.17 Trial 5 (s) ± 0.17 60.0 ±1.22 3.43 3.98 3.50 4.02 3.62

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80.0 ±1.61 4.20 4.38 4.54 3.99 4.17 100. ±2.03 4.79 4.48 4.61 4.76 5.12 120. ±2.41 4.81 4.98 5.54 5.20 5.00 140. ±2.84 5.22 5.05 5.08 5.51 5.59 (Table 2 - Raw Data) The Uncertainty of the time was obtained through recording the smallest increment of measurement. For the apple stopwatch on the clock app, the smallest increment is 0.01 making the uncertainty 0.01. The uncertainty of the human reaction time was ± obtained through calculating my average reaction time out of 5 trials which resulted in an uncertainty of ± 0.16s. Furthermore, the product description disclosed that the pressure gauge used has an uncertainty of 2%. 5.1 Processed Data Processed Data Table Pressure of tire (psi) Average time of deceleration (s) Rage of time taken to decelerate (s) in velocity (m/s) △ Average Acceleration (m/s 2 ) Coef±icient of rolling resistance ( C sr ) 60.00 ±1.221 3.713 ±0.312 0.59 -2.230 ±0.122 -0.602 ±0.078 0.061 ±0.008 80.00 ±1.613 4.246 ±0.275 0.55 -2.230 ±0.122 -0.523 ±0.061 0.053 ±0.006 100.0 ±2.031 4.751 ±0.321 0.64 -2.230 ±0.122 -0.472 ±0.062 0.048 ±0.006 120.0 ±2.411 5.103 ±0.374 0.74 -2.230 ±0.122 -0.441 ±0.053 0.045 ±0.006 140.0 ±2.842 5.294 ±0.271 0.54 -2.230 ±0.122 -0.424 ±0.043 0.043 ±0.005 (Table 3 - Processed Data) Average time of deceleration This was calculated by adding the values of all 5 trials for each pressure separately and dividing by 5 (the number of trials)

Example calculation for pressure 80psi: 46s 4.2?+4.83?+4.54?+3.99?+4.17? 5 = 4. 2 The range of data divided by 2, represents random uncertainty of the average which can present the spread from the data collected Example Calculation for pressure 80psi: s 4.54?−3.99? 2 = 0. 275 Change in Velocity The Sparta A-shine ultra M5B manual booklet provides that there's a 5% uncertainty ± in the change in velocity, as well as the Uncertainty of the screen displaying the velocity 0.01. ± Example Calculation: + 0.01= 0.1215 − 2. 23?/?( 5% 100 ) = 0. 1115 ± Acceleration The equation of acceleration is the change in velocity over the change in time ( ) . For each pressure, I divided the change of velocity by the average 𝑎 = 𝑣−𝑣 0 ? = △𝑣 △? time taken to decelerate. Example calculation for pressure 80psi: m/s 2 −2.23?/? 4.26? =− 0. 523 To calculate the uncertainty of acceleration, the % uncertainty has to be discovered and consequently added together. After the % uncertainty is found, simply convert it back to absolute uncertainty.

% ?𝑒??𝑐𝑖?? : ±0.11?/? −2.23?/? × 100% =± 4. 93 % 𝑇𝑖?𝑒 : ±0.35? 3.71? × 100% =± 8 m/s 2 8%+4.93% 100 ×− 0. 6?/? 2 =± 0. 078 The Force ( F ) acting on the wheel can be determined by having discovered the acceleration. For this investigation, F is equal to the F net on the tire . Drag is neglected which is the force which acts opposite to the relative motion of a moving object, with respect to a ±luid surrounding (Aakanksha Gaur). Since the bike is in a controlled environment ( stationary and not moving and within a fenced area blocking wind), the only component of drag relevant would be the rotation of the tire, where the spokes push away air particles as it is moving relative to the fuid. The bike tire is rotating at a rate where drag would be so insigni±icantly small it will be neglected within this investigation (Shital Shah). You can rearrange the equation F=C sr W with F net =ma , as forces like kinetic frictional force which is exerted on the wheel as it moves along a surface can be neglected due to the various properties of C sr such as surface roughness. All other forces being constant allows only pressure to change F and F net . For the rearranged equation ( C sr = a/g ), the acceleration due to gravity (g)is equal to 9.81m/s 2 . 𝐹 = 𝐶 ?? ? −> 𝐹 ?𝑒? = ?𝑎 −> 𝐹 = 𝐹 ?𝑒? −> 𝐶 ?? ? = ?𝑎 −> 𝐶 ?? = ?𝑎 ?𝑔 −> 𝐶 ?? = 𝑎 𝑔 Example Calculation for pressure 80psi: 0.6 9.81 = 0. 061 To obtain the absolute uncertainty of the coef±icient of rolling resistance, the sum of the fractional uncertainties has to be taken. Example calculation for pressure 80psi: 0.061 −0.523 × 100 =− 11. 6% −11.6 100 × 0. 078 =± 0. 0062

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5.2 Graph (Figure 2 - Graph of C sr vs change in pressure) The graph represents the relationship between the coef±icient of rolling resistance (Y-axis) as a result of a change in the pressure within the bike tire (X-axis). The data presents a negative correlation, as the air pressure of the tire increases, the coef±icient of rolling resistance decreases. The curve ±it line proposes possibilities of a horizontal asymptote of the relationship C sr = y = 0. This can further be evaluated by ±itting the line of best ±it to the power function . With a correlation value of 0.9960, the data ? = 0.3428 ? 0.4237 appears to ±it very well. Applying the In±inity property, , to the ? ∞ lim → 𝑐 ? ? = 𝑐 ∞ = 0 power function indicates the relationship of the horizontal asymptote is ? ∞ lim → 0.3428 ? 0.4237 y=0. The vertical asymptotes with a relationship x=0 is indicated by taking . As C sr is a characteristic property of the bike tire, negative air ? 0 lim → 0.3428 ? 0.4237 = ∞ pressure in a tire is impossible, making both asymptotes reasonable. There would have to be zero tire for the C sr to equal 0, which cannot be achieved through manipulation of the tire pressure. The increasing air pressure causes the coef±icient to decrease to a point where the decline becomes unnoticeable re±lected on the right side of the graph, which is realistic as the existence of a tire cannot be erased as a result of an increase in pressure. The asymptote can be questioned as it suggests that a coef±icient of rolling

resistance near 0 is obtainable. Various factors, previously explored in section 3.3, show how diverse factors besides the air pressure (surface roughness, tire material) affect the coef±icient of rolling resistance suggesting that the asymptote would not be zero. The other factors are independent of the air pressure which causes them to act as a minimum for how low the coef±icient of rolling resistance can go when manipulating the tire's air pressure. 5.3 Linearisation The relationship can be linearised through C sr to power ( ), or − 1 0.4237 ?ℎ 0. 061 − 1 0.4237 by taking the change in pressure to power ( 60psi - 0.4237 .) − 0. 4237 ?ℎ (Figure 3 - Graph of C sr to power vs change in pressure) − 1 0.4237 ?ℎ The acquired data indicates that to a large extent it follows the linearisation of function . Therefore the appropriacy of the previous function is supported as a model ? = 0.3428 ? 0.4237 for the data. The vertical order of the starting points for the separate trending lines corresponds with the vertical order of titles and equations for each trendline. Although multiple error bar edges rest outside the max-min lines, the overall linearisation of points ±its well within the max-min margin. The majority of error bars pass through the max-min area, which suggests no major outliers within the data, disrupting the model. Overall the trendline is suitable for the data, even though the error is fairly large. Furthermore, a positive correlation is displayed between the tire pressure and coef±icient of rolling resistance respective to the power, indicating an − 1 0.4237 ?ℎ accurate relationship modelled.

6.0 Conclusion The data obtained indicates that as the pressure of the bike tire increases, the coef±icient of rolling resistance consequently decreases. At 60 psi the coef±icient of rolling resistance was 0.061, at 100 psi the coef±icient of rolling resistance was 0.048 and at 140 psi, 0.043. The experimental data clearly underlines a decline representing an inverse relationship between the coef±icient of rolling resistance and pressure. Although there is a decline, it begins to diminish in correlation with an increase in pressure. The difference in the coef±icient of rolling resistance between 60psi and 80psi was 0.008, whereas the difference between 120psi and 140psi was 0.002, which is a fourth of the previous decline further reinforcing the possibility of an asymptote. This occurs as a result of the change in pressure correlating to a change in the surface area of the tire as a consequence of the tire deformation. The increased pressure would correspond to a reduced surface area. The greater the cross-sectional area of the tire, the greater the amount of rolling resistance it encounters since it collides with more particles on the surface (Simon von Bromley). Two minor outliers can be identi±ied in the raw data set for 100psi at trial 5 and 120psi at trial 3. These outliers can be justi±ied through multiple components, reaction time and subjectivity. As the human reaction time varies, the data measurements will be in±luenced by this human error. Subjectivity can relate to the human perception upon a situation in this case when the tire reached a velocity of 0. Throughout data collection, I based my perception on the tire's rotation however for a few trials I based my perception on the digital software of the bike displaying the velocity which could have altered the data and created these minor outliers. The reliability of the data can be re±lected through the 0.9944 The results, therefore, are consistent with the hypothesis as I hypothesised the inverse relationship that as the pressure of the bike tire decreases, the coef±icient of rolling resistance begins to increase, as a result of a larger surface area generating a greater resistant force, further reinforced by the linear graph (±igure 3) presenting a fairly strong correlation. Comparing my results to a study conducted by Greg Kopecky, published on Wednesday 04/09/2019, his data suggests a similar inverse relationship, further validating my investigation. Through this my real-life understanding of biking is re±lected, where an underin±lated tire will feel more dif±icult and slower to bike. 7.0 Evaluation 7.1 Evaluation of Hypothesis

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The results which I acquired from my experimental procedure were suf±icient and consistent enough to reinforce my hypothesis. Any errors did not signi±icantly alter the conclusion of the investigation, despite some uncertainties being consequentially higher than ideal, however reason as of this was mentioned in section 6.0. For both original and linearized data, no overlap occurred with the uncertainties, exhibiting how the experimental data acquired from the investigation supports the overall conclusion, despite the uncertainties. 7.2 Strength A major strength of the investigation was controlling the mass and keeping it constant. This was effectively controlled by placing the bike on a ±lat stone for elevation allowing the bike to tilt forwards and make contact with the surface without any additional mass. A constant mass ensures the weight is constantly allowing no alteration of the surface area displaced by the tire which is the key component in±luencing the C sr , therefore maintaining reliable results. Another strength was being able to accurately control the initial velocity of the wheel to an uncertainty of 0.01 through the bike's software ± settings. During the data collection, the ef±iciency of data collection largely contributed to a major strength. This allowed for minimal error when processing data and obtaining the C sr value. By ensuring to conduct each trial through a planned and controlled time period, I obtained all the needed results within a 30min time period without rushing results. An ef±icient data collection was essential to lower the error of temperature affecting the pressure discussed in (section 7.3). 7.3 Weakness Error’s during the experimental procedure which could have affected the reliability of the data would be my perspective, as I was the one who identi±ied when the tire’s velocity reached 0 and stopped the stop-watch. Human error could have occurred both through the millisecond delay of a human's reaction time as well as misinterpreting when the tire has reached a complete stop (0m/s) mentioned in (section 6). To minimise the random error, throughout the scienti±ic procedure I took 5 trials for each variation in pressure, therefore using the mean as a value. Due to a smaller range of data being collected, random error can be argued to be the largest source of uncertainty. The investigation was likewise done outside, allowing ±luctuations in the temperature, affecting the pressure within the tire. This is because the Ideal gas law states that as temperature increases, the velocity of Gas particles collides against the tire due to the average kinetic energy increasing. The pressure being the force exerted by particles per unit of area, Temperature is proportional to pressure (Patricia Shapley). To signi±icantly reduce this random error, if this investigation was repeated, conducting the experimental procedure indoors would allow a controlled room temperature which

would maintain constant through a controlled home thermometer, therefore possible temperature ±luctuations can be neglected. 7.4 Extension To reduce the effect of human error, if the investigation was repeated I would increase the initial velocity. This would ultimately enhance the distinction of times more enunciated and allow the reaction time to be more constant correlating to an increased focus. Likewise focusing on the digital screen displaying the velocity would reduce the human error of perception as the stopwatch would only be stopped once 0m/s is displayed. When evaluating how this investigation can be applied in real life, although variables such as propelling force, temperature and surface roughness are controlled, in real life these variables will never be constant. The investigation does not identify the relationship between the coef±icient of rolling resistance and various terrains/temperature. Furthermore, rougher terrain such as forests potentially would cause lower-pressure tires to lose less energy as reducing tire pressure would expand its tread and increase the surface area of the tire causing increased traction as the tire can dis±igure to match the surface roughness (Dean Mellor). For future investigations, these variables can be measured in various ways. To investigate the effect of various terrains, my method (section 4.3) can be conducted on various surfaces (sand, tarmac, rock), with a possible research question “How does the terrain (sand, tarmac, rock) affect a change in C sr of a bike tire”. A second possible investigation could consider the effect of temperature as pressure is heavily in±luenced by it (section 6.0), with a possible research question “How does the temperature of air (5 °C , 15 °C , 25 °C , 35 °C , 45 °C ) affect the C sr of a Bike tire” Works Cited

Brothers, Burt. “How Temperature Affects Tire Inflation.” Burt Brothers , 5 Sept. 2018, burtbrothers.com/blog/how-temperature-affects-tire-inflation/#:~:text=The%20inflatio n%20pressure%20in%20tires,to%2020%20minutes%20you%20drive.. Accessed 11 Sept. 2022. “FIFTY-TWO ADDITIONAL APPLICATIONS.” Applied Dimensional Analysis and Modeling , 2007, pp. 527–657, www.sciencedirect.com/topics/engineering/rolling-resistance-coefficient#:~:text=Not e%201%3A%20Rolling%20Resistance%20Coefficient,on%20a%20flat%20horizonta l%20surface., 10.1016/b978-012370620-1.50024-1. Accessed 11 Sept. 2022. https://blog.flocycling.com/author/floadmin. “Flo Cycling Blog | Carbon Bike Wheels Articles.” FLO Cycling , 23 May 2015, blog.flocycling.com/aero-wheels/bike-tire-pressure-and-temperature/. Accessed 11 Sept. 2022. “Khan Academy.” Khanacademy.org , 2022, www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-a cceleration-tutoria/a/what-is-centripetal-acceleration. Accessed 11 Sept. 2022. “Rolling Resistance.” Engineeringtoolbox.com , 2022, www.engineeringtoolbox.com/rolling-friction-resistance-d_1303.html. Accessed 11 Sept. 2022.

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unknown. “Figure 4. Rolling Resistance Coefficients vs. Inflation Pressure.” ResearchGate , ResearchGate, 5 Apr. 2016, www.researchgate.net/figure/Rolling-Resistance-Coefficients-vs-Inflation-Pressure_fi g3_299639874. Accessed 11 Sept. 2022.