The physics behind cycling
To move forward with constant speed \(V\) you have to provide energy (power) to overcome total resistive force:
\[P = F_r \cdot V = ( F_{downhill} + F_{rolling} + F_{drag} ) \cdot V\]Gravity
Cycling uphill or downhill force:
\[F_{downhill} = m \cdot g \cdot sin(\theta)\]where:
- \(m\) - weight of cyclist and bike;
- \(g = 9.80665~m/s^2\) - earth-surface gravitational acceleration.
Rolling resistance
\[F_{rolling} = C_{rr} \cdot m \cdot g \cdot cos(\theta)\]where:
- \(C_{rr}\) - coefficient of rolling resistance.
The coefficient of rolling resistance of the air filled tires on dry road:
\[C_{rr} = 0.005 + \frac 1 p \left( 0.01 + 0.0095 \left(\frac V {100}\right)^2 \right)\]where:
- \(p\) - the wheel pressure (Bar);
- \(V\) - the velocity (km/h).
The angle \(\theta\) can be calculated using elevation gain and total distance:
\[tan(\theta) = \frac H L \Rightarrow \theta = arctan\left(\frac H L\right)\]where:
- \(H\) - height (opposite side);
- \(L\) - length (adjacent side).
Aerodynamic Drag
Drag force:
\[F_{drag} = \frac 1 2 \cdot \rho \cdot (V - V_w)^2 \cdot C_d \cdot A\]where:
- \(\rho\) - the density of the air;
- \(V\) - the speed of the bike;
- \(V_W\) - the speed of the wind;
- \(A\) - the projected frontal area of the cyclist and bike;
- \(C_d\) - the drag coefficient.
Approximated body surface area can be estimated from the measurement of the body height and body mass (Du Bois & Du Bois, 1916; Shuter & Aslani, 2000):
\[A = 0.00949 \cdot (H/100)^{0.655} \cdot m^{0.441}\]where:
- \(H\) - the body height in \(m\);
- \(m\) - the body mass in \(kg\).
Drag coefficient in cycling can be related to the body mass also and depends on cyclist position.
Density
The density of the air is its mass per unit volume:
\[\rho = \frac m V\]where:
- \(m\) - the mass;
- \(V\) - the volume.
It decreases with increasing altitude and changes with variation in temperature or humidity.
The density of dry air:
\[\rho = \frac {p_0 M} {R T_0} \left(1 - \frac {Lh}{T_0}\right)^{gM/RL-1}\]where air specific constants:
- \(p_0 = 101325~Pa\) - sea level standard pressure;
- \(T_0 = 288.15~K\) - sea level standard temperature;
- \(M = 0.0289654~kg/mol\) - molar mass of dry air;
- \(R = 8.31447~J/(mol \cdot K)\) - ideal gas constant;
- \(g = 9.80665~m/s^2\) - earth-surface gravitational acceleration;
- \(L = 0.0065~K/m\) - temperature lapse rate.
Density close to the ground is:
\[\rho_0 = \frac {p_0 M} {R T_0}\]At sea level and at 15℃, air has \(1.225~kg/m^3\).
Using exponential approximation:
\[\rho = \rho_0 e^{(\frac {gM}{RL} - 1) \cdot ln(1 - \frac {Lh}{T_0})} \approx \rho_0 e^{-(\frac {gMh} {R T_0} - \frac {Lh} {T_0})}\]Thus:
\[\rho \approx \rho_0 e^{-h / H_n}\]where:
\[\frac 1 H_n = \frac {gM} {R T_0} - \frac L T_0\]So \(H_n = 10.4~km\).
Coefficients Table
Rolling resistance coefficient:
| Tire type | \(C_{rr}\) |
|---|---|
| Bicycle | 0.006 |
| Road bike | 0.004 |
Surface area and drag coefficient of cyclist:
| Position | \(A~m^2\) | \(C_d\) | \(C_d A\) |
|---|---|---|---|
| Back Up | 0.423 | 0.655 | 0.277 |
| Back Horizontal | 0.370 | 0.638 | 0.236 |
| Back Down 1 | 0.339 | 0.655 | 0.222 |
| Back Down 2 | 0.334 | 0.641 | 0.214 |
| Elbows | 0.381 | 0.677 | 0.258 |
| Froome | 0.344 | 0.677 | 0.233 |
| Top Tube 1 | 0.371 | 0.644 | 0.239 |
| Top tube 2 | 0.355 | 0.611 | 0.217 |
| Top Tube 3 | 0.345 | 0.588 | 0.203 |
| Top Tube 4 | 0.333 | 0.604 | 0.201 |
| Pantani | 0.343 | 0.618 | 0.212 |
| TT* & TT Helmet | 0.370 | 0.641 | 0.237 |
| TT Top Tube | 0.331 | 0.568 | 0.188 |
| TT & Helmet | 0.374 | 0.679 | 0.254 |
| Superman | 0.244 | 0.615 | 0.150 |
* Time trial
Conclusion
Based on all this formulas we are able to calculate power effort, burned calories and fat loss of bike ride activity.
Links
- http://bikecalculator.com
- http://thecraftycanvas.com/library/online-learning-tools/physics-homework-helpers/incline-force-calculator-problem-solver/
- https://www.gribble.org/cycling/power_v_speed.html
- https://www.researchgate.net/publication/51660070_Aerodynamic_drag_in_cycling_Methods_of_assessment
- https://www.sciencedirect.com/science/article/pii/S0167610518305762