# Cycling Power

## 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