Cycling Power

3 minutes read

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.