Cycling power calculation
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# Cycling power calculation

## 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$$C_{rr}$$
Bicycle0.006

#### Surface area and drag coefficient of cyclist:

Position$$A~m^2$$$$C_d$$$$C_d A$$
Back Up0.4230.6550.277
Back Horizontal0.3700.6380.236
Back Down 10.3390.6550.222
Back Down 20.3340.6410.214
Elbows0.3810.6770.258
Froome0.3440.6770.233
Top Tube 10.3710.6440.239
Top tube 20.3550.6110.217
Top Tube 30.3450.5880.203
Top Tube 40.3330.6040.201
Pantani0.3430.6180.212
TT* & TT Helmet0.3700.6410.237
TT Top Tube0.3310.5680.188
TT & Helmet0.3740.6790.254
Superman0.2440.6150.150

* Time trial

### Conclusion

Based on all this formulas we are able to calculate power effort, burned calories and fat loss of bike ride activity.

• 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