Ride a bicycle

Classical Mechanics Level 3

What is the maximum achievable driving force F ⃗ wheel \vec F_\text{wheel} (in units of Newton) of a bicycle when one pedal is loaded with the full weight m = 70 kg m = 70 \text{ kg} of the driver? The gear shift is set to the lowest gear ratio.


Details and Assumptions:

  • The crank length is l = 17.5 cm l = 17.5 \, \text{cm} and the wheel radius is R = 35 cm R = 35 \, \text{cm} .
  • The numbers of the teeths of the chainring ( n 1 ) (n_1) and rear cogs ( n 2 ) (n_2) take the values n 1 ∈ { 24 , 48 } and n 2 ∈ { 12 , 24 , 36 } , n_1 \in \{24,48\} \quad \text{and} \quad n_2 \in \{12,24,36\}, depending on the gear selected.
  • Gravity acceleration is g = 10 m/s 2 . g = 10 \, \text{m/s}^2.


The answer is 525.

Markus Michelmann by Markus Michelmann , 35, Germany

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1 solution

Markus Michelmann
Oct 22, 2017

The cogs have each a circumference C i = 2 π r i = n i d ⇒ r i = d 2 π n i , i = 1 , 2 C_i = 2 \pi r_i = n_i d \quad \Rightarrow \quad r_i = \frac{d}{2 \pi} n_i \,,\qquad i = 1,2 with a radius r i r_i of the the cog and distance d d between two teeths, that is equal the length of one chain link. The torque acting on the pedals and chainring results T 1 = r 1 F 1 = m g l T_1 = r_1 F_1 = m g l The torque acting on the rear wheel and on the pinion results T 2 = r 2 F 2 = R F wheel T_2 = r_2 F_2 = R F_\text{wheel} With F 1 = F 2 F_1 = F_2 the equations can solved for the driven force: F wheel = l R r 2 r 1 m g = l R n 2 n 1 m g F_\text{wheel} = \frac{l}{R} \frac{r_2}{r_1} m g = \frac{l}{R} \frac{n_2}{n_1} m g The force is maximal for the teeth numbers n 1 = 24 n_1 = 24 and n 2 = 36 n_2 = 36 : F wheel = 17.5 cm 35 cm ⋅ 36 24 ⋅ 700 N = 3 4 ⋅ 700 N = 525 N F_\text{wheel} = \frac{17.5 \,\text{cm}}{35 \,\text{cm}} \cdot \frac{36}{24} \cdot 700 \,\text{N} = \frac{3}{4} \cdot 700 \,\text{N} = 525 \,\text{N}

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Do they really make bikes so that the rear has more cogs than the front? The n_1=24 seems too small for the front. Perhaps it is possible for a mountain bike....

Laszlo Mihaly - 3 years, 7 months ago

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You're right, the rear has normally less cogs than the front. This bike is a custom-build model with which you can cope with the biggest climbs. (Originally, I wanted to ask in the problem, if you can bike up a slope with 100% slope.)

Markus Michelmann - 3 years, 7 months ago

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