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Bicycle Propulsion

When your legs push on the pedals, you are expending energy to propel the bicycle forward. This energy is transferred via the chain to the rear wheel, and friction with the ground pushes the bicycle forward. A key concept to understand in this system is torque, the rotational equivalent of linear force. You might already have an instinctive understanding of torque. For example, it’s easier to open a door when you push on it (with a constant force) far away from the hinges rather than close to them. It’s also easier to loosen a nut with a long wrench rather than with a short one. The formula for torque is given by: 


where F is the force applied about the center of rotation and d is the perpendicular distance between the center and the force. Now let’s look at the bicycle drive system.


The input torque that your legs apply about the center of the crank is given by the product of the downward pedal force F(in) and the crank length L.


This input torque is resisted by the chain with a tension force F(chain). The radius of the chainring, R(chainring), determines the perpendicular distance.


Combining Equations (2) and (3) to eliminate the input torque, we have:


At the rear wheel, the chain tension force will create an output torque. The radius of the sprocket, R(sprocket), determines the perpendicular distance and thus the magnitude of the output torque.


This output torque is resisted by the drive force acting on the ground at a perpendicular distance equal to the radius of the wheel, R(wheel).


As before, we can combine Equations (5) and (6) to eliminate the output torque. We then rearrange to isolate the chain tension force.


Next, we combine Equation (7) with Equation (4):


We see that the input and drive forces are related by four terms. For a given bicycle, the crank length and wheel diameter will (hopefully) remain constant, so we can ignore the L and R (wheel) terms. This means that the ratio of R(chainring) to R(sprocket), known as the gear ratio, determines how the input force will be converted into output force.


Although the gear ratio pertains to the radii of the chainring and sprocket, it is more useful to think in terms of the number of teeth N, which are directly proportional to the radii (a gear with twice the radius of another gear will have twice the number of teeth).