|MadSci Network: Engineering|
Scaling the crank and pedals for the height of the rider can generate an advantage for a tall cyclist.
We can derive this advantage by looking at the simple balance of torques and forces involved in the pedal/crank and back sprocket/tire system. This is the drive system of a standard bicycle. This analysis is borrowed from The Physics of Sports edited by Angelo Armenti, Jr. The article, Bicylces in the Physics Lab was written by Robert G. Hunt and originally appeared in The Physics Teacher 27, 160-165 (1989).
You might also find this analysis in a high school or college general physics textbook. I know I saw it back in a science class in the early 1980’s.
Looking at pedal and front sprocket, collectively the crank, the force provided by the cyclist on the pedal times the length of the pedal arm produces a torque:
Torquepedal = Forcecyclist * Lpedal-arm
where Forcecyclist is the drive force provided by the cyclist’s leg, and Lpedal-arm is the length of the pedal arm measured from the center of the crank to the center of the pedal.
This torque is transferred to the front sprocket by the chain to produce a torque on the chain given by:
Torquefront-chain = Forcefront- chain * Lfront-sprocket
where Forcefront-chain is the force on the chain, and Lfront-sprocket is the radius of the sprocket (measured from the crank axis to the chain).
These torques are equal since the pedal and sprocket move together.
Torquepedal = Torquefront- chain
Forcecyclist * Lpedal-arm = Forcefront-chain * Lfront-sprocket
A similar argument can be made with the rear axle.
Torquerear-chain = Forcerear-chain * Lrear-sprocket
Torquerear-tire = Forceon-road* Ltire-radius
where the Forcerear-chain is the force on the rear chain, Lrear-sprocket is the radius of the rear sprocket, Forceon-road is the propulsion force toward the road, and Ltire-radius is the radius of the rear tire.
Again because the tire and sprocket move together during forward motion,
Torquerear-chain = Torquerear- tire
Forcerear-chain * Lrear-sprocket = Forceon-road* Ltire-radius
The force on the rear part of the chain should be the same as the force on the front part of the chain or it will be torn apart. So,
Forcefront-chain = Forcerear- chain
and using this equation we can relate the force on the pedal with the force produced to propel the bike to arrive at:
Forceon-road / Forcecyclist = Lpedal-arm * Lrear-sprocket / ( Lfront-sprocket * Ltire- radius).
The force to the road represents the force used to overcome friction and accelerate the bicycle. More Forceon-road means a faster ride. The ratio of Forceon-road / Forcecyclist at cruising speed (constant speed), is the mechanical advantage of the bike.
Now, let’s look at the advantages of changing the length of the pedal arm while leaving other parts fixed. Let’s assume that the force produced by the cyclist does not change.
As equation let’s express the new pedal arm as:
L’pedal-arm = K * Lpedal-arm
where K is a number that represents scaling size. For example, a pedal arm twice as long, K = 2.
Assuming everything else stays the same, the new force on the road Force’on-road:
Force’on-road = K * Forceon-road.
In words, the force passed to the road scales with the length of the pedal arm, if all other parameters remain fixed.
Therefore a longer pedal means more force to the road, so the bike accelerates faster for the same force provided by the cyclist.
When cruising at constant speed,
(Forceon-road / Forcecyclist)’ = K * (Forceon-road / Forcecyclist).
So the cyclist can provide less force to achieve the same speed.
A longer pedal arm is an advantage from this simple argument.
Tom “Stubby Legs” Cull
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