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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:

Torque_{pedal} = Force_{cyclist} *
L_{pedal-arm}

where Force_{cyclist} is the drive force
provided by
the cyclist’s leg, and L_{pedal-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:

Torque_{front-chain} = Force_{front-
chain}
* L_{front-sprocket}

where Force_{front-chain} is the force on the
chain,
and L_{front-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.

Torque_{pedal} = Torque_{front-
chain}

Force_{cyclist} * L_{pedal-arm} =
Force_{front-chain}
* L_{front-sprocket}

A similar argument can be made with the rear axle.

Torque_{rear-chain} = Force_{rear-chain}
* L_{rear-sprocket}

Torque_{rear-tire} = Force_{on-road}*
L_{tire-radius}

where the Force_{rear-chain} is the force on the
rear chain, L_{rear-sprocket} is the radius of the rear sprocket,
Force_{on-road}
is the propulsion force toward the road, and L_{tire-radius} is the
radius of the rear tire.

Again because the tire and sprocket move together during forward motion,

Torque_{rear-chain} = Torque_{rear-
tire}

Force_{rear-chain} * L_{rear-sprocket}
= Force_{on-road}* L_{tire-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,

Force_{front-chain} = Force_{rear-
chain}

and using this equation we can relate the force on the pedal with the force produced to propel the bike to arrive at:

Force_{on-road} / Force_{cyclist} =
L_{pedal-arm}
* L_{rear-sprocket} / ( L_{front-sprocket} * L_{tire-
radius}).

The force to the road represents the force used to
overcome
friction and accelerate the bicycle.
More Force_{on-road} means a faster ride. The ratio of
Force_{on-road} / Force_{cyclist}
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 * L_{pedal-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 * Force_{on-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,

(Force_{on-road} / Force_{cyclist})’ =
K *
(Force_{on-road} / Force_{cyclist}).

So the cyclist can provide less force to achieve the same speed.

A longer pedal arm is an advantage from this simple argument.

Sincerely,

Tom “Stubby Legs” Cull

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