Shifting Gears, Part 3

by David Wolpe on October 3, 2014

If you can’t explain it simply, you don’t understand it well enough. ~Albert Einstein

Imagine yourself on an early bicycle,  with no chain and a big front wheel, as pictured below. The radius of the circle described by the movement of the pedals is smaller than that of the wheel on which they reside — the front wheel.   So if the rider pedals once around, the cycle moves a distance equal to the circumference of the front wheel. The larger the wheel, the farther the rider goes, and the more difficult it is to pedal. This is why early bicycles had such large front wheels.

 

An early bicycle

And this is a simpler system than our modern day chains. This suggests that a small radius ring — equivalent to the circle described by the pedals — will propel the bicycle faster. The smaller the ring relative to the wheel, the greater the mechanical advantage and the faster the bicycle will go — which means it is a high gear.

This is exactly what happens on the back wheel of a modern bike. It is the ‘drive’ wheel — the wheel that is connected to the gears.  On that rear wheel, a full revolution of the little center gear is the equivalent of a revolution of the tire — the same scenario as the fixed wheel, above.  It then makes sense that as the gear ring size increases, the bike goes farther a lesser amount — but is easier to pedal.

The chain is what allows you to shift gears — by moving to a smaller, or larger, ring.  (And remember, we’re only talking about the back right  now.) The chain can be thought of as a direct translator — it directly translates the motion of the pedals into the gear on the back.  It is, in a sense, invisible, like the disappearing middle term of a mathematical equation: it is easy to think of it not being there at all.  It is just the circumference described by that gear relative to the circumference of the wheel like in the old bicycle.

With this concept in mind, we can now think about the front wheel’s gears. Say we are on the smallest ring in the back — the one which, we now know, will propel the bike forward the farthest.  Why would a larger ring in front make the bike go farther — the ‘longer stride’ analogy?  Not sure yet.  But we’re getting there.

The larger the ring, the greater the number of sprockets on it.  Let’s say that that smallest rear ring has 15 sprockets.  And that we now introduce the chain.  The chain connects the front to the back. Let’s now say that the chain is on a large ring in the front which has 30 sprockets.  In order to make that rear ring go around one full revolution, we would have to make only a half-circle of the front ring, because (assuming that sprockets are spaced the same in the front as in the back) that is all we need to engage 15 sprockets.  A full revolution would engage 30 sprockets, which would mean that the back ring went around two full revolutions.

And what if the front ring has 45 sprockets?  A full turn of the front ring would propel us forward three turns of the rear ring. 60? Four turns. And so on. In other words, the larger the front ring size, the harder it is to pedal, because the greater distance the cycle is propelled, and the greater the number of full revolutions of the wheels.

Gears and the chain

What have we done here? Why is it now easier to understand? Why did we not understand it before? What is it that occurs when understanding occurs? Interestingly, the same principle was applied to both front and back: the circumference of the ring relative to that which it is propelling. In the case of the rear ring, it is propelling that back wheel.  In the case of the front ring, it is propelling that rear ring (which in turn propels the wheel).

This process of detachment — or, in programmer terms, of decoupling — is one we will come across repeatedly. It seems to be a critical technique in the execution of an increasingly intelligent machine.

At least a part of our earlier confusion was due to the fact that it was difficult to comprehend, to grasp, the system.  Which is the ‘drive’ gear? Is it the front one or the rear? Sometimes the head hurts and the mind wants to take a vacation. Gears on the left, gears on the right, stand up sit down fight fight fight!

Vacation over.

We had size varying in two places, described in conflicting and ambiguous terms, connected by a chain, which binds two subsystems together. We disassembled and simplified the system (we took off the chain, for one thing) in order to examine its parts. We have to have something to hang on to — a reference point — from which we can proceed. That something was the circumference of the pedal circle relative to the wheel circle. We eliminated complicating factors in order to be able to see one thing clearly.

Understanding bears some relation to sight. It is probably no accident that when we understand something, we say, “I see.” It also bears a relation to a less precise sense: touch.  That is a concept which we hope to grasp by illuminating similar moments of confusion. That is the subject of the next post.

 

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