How does changing gears make a bicycle easier or harder to ride?
Q. Changing gears on a bicycle makes it easier or harder to pedal. Changing the front gears to bigger gears will make the bicycle harder to ride, while changing the back gears to bigger gears makes it easier to ride. Why are they different?
A. If we first look at the front gears, we notice that they all turn one full rotation when we push the pedals through one rotation. The difference comes in their size. The big gear has a larger circumference than the smaller gears and so pulls more length of chain for every full rotation than the smaller ones. Since the chain is, to a good approximation, under uniform tension, the length of chain pulled corresponds to an amount of work done. Thus the larger gear must be doing more work per rotation of the pedals, making it harder to turn.
While the front gears get harder with increasing size, the back gears perform the opposite, making it easier to turn the pedals with the larger gears. The way to understand this is that the front gears are pulling the chain, while the chain is pulling the back gears. We can understand this in just the way we understood the front gears:
The back gears turn one full rotation when the wheel turns one full rotation, not when the pedals turn one rotation. If we imagine that the wheel takes a set amount of work to turn it one full rotation, then the smaller gears are doing the same amount of work as the larger gears, but are doing so with a lesser amount of chain. This means that the chain must be pulling with a greater force on the smaller gears, meaning that we have to push harder on the pedals.
If this still seems confusing, try to separate the front and back gears into two separate systems. The front gears interact with the pedals and the chain, while the back gears interact with the chain and the wheel.