Changing Gears

Let $r_p$ be the distance of pedal from its axis, $r_f$ be the radius of front gear, $r_b$ be the radius of back gear, and $r_w$ be the radius of the back wheel.

Suppose we apply a force of $F_p$ on the pedal. Then the torque on the front gear is $\tau_f = F_pr_p$ . The force in the chain due to this is $F_c = \dfrac{\tau_f}{r_f} = \dfrac{r_p}{r_f}F_p$ .

The chain applies a torque of $\tau_b = F_c r_b = \dfrac{r_pr_b}{r_f}F_p$ on the back wheel. This torque is balanced by the torque due to static friction between the wheel and the ground (assuming the wheel does not slip). We get

$F_d = \dfrac{r_pr_b}{r_fr_w} F_p$

Here, $F_d$ is called the driving force. It is provided by static friction between the wheel and the ground. It is the external force on the cycle which pushes it forward.

The values $r_p$ and $r_w$ are fixed in a given cycle, so for a given $F_d$ , the value of $\dfrac{r_b}{r_f} F_p$ is also fixed. Therefore $F_p$ is directly proportional to $\dfrac{r_f}{r_b}$ . To decrease $F_p$ , the amount of force required to pedal, we should decrease $r_f/r_b$ . We can cause the maximum decrease in $r_f/r_b$ if we decrease the radius of the front gear and increase radius of the back gear.

This gear configuration is called low gear, because it requires the least amount of force to pedal, and also because each full turn of the pedal results in a smaller distance moved by the cycle. If we increase the gear ratio $r_f/ r_b$ by using a larger front gear and/or smaller back gear, then the configuration is called high gear. It requires more force to pedal, but each turn of the pedal results in greater forward motion by the cycle.