# Relation between force and torque for a set of gears/bicycle

If there are 2 gears meshed together and they are of different sizes, then rotating the smaller one will make the larger one spin with a smaller angular velocity but with more torque. And the opposite happens when you spin the larger one. Using a lower gear ratio in a bicycle for example, makes it easier to go uphill. How does the increased torque from the lower gear ratio help in this? Like how does the higher torque equate to a greater force to move the bike forward?

Changing to an easier gear reduces the ratio of torque required at the crankset over torque at the wheel. Given constant torque at the wheel (riding up a hill of a given grade at a given speed) this means the torque, and pedal forces, required at the crankset are lower. The rider thus feels it is easier to turn the pedals.

I do not want to put too much focus on the gear systems.

In general, $P_{in} = P_{out}$ (assuming no power loss). Using, $P = F v$, one gets $F_{in} v_{in} = F_{out} v_{out}$. That is, one can "scale up" the output force by moving through a greater distance per unit time (i.e. since $F_{out} = F_{in} (v_{in}/v_{out})$, increase $(v_{in}/v_{out})$)

What with gear systems? Replace forces with torques, velocities with angular velocities etc.

• So for a gear system, how does the increased torque make the bicycle easier to move uphill? – BlurryPic Nov 18 '15 at 7:01