# 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?

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 $(v_{in}/v_{out})$)