# Applying Force to a Wheel

I am sure this is a simple problem, but I am posting this thread to make sure I understand it properly:

If a motorcycle requires 3000N of force in order for it to achieve a certain speed over specific terrain and allowing for wind drag, etc. (I.E to counter the sum of the resistive forces acting against it), then it will need 3000N of thrust. This could be accomplished by an immensely strong person pushing the motorcycle or by a shaft and bevel drive connected to one of the wheels creating 3000N of ‘push’ where the tyre contacts the ground (directly below the axel).

However, if the wheel has a radius that is three times that of the bevel cog (the cog attached to the wheel and driven by the drive shaft) then in order for the wheel to have 3000N of push against the ground (road surface) the shaft drive would need to input three times this 3000N into the cog. E.g. if the cog, that is directly mounted to the center of the wheel, has a radius of 0.1m and the wheel, that is three times larger, has a radius of 0.3m then the drive shaft should exert a force of 9000N upon the cog in order to achieve the required 3000N of thrust at the ground/tyre contact point.

Is this correct?

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berkeman
Mentor
Yes, that sounds correct. You can think about it in terms of torques. 1/3 the lever arm requires 3x the force in order to produce the same torque. The sum of all forces and torques on an object must equal zero if it is not accelerating linearly or rotationally.

I have just been thinking about this; if a racing cyclist had a rear cog the same size as the wheel to which it is attached then the cyclist would be able to transmit more force into the wheel and thus be able to go faster. Of course, in practice, there are other factors that get in the way, such as the size of the driver cog (attached to the cranks), effects of having such a large cog on a wheel, etc., but if these can be overcome then, in theory, a cyclist could transmit several times the power that they currently transmit using standard gear setups.

This must be why many advocates of small-wheeled bicycles claim their bikes to be faster and more efficient than conventionally wheeled bikes. The reports I have read talk only of effects of rolling resistance of various wheel sizes, concluding that smaller wheels fare better (although many others claim that larger wheels endure lesser effects of rolling resistance). The authors of these reports said nothing of mechanical advantage, or disadvantage, concerning the respective sizes of the driven wheel and cog/sprocket.

I find that to be rather odd.

berkeman
Mentor
Road racing bicycles have gear ranges that allow the rider to target the most efficient cadence for the conditions. It is no help to put more power to the ground if you can only sustain that power output for a brief period.