Simple Gear Ratio Problem: Solving for Wheel Rotation with Formula

[alexcc17]

Homework Statement

In a bicycle, the ratio between the size of the wheel sprocket to the size of the crank sprocket is 2.5 which means...

An example of one of the answer choices is:

The rear wheel turns ___ times for every full rotation of the pedals.

Homework Equations

The Attempt at a Solution

I feel like that should be a really simple question if I had formula. We were given that:
ωwheel=(R1/R2)ωpedal
R1 is the radius of the crank sprocket
R2 is the radius of the wheel sprocket

This would mean that ωwheel=(2/5)ωpedal, but that doesn't really help.

 

[Pkruse]

No complicated math required. The wheel turns 2.5 times for every one turn of the crank.

 

[CWatters]

ωwheel=(R1/R2)ωpedal

Rearrange to give..

ωwheel/ωpedal = R1/R2

But perhaps it would help to understand it from first principles? Let's say TP is the Tooth Pitch in inches. If the crank sprocket had 50 teeth each revolution of the crank advances the chain a distance of..

50 * TP

Then if the wheel sprocket had 20 teeth how many revolutions would it make...

= 50*TP / 20*TP

TP cancels

= 2.5

It's quickly obvious that the gearing depends on the ratio of the number of teeth on each.

In fact the ratio depends on the size of the gears regardless of how you specify the size. For example you could specify the size in terms of:

Teeth
Circumference
Radius

It doesn't matter because they are all proportional to each other. In each case when you work out the ratio either ∏ or the tooth pitch will cancel.

 

[Pkruse]

It is common in the bike industry to express the ratio as the chain ring size divided by the cluster sprocket size and multiplied by the wheel diameter. If you multiply that by pi, then you have distance traveled per turn of the crank. That is a more meaningful metric.