# Simple Gear Ratio Problem: Solving for Wheel Rotation with Formula

• alexcc17
In summary, the ratio between the size of the wheel sprocket to the size of the crank sprocket in a bicycle is 2.5, meaning that the wheel will turn 2.5 times for every one turn of the crank. This ratio is dependent on the number of teeth on each gear, regardless of how the size is specified. In the bike industry, it is common to express this ratio as the chain ring size divided by the cluster sprocket size and multiplied by the wheel diameter, which gives a more meaningful metric for distance traveled per turn of the crank.
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.

## 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.

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

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

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.

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.

I think the answer should be that the rear wheel turns 2.5 times for every full rotation of the pedals, but I'm not sure. I would need more information about the rotational speed of the pedals to accurately determine the answer.

## What is a gear ratio and why is it important?

A gear ratio is the ratio between the number of teeth on two interlocking gears. It is important because it determines the speed and torque of a mechanical system, and can be used to increase or decrease the output speed or force.

## How do I calculate the gear ratio?

To calculate the gear ratio, divide the number of teeth on the driven gear by the number of teeth on the driving gear. For example, if the driven gear has 20 teeth and the driving gear has 10 teeth, the gear ratio is 20/10 = 2.

## What is the difference between a simple gear ratio and a compound gear ratio?

A simple gear ratio involves two gears directly connected to each other, while a compound gear ratio involves multiple gears connected in a series. This allows for a more complex gear ratio and can achieve higher or lower speeds and torques.

## How does gear ratio affect the performance of a machine?

Gear ratio affects the performance of a machine by determining the speed and torque at which it operates. A higher gear ratio will result in a slower but more powerful output, while a lower gear ratio will result in a faster but weaker output.

## Can gear ratio be changed in a mechanical system?

Yes, gear ratio can be changed in a mechanical system by replacing the gears with ones that have a different number of teeth. This will alter the gear ratio and therefore the speed and torque of the system.

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