Explanation of Bike Odometer Readings for Different Wheel Sizes

[rcmango]

Homework Statement

There is a bike odometer that measures distance traveled for 27" wheels.

So what heppens if you use it on a bike with 24" wheels?

Will the odometer show the same, a larger or smaller distance traveled?

Explain.

Homework Equations

The Attempt at a Solution

My guess is that since the wheels are smaller it will measure a smaller distance?

The odometer will show a less distance travelled.

Not sure why though.

 

[Kurdt]

A bike odometer works by counting the revolutions of the wheel. You program it with the size of the wheel and from that it can work out how far you've traveled by simply multiplying the circumference of the wheel by the number of revolutions. If the computer is put on a bike with smaller wheels and is still programmed for the bigger wheel, what do you think will happen?

 

[rcmango]

it will probably compute a longer distance for each revolution since the wheels are supposed to be big.

 

[Bill Foster]

The odometer gets its reading from the number of wheel rotations.

Let $$x$$ be the distance traveled.

Let $$n$$ be the number of revolutions the wheel made.

Relating the two:

$$x=n2\pi r=n2\pi \frac{d}{2}$$ where $$r$$=radius and $$d$$=diameter.

If a bike travels a certain distance with a wheel of the first size, then...

$$x=n_1 2\pi r_1=n_1 2\pi \frac{d_1}{2}$$

And if it travels the same distance with a wheel of the second size, then...

$$x=n_2 2\pi r_2=n_2 2\pi \frac{d_2}{2}$$

Now we can write...

$$n_1 2\pi \frac{d_1}{2}=n_2 2\pi \frac{d_2}{2}$$

Reducing this, we get...

$$n_1 d_1=n_2 d_2$$

The odometer reading is some function of $$n$$ as stated before.

So... $$n_2=n_1 \frac{d_1}{d_2}$$

So plugging in the numbers...

$$n_2=n_1 \frac{27}{24}=1.125n_1$$

The odometer will read higher by 12.5%.