Math Is Fun Forum

prakas24

Member

Registered: 2012-03-06

Posts: 2

Urgent Help

The pedal shafts on a bike are 8 inches long. The front gear has a diameter of 10 inches, the rear gear has a diameter of 5 inches. The wheels, including the tires, have a radius of 14 inches. How fast (in rpm) does the rider have to pedal in order for the bike to travel 15 miles per hour?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Urgent Help

Hi prakas24;

Have not worked on a bike in a long time so I have forgotten most of the mechanism. But from what I do remember, I would say such a bike is infeasible. Here is why.

For every pedal revolution the rear wheel turns twice because the gearing is two to one. That means the bike travels forward 28 π  inches or 87.964594 inches. From that I derive this formula:

That gives the mph for any revolutions per hour. So we solve:

We get 10804.346993 revolutions per hour or rph. We divide that by 60 to get the rpm.

So we need about 180 rpm or about 3 rps. That is two much pedalling.

Welcome to the forum.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

prakas24

Member

Registered: 2012-03-06

Posts: 2

Re: Urgent Help

Thanks a lot!

The student gave the answer as 90 rpm. Could you please verify it? Also please explain how you derived that formula.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Urgent Help

The front gear makes one revolution for each revolution of the pedals. They are connected. The rear gear spins twice for every revolution of the front gear.

Each revolution propels the bike 28 π  inches forward. This is because the circumference is π * d. d = 2r.

I put everything in terms of revolutions per hour or rph. The distance travlled per hour in inches is

28 π * rph

To get the feet travelled you divide by 12.

To get the miles travelled you divide that by 5280.

The right hand side is a simplification.

To find how many revolutions per hour you form the equation

When you solve that I get 10804.346993 revolutions per hour. You divide that by 60 to get180.07244 rpm.

Your student may have used 14 π instead of 28 π. Or else I have got my gearing wrong.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,649

Re: Urgent Help

hi bobbym and prakas24

I get 90 rpm.  Explanation follows.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,649

Re: Urgent Help

hi again,

Here we go:

The length of the pedal crank is irrelevant; it's the rate of turning of the crank wheel that determines the speed.

1 revolution of the crank wheel causes 2 revolutions of the rear sprocket.

That moves the rear wheel 2 x 2 x pi x 14 inches along the road. = 56 pi

= 56 pi /(12 x 3 x 1760) miles along the road.

So let's say you need x revs/hour to go at 15 mph.

so

hope that helps

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Urgent Help

Agreed, I forgot to multiply by 2.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.