Bicycle

Date: 03/03/99 at 23:26:54
From: Maureen
Subject: Angular and Linear Speed

Speed of a Bicycle: The radii of the sprocket assemblies and the wheel 
of the bicycle are 4 inches, 2 inches, and 13 inches, respectively. If 
the cyclist is pedaling at the rate of one revolution per second, find 
the speed of the bicycle in a) feet per second and b) miles per hour.

(The back wheel has a radius of 13 inches. The chain is connected to 
the center part of the back tire, with a radius of 2 inches, and is 
also connected to the pedal, with a radius of 4 inches.) 

I have the formula that speed = r/t. I was also told that Velocity = 
Wt. I am not sure where omega comes into the problem, and do not know 
if I am supposed take the average of the three circles or what. 
Whatever helpful hints you can give me will sure help. 

Thank you!

Date: 03/04/99 at 16:16:03
From: Doctor Rick
Subject: Re: Angular and Linear Speed

What you need to do is to work step by step, following the "power 
train" of the bike.

The front sprocket wheel (turned directly by the pedals) has a radius 
of 4 inches. I would not put in the actual number, but just call the 
radius r1.

The pedals are turning 1 revolution per second. This means they turn 
through an angle of 2pi radians per second; this is the angular 
velocity W (omega).

The linear velocity of the chain (at radius r1) is W * r1. (Your 
formula was incorrect; linear velocity is angular velocity times 
radius, not time.)

Each link in the chain is moving at the same linear velocity (though 
the direction changes). You can use this fact to find the linear 
velocity v2 of the rim of the rear sprocket wheel (radius 2 inches, I 
would call it r2). Then use v = W * r to find the angular velocity W2 
of the sprocket.

The wheel turns with the rear sprocket, so its angular velocity is the 
same, W2. Knowing the radius of the wheel (13 inches), you can find 
the linear velocity of the outer edge of the wheel (relative to the 
hub). And this is equal to the speed of the bike.

You will have to do some conversions to get the speed in feet per 
second and miles per hour, because up to this point you have been 
working in inches and seconds. Keep the units with your numbers when 
you put everything together, and you will be able to see how to do the 
conversion.

That is an outline of the method I would use to solve this problem. I 
hope it helps you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/