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Speed of a Rotating Disc

Date: 03/01/98 at 04:13:28
From: Justin Yeo
Subject: Re: Average speed

Dear Doctor Math,

I have a problem with this question:

You have a compact disc rotating at a constant speed. The outer-most 
edge of the compact disc is rotating at, say, 20 km/h. Thus, the 
inner-most edge of a compact disc is also rotating at 20km/h. 

It is easily noticeable that the outer edge is travelling a longer 
distance than the inner edge, but if this is true, then the speeds 
should be different (the outer edge should spin faster, as it has to 
cover more distance in the same period of time).

But saying that the outer edge spins at a higher rate is also wrong. 
I tried this out with a paper disc and made two markings (one on the 
inner edge and one on the outer edge) on a straight line, and after 
spinning the disc for a while, the markings were still in line with 
each other. If one of the edges had spun faster than the other, the 
paper would have been ripped to pieces! This simple experiment proves 
that the outer edge spins no faster than the inner edge.

Dr Math, this question seems unsolvable to me. I know that there must 
be a mathematical explanation, but what? 

Date: 03/01/98 at 08:43:40
From: Doctor Jerry
Subject: Re: Average speed

Hi Justin Yeo,

Maybe you are mixing two kinds of speeds. Every point on the disk has 
the same angular speed, usually measured in radians per second. If 
the disk is turning at 10 rpm, then its angular speed is 

       10 rev     2*pi      1 min
       ------ * -------- * -------- = (pi/30) radians/sec
        1 min    1 rev      60 sec

The linear speeds of points on the disk depend upon how far they are 
from the center. You need the fact that the length s of the arc of a 
sector with central angle w (radians) in a circle of radius a is 

       s  =  a * w

In one second, the central angle will have increased by pi/30 radians.  
A point 2cm from the center will have moved 2*(pi/30) cm; a point 4 cm 
from the center will have moved 4*(pi/30).

The relation between linear speed and angular speed is that

    linear speed = a* angular speed. 

-Doctor Jerry,  The Math Forum
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