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Topic: [ap-calculus] Units of RADIANS per Second--Related Rates Problem
Replies: 2   Last Post: Oct 21, 2012 6:18 AM

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Paul A. Foerster

Posts: 892
Registered: 12/6/04
Re: [ap-calculus] Units of RADIANS per Second--Related Rates Problem
Posted: Oct 20, 2012 6:21 PM
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------------------------------------------------------------------------------------------------
Leanne, et al. -

I agonized for years over the sudden appearance of "radians" in angular
velocity problems. The best I could do was "wave my hands" and say,
"Radians are dimensionless, so you can put them in anywhere you need
them." This satisfied neither me nor my students.

About ten years ago I had an insight that clears up this dilemma. The
radius of rotation really tells you how far a point at the end of the
radius travels when the object rotates through an angle of 1 radian. If
the radius is in centimeters, then the proper radius units are centimeters
per radian. Thus all the formulas submit to dimensional analysis.

A = r theta has units cm = cm/radian * radians

V = r omega has units cm/second = cm/radian * radians/second

Since then the mystery of the appearing and disappearing radian is no
longer mysterious to my students.

Best wishes to you and your students!

Paul
Teacher Emeritus of Mathematics
Alamo Heights High school
San Antonio

--------------------------------------------------------
"Leanne Wied-Brusky" <wiedbrul@elmbrookschools.org> writes

> I have a question about the units in a related rates problem. I insist on
> using units throughout a related rates problem to reinforce the proper
> set-up and understanding of an application problem.
>
> When dealing with a rate of change of angle problem, how do units of
> radians/second suddenly appear? For instance, when we have the classic
> falling ladder problem:


> cosA = (x ft)/(13 ft) and dx/dt = 2 ft/sec at the moment when x = 5 ft

> How does dA/dt acquire units of RADIANS/sec?

> Leanne Wied-Brusky
> Mathematics Department
> Brookfield East High School
> Brookfield, Wisconsin 53005




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