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Re: [apcalculus] Units of RADIANS per SecondRelated Rates Problem
Posted:
Oct 20, 2012 6:21 PM


NOTE: This apcalculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus Teacher Community Forum at https://apcommunity.collegeboard.org/gettingstarted and post messages there.  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 WiedBrusky" <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 > setup 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 WiedBrusky > Mathematics Department > Brookfield East High School > Brookfield, Wisconsin 53005
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