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Re: Radians, dimensions, & explanations



Mike Wilson wrote:

From <Mike Wilson mwilson@colosys.net>

Each year as I go through angular motion I realize
I do not have a simple explanation for why radians are a dimensionless
measurement.

I am always forced back into some discussion of the circumference of
a circle as 2 pi radi so the distance drops out.

Do you mean dimensionless or unitless? I suspect people with stronger
opinions will wade in soon, but let me muse a bit about this.

A radian is usually defined as measuring the angle subtended by an arc
as the ratio of the arc length to the radius of the arc. The dimensions
are distance divided by distance, so the dimensions cancel and you could
say it is dimensionless. This is unsettling, why can't you say the
dimensions are angle? Isn't this the dimension you are measuring if you
work in degrees... why would two unit systems for the same quantity have
different dimensions?

Perhaps you mean it is unitless? You have meters divided by meters, or
feet divided by feet, so the unit is one. If we don't mean that
"radian" is a unit when we say "360 degrees = 2pi radians", then why are
we treating it as a convertable unit by writing this equation? Finally,
isn't calling 'one' a 'radian' in the right context pretty much the same
thing as calling the same unit 'Nm' or 'J' depending on whether the
dimension of the quantity we are talking about is torque or energy?

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

Doug Craigen "Technology with purpose"
http://www.dctech.com