Re: when to write radians

[Joe Heafner <heafnerj@VNET.NET>]

On Monday, Jun 30, 2003, at 22:22 US/Eastern, thomas pfeiffer wrote:

> You make an excellent point. We want students to
> understand the mathematical angle and the operational
> angle. However, how do you make the radians disappear
> when using v = r*(omega) when calculating linear
> velocity from angular velocity and radius?

That question is what started all this for me. All through my undergrad
and grad school years, no one could explain "where the radians go" in
such expressions. Finally, in 2001, Anthony P. French from MIT sent me
a copy of a paper, an unpublished one I believe, that he wrote that
addresses this issue. Here's the answer.

v is a linear speed, the ratio of a linear displacement to a time
interval. Linear displacement, ALL linear displacement, is measured
with a stick (e.g. a meter stick) and its numerical value has
absolutely no dependence on which unit is used for angular measure. The
numerical value of angular speed, on the other hand, depends explicitly
on the unit used for angular measure. For example, consider a sphere of
radius 1 m rotating at an angular speed of 2pi rad/s. In a time
interval of 1 s, a point on the sphere's equator will sweep out an
angle of 2pi rad or 360 deg. Note that the numerical value of the
angular displacement depends directly on the angular unit used.
However, the linear displacement of the same point, when "unwrapped"
from the sphere's equator will have, and indeed MUST have, the same
linear length (measured with a stick operationally) regardless of
whether we used radians or degrees for the angular measure.
Numerically, the linear displacement is 2pi m, independent of the
angular unit.

The rule is this: If the numerical quantity depends on the angular
unit, then explicitly write the angular unit. If the numerical quantity
doesn't depend on the angular unit, then do not write the unit.
Therefore, we don't explicitly write the radian when we articulate a
value for a linear speed. The "radians" disappear because they were
never there in the first place!

I used to believe that all ratios are dimensionless and have units of
radians, but this is incorrect. Ratios are all dimensionless, but are
also unitless unless there is an angle involved. In such cases, the
preferred unit is the radian. It doesn't make sense, for example, to
compare the lengths of two sticks and slap a unit of radian on the
answer. The ratio of two speeds is dimensionless and unitless because
there's no angle involved.

One thing I'm struggling with now is what to call the argument of a
trigonometric or exponential function. Is it an angle in radians or a
pure (unitless) number? For example, sin(wt) is an angle but what about
exp(-t/RC)? Is the argument an angle or just a pure number? From my
explanation above, it appears that trig functions must take angular
arguments in radians and other numerical functions (exponentials,
logarithms, etc.) must take pure numbers, both dimensionless and
unitless. I welcome commentary.

Things like this are a much bigger issue that we traditionally realize.
Unit conversions are drilled into our students in basic math and we
attempt to carry them over into introductory physics. Students hit the
proverbial brick wall with radian measure and frequently are afraid to
openly admit it. Arons and French give the only coherent explanations
of this issue that I've come across.

Cheers,
Joe Heafner

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I don't have a Lexus, but I have a Mac. Same thing.