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Re: Radians, dimensions, & explanations



On Thu, 10 Dec 1998, Jerome Epstein wrote:

ALL measures of angles, not just radians, are dimensionless. They come
essentially from ratios of lengths, like the sine or tangent.

The radian is just the ratio of arc length (in say meters) to radius (in
meters). This ratio stays the same whether one measures in meters, or
centimeters or feet. Thus the angle is dimensionless. The number you get
is independent of what system of units you use.
J. Epstein


No. Measures of angle from the definition angle = arc length / arc radius
are untiless, being ratios of lengths. Units are a convenence we tack on.
Since there are several scales used for angles (radian, grad, degree) we
tack those unit names onto the value to distinguish them.

Another place (in the introductory course) where it's plain that units and
dimensions aren't synonymous is in the distinction between work and
torque. Of course torque is a vector and work is a scalar, but that
doesn't show up in the dimensions. Both have the dimensions of force x
length, which would be expressed in proper dimensional form as always, in
terms of the dimensions of the fundamental measurables, M, L, T, (mass,
length and time) always upper-case.

2 -2
M L S

Its units can be expressed (always lower-case)

2 -2
kg m s

Somewhere along the line people gave this the convenience name "joule"
(J). Torque, which has the same dimensions, is never given the unit name
"joule". This example alone should caution intro students that dimensions
and units aren't synonyms.

Back when schools used English units, we were taught that the unit name of
work was the foot-pound, while the unit name of torque was the pound-foot.
This was a naming convention designed to remind us that the two are not
physically the same thing.

Check this one. A one candela source emits 4 pi lumen, or F = 4 pi C where
C has units candela and F has units lumen. Yet C and F have the same
dimensions, since pi is dimensionless. And of course, the two physical
concepts are quite different.

-- Donald

.....................................................................
Dr. Donald E. Simanek
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
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