Jack wrote:
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What does dimensionality tell us? When we say that torque and
work have the same "dimensionality", what information are we conveying?
In my view, the dimensionality carries information about the
scaling of value. When we say that torque and work have the same
dimensionality (M*L^2/T^2), we mean that the values of torque and work
scale in the same way when we change the units. So if I change from
grams-cm-s to kg-m-hr, then my values of torque and work will each
be multiplied by a factor of .1296.
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Good point. Which gives some meaning to statement that radians are a
dimensionless unit in the standard scheme of things. If we change the scale
for length, say going from meters to feet. The radian does not change.
I.e. being dimensionless you don't have to change anything for it (the
scale factor is unity). I think this is part of what is meant when people
say that the definition of the radian is a "natural" definition, rather than
an arbitrary definition of dividing up the circle into some arbitrary number
of equal parts.