Bill Hooper wrote at 16:50 today: >Joe, > >I had asked: >>> How does the use of N.m/rad as a torque unit tell you that torque is a >>> vector? > >and you replied: >> N,m = J, a scalar, is energy >> N,m/rad = J/rad, a vector, is torque > >I don't see how the use of N.m/rad shows that this is a vector. The >inclusion of the "unit" radian in a combination of units does not seem to me >to indicate a vector characteristic.
In USMA 15998 Bill Hooper wrote: >The problem is that the radian is defined as having the very peculiar >property that it is equal to one (1) so that (among other things) its square >is equal to itself: > (1 rad) squared = 1 rad I might add that 1 rad = (1 rad) squared = 1/(1 rad), but it is always about an axis which means that it is always a vector quantity. >But I still don't see the NEED to use the unit to tell whether something is >a scalar or a vector. The footnotes to Table 3 in the metric bible, "The International System of Units", say: "(a) The radian and steradian may be used with advantage in expressions for derived units to distinguish between quantities of different nature but the same dimension. Some examples of their use in forming derived units are given in Table 4 "(b) In practice, the symbols sr and radare used where appropriate.... "(c) In photometry, the name steradian and the symbol sr are usually retained in expressions for units." My first job was in photometry, which may explain my preference for the use of rad in a statement of units. ............................................................................... >The only argument I CAN understand for inserting the unit radian into the >torque unit as you suggest is the use in the equation for the work done on a >rotating body by a torque. That equation is: > >E = t . a > >where E = energy, t = torque (a vector), a = angle (a vector), and the dot >indicates the scalar product so E is a scalar. This equation can be inverted >to give: > >t = E / a > >which is NOT a vector equation (since division by a vector is an undefined >process). 1/(5 rad) = 0.2 rad. Both expressions in your equation are vectors. Your equation IS a vector equation. ............................................................................... >Simpler yet, I would use the coherence of SI to insist that one does not >need to do ANYTHING with the units in the calculation steps. It is >sufficient to know that, when all units substituted into the equation are in >the basic SI units (including the angles being in radians), then the answer >AUTOMATICALLY comes out in the basic SI unit for that kind of quantity: J if >it is energy or N.m if it is torque. How is the reader to know that you measure angles in radians? In engineering it is quite common to use degrees or revolutions. Joe Joseph B.Reid 17 Glebe Road West Toronto M5P 1C8 TEL. 416-486-6071
