## Sunday, March 8, 2009

Archimedes is accredited with saying "Give me a place to stand and I will move the earth". This statement reflects his profound understanding of the lever principle which lead to the invention of block and tackle pulley system, which apart from saving many backs since it's inception, represents a significant milestone in the development of machines. Levers and pulleys illustrate the mechanical advantage gained by applying a force over a distance. In vectorial form, we write the relationship as:

M = F x r = "moment" vector or "torque" provided by applying a force over a distance.

where F is the force vector and r is the postions vector, and the "x" symbol represents the operations of "cross products" or "vector products". "Torque" and "moments" are imporant mechanical concepts to understand, as they underpin our understanding of machines. The starting point is the simple lever. Appreciate how a lever works (whether it be a spanner or a see saw) and you can start to appreciate the notion of torque. A "moment" is in effect the same as "torque", except the later is generally reserved for scenario's where the moment is induced by circular motion (e.g. the torque induced in an axle by a force being applied to a wheel).
Further analysis also shows:

M = F x r = Fr sin k n

where k is the angle between F and r, F and r are the magnitudes of the two vectors, and n is the unit vector perpendicular to the plane containing F and r.

Very conveniently F x r = det( i j k, F1 F2 F3, r1 r2 r3)
(if you haven't done matrix algebra see http://en.wikipedia.org/wiki/Determinant for a quick explanation)

which is the same, as saying:

F x r = (F2r3-r2F3)i - (F1r3-r1F3)j + (F1r2-r1F2)k

where F = F1i + F2j + F3k and r = r1i + r2j + r3k

Thus, the cross product functions allows important mechanical calculations to be performed and for vectors perpendicular to a particular plane to be easily calculated.