[AntonioLao]

Pulleys and dollies all provide mechanical advantages when lifting or moving loads against the forces of gravity and inertia, but as the number of strand or pulley increases the distance traveled by the load decreases. This phenomenon disguises the fundamental working principle behind Archimedes’ lever.

In 2D, Archimedes’ lever is denoted by F1·R1=F2·R2, a scalar inner dot product. In 3D, its description is the equivalent of a torque t=F´R, a vector outer cross product. Since vector outer cross product is not commutative, F´R=-R´F. This is analogous to a vector potential. The scalar potential is just F·R and it is commutative that is F·R= R·F. In order to transform these into each other, it is necessary to take the squares of both expressions. The math is easy to do but the physics becomes complicated. However, the square of energy can be expressed as E= F1´R1·F2´R2. When expanded by Lagrange’s identity E=(F1·R2)(F2·R1)-(F1·F2)(R1·R2) and E=(F1·F2)(R1·R2)-(F1·R2)(F2·R1). The physical meaning of each scalar product needs to be explained further.