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mathforfun@yahoogroups.com, bqllpd <no_reply@...> wrote:

>

> Suppose you have two different masses at the end of a lever and a single gravitational point. The masses are attracted to this point with weight vectors V1 and V2. Is the fulcrum of this lever at a point where a line extending V1+V2 from the central gravitational point intercepts the lever?

>

Yes - nice problem. Here's a proof:

Choosing the fulcrum as the origin, let S,P1,P2 denote the displacement vectors from the fulcrum to the gravitational source point and to the two endpoints of the lever respectively.

Since the lever is in equilibrium, the torques produced by the two forces are equal and opposite:

P1 x V1 = -P2 x V2

where x denotes the vector cross product.

Because V1 is parallel to S-P1 and V2 is parallel to S-P2,

(S-P1) x V1 = 0

(S-P2) x V2 = 0

==>

S x V1 = P1 x V1

S x V2 = P1 x V2

Substituting into the previous equation gives:

S x V1 = -S x V2

==>

S x (V1+V2) = 0

This says that the vector S is parallel to V1+V2. The direction of the line connecting the source and fulcrum is parallel to V1+V2.

Mark