Question

A NON-UNIFORM board of length L and mass M has a center of mass at 5/6L. If the fulcrum is placed at the center of the board and a block of mass m is placed at the right of the board, then where must a block of mass 3m be placed to balance the board?

I assumed new variables for the distances mass m and mass 3m are from the fulcrum, but the non-uniformity/center of mass is throwing me off. Help please?

Answer 1

You do not have to know how the mass is distributed, only that the center of mass is 5/6 of the way from one end.

Require that the total moment about the fulcrum be zero. The lever arm to the center of mass from the fulcrum will be
L/2 - L/6 = L/3
Let x be the location where the mass 3M is placed on the opposite side of the fulcrum from the center of mass. QWe will measure the distance x from the fulcrum (center of board).

L/3 * M = 3M*x for equilibrium
x = L/9

Answer 2

Gotcha, thank you!