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?
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
Gotcha, thank you!
To find the position where the block of mass 3m should be placed to balance the board, we can start by considering the center of mass of the combined system of the board and the two blocks, m and 3m.
Given that the center of mass of the board is located at 5/6L, we can calculate the total mass and the center of mass of the combined system.
Let's assign variables for the distances of m and 3m from the fulcrum as x and y, respectively.
To calculate the total mass of the system, we sum the masses of the board and the blocks:
M_total = M + m + 3m
M_total = M + 4m
Next, we calculate the center of mass of the combined system using the principle of moments:
(M * (5/6L)) + (m * x) + (3m * y) = 0
Since the system is in equilibrium, the sum of the moments about any point must be zero.
We can rearrange the equation to solve for y:
3m * y = -[(M * (5/6L)) + (m * x)]
y = -[(M * (5/6L)) + (m * x)] / (3m)
y = -[(5/6L * M) + x] / 3
Therefore, the block of mass 3m should be placed at a distance of -[(5/6L * M) + x] / 3 from the fulcrum to balance the board and the block of mass m.
Note: The negative sign indicates that the block of mass 3m should be placed on the opposite side of the block of mass m.