Calculate the center of mass on the force plate

[Jim Newt]

Homework Statement

The attached jpeg shows a force plate with four measured masses each corresponding to a corner node. Calculate the center of mass on the force plate.

Homework Equations

I think they would be:

xcg = (x1M1 + x2M2 + x2M3 + x4M4) / (M1 + M2 + M3 + M4)

The ycg would be the same except you put yn in for xn.

The Attempt at a Solution

The textbook I've seen will place the lower left node at (0,0), thus zeroing out two of the nodes in each calculation. How does this give an accurate answer?

So for this method, the xcg would be:

xcg = (19*20 + 9*20) / (20 + 42 + 9 + 19)

Is this correct?

 

[Doc Al]

The textbook I've seen will place the lower left node at (0,0), thus zeroing out two of the nodes in each calculation. How does this give an accurate answer?

They are just choosing a reference point for describing the coordinates. They chose the origin to be at the lower left. No problem--it doesn't change where the center of mass is, just how you describe it.

So for this method, the xcg would be:

xcg = (19*20 + 9*20) / (20 + 42 + 9 + 19)

Is this correct?

Sure. Why not? Note that this is consistent with the formula you gave.

 

[tomcue]

Your attempt at a solution is mostly correct. However, the center of mass calculation should also take into account the z-coordinate, as the force plate is a three-dimensional object. The formula for the center of mass in three dimensions is:

xcm = (x1M1 + x2M2 + x3M3 + x4M4) / (M1 + M2 + M3 + M4)

ycm = (y1M1 + y2M2 + y3M3 + y4M4) / (M1 + M2 + M3 + M4)

zcm = (z1M1 + z2M2 + z3M3 + z4M4) / (M1 + M2 + M3 + M4)

Where x, y, and z represent the coordinates of each node, and M represents the mass at each node. So for your example, the center of mass coordinates would be:

xcm = (0*20 + 20*20 + 20*42 + 0*9) / (20 + 42 + 9 + 19) = 32.5

ycm = (0*20 + 0*20 + 24*42 + 24*9) / (20 + 42 + 9 + 19) = 27.4

zcm = (0*20 + 0*20 + 0*42 + 0*9) / (20 + 42 + 9 + 19) = 0

Therefore, the center of mass on the force plate would be located at (32.5, 27.4, 0). This calculation takes into account all three dimensions and provides a more accurate representation of the center of mass.