Three spheres are placed around a coordinate grid: one of mass m at the bottom-left, one of mass 3m a distance of 3L above the first, and one of mass 2m a distance of 4L to the right of the first.
4 points between these three spheres are labeled: A near (1.3L, 1.5L), B near (2L, 1.3L), C near (1.1L, 0.7L), and D near (0.4L, L).
At which point would the center of mass be located?
Here is a video explanation of the solution, but I am totally confused on how this was calculated.
The relevant equations that were derived in a prior video to this example were x (center of mass) = mass(i)*x-position(i) / m(i) , and this was rewritten as First Moment of the mass in x direction, or FM(x) = Σ m(sub i) * x(sub i) all over the Zeroth Moment of Mass in x direction, Σ m(sub i)
So this total formula was written as r (center of mass) = FM (r) / M
The Attempt at a Solution
I attempted the solution by looking at the other relevant equations, and must be confusing myself terribly. I know that this example is a two-dimensional system. So when I tried to apply the total r(center of mass) formula in the y-direction, what I'm doing is takign the mass of the three spheres and their relevant position in the y-direction and multiplying them together. Then adding their totals and dividing by the total mass of the system M.
In the x-direction, following the explanation given, I understood it as it was:
FM(x) = m(0) + 3m(0) + 2m(4L) = 8mL
And M = 6m
So if I divide 8mL by 6m, then that's (4/3)L in the x-direction.
I then attempted to do the same thing in the y-direction.
So FM(y) = m(0) + 2m(0) + 3m(3L) = 9mL
And M = 6m
So if I divide 9mL by 6m, I get (3/2)L as the position of the center of mass in the y-direction. However, the correct answer is saying it should be (1/2)L in the y-direction. How is this? What am I doing incorrectly with this formula?
I appreciate any help with this.