# Finding the POSITION of the center of mass

• jigsaw21
In summary, three spheres of masses m, 3m, and 2m are placed at specific positions on a coordinate grid. The center of mass of this system is located at (4/3)L in the x-direction and (1/2)L in the y-direction. This can be calculated by using the formula r (center of mass) = FM (r) / M, where FM(x) = 8mL and FM(y) = 9mL. The wording in the explanation of the solution may have caused confusion, but the correct answer is (4/3)L in the x-direction and (1/2)L in the y-direction.
jigsaw21

## Homework Statement

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.

## Homework Equations

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.

You have the right answer, where did you see that L/2 in the y direction is the right answer? It isn’t even a choice! He said halfway between 0 and 3L. He probably could have been a bit more clear in his explanation , though.

omg... I feel so dumb sometimes.

The (halfway) I see now meant HALFWAY BETWEEN 0L and 3L in the y-direction which is 1.5L. Which I now see is equivalent to 9mL /6m.

The wording of these things always gets me confused. Thanks for helping to clear this up for me.

## 1. How do you find the position of the center of mass?

The position of the center of mass can be found by taking the weighted average of the positions of all the particles in the system. This can be calculated using the formula:

xcm = (∑mixi) / (∑mi) , where xcm is the position of the center of mass, mi is the mass of each particle, and xi is the position of each particle along a particular axis.

## 2. What is the significance of finding the center of mass?

The center of mass is an important concept in physics as it represents the average location of all the mass in a system. It can help determine the overall motion and stability of a system, and is also useful in solving problems involving collisions and rotations.

## 3. Can the center of mass be located outside of an object?

Yes, the center of mass can be located outside of an object. This is especially true for irregularly shaped objects where the mass is not evenly distributed. The center of mass can also be located in empty space, for example, in a system of two objects orbiting each other.

## 4. How does the shape of an object affect the position of its center of mass?

The shape of an object can greatly affect the position of its center of mass. For symmetrical objects, the center of mass will be located at the geometric center. However, for asymmetrical objects, the center of mass will be shifted towards the heavier side of the object.

## 5. Is the center of mass always stationary?

No, the center of mass can move if there is an external force acting on the system. However, in the absence of external forces, the center of mass will remain stationary or move with a constant velocity according to Newton's first law of motion.

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