# Center of mass on plank problem

• reshmaji
In summary: So if you have a mass at (x1, y1) and a mass at (x2, y2), the center of mass would be (x1+x2/2, y1+y2/2).
reshmaji
Problem
A 60kg woman walking at a speed of 1.0 m/s steps onto a long plank that has a mass of 60kg. Upon stepping on the plank, both she and the plank begin to slide with speed v and spin with angular rate ω on a frictionless surface.

1. How far from the woman is the center of mass of the woman + plank system?
a. 0.0m, b. 2.5m, c. 5.0m, d. 7.5m, e. 10m

Relevant equations
regarding center of mass m1x1 = m2x2, x is center of mass from side of plank where woman is standing

The attempt at a solution

for woman+plank: [60 kg + (x/10)*60kg] * x
& for plank on other side: [60 kg + ((10-x)/10)*60kg] * (10 - x)
These 2 equal each other by definition of center of mass
We get 60x + 6x2 = 1200 - 120x - 60x + 6x2
60x = 1200 - 180x
240x = 1200
x = 5m

This doesn't conceptually seem right to me, but I'm not sure how else to go about this. 5m is the center of the 10m plank, that would be the center of the mass without the woman standing on one end, no? So I'm thinking it wouldn't be with her there? But what is wrong with my calculations if this is true?

You seem to know that the plank is 10M long, but you have not stated that in you problem. Nor have you described where on the plank the woman stepped, or at what angle she approached the length of the plank.

I suspect that there is a diagram that you have not included.

Oh my, I did leave out that a given is that the plank is 10m long. Sorry about that. I'll attach the diagram from the problem here. Some of the extra information is because there are follow up problems after the one I'm asking that use the center of mass value.

Basically what you do to calculate center of mass in one dimension is you pick an origin point (anywhere within that one dimension), then sum the moments about that origin (mass of each object times its distance from the origin), then divide that sum by the total mass. The result is the center of mass.
In other words, the distance from the origin of the center of mass is:
xcm = (x1m1 + x2m2 + . . . x1mn)/(m1 + m2 + . . . mn)

So yes, you are right. If the center of mass of the plank is 5.0 m, and the woman's position is anywhere other than that point, then the center of the mass of the woman + plank system simply cannot be at 5.0 m. It has to move.

Another tidbit that should be fairly obvious is that if you have 2 point masses that are equal (in mass), then the center of mass has to be midway between them.

## 1. What is the center of mass on a plank?

The center of mass on a plank is the point at which the entire mass of the plank can be considered to be concentrated. It is the average position of all the mass of the plank.

## 2. How is the center of mass calculated on a plank?

The center of mass on a plank can be calculated by finding the average position of all the mass on the plank. This can be done by dividing the total mass by the total length of the plank.

## 3. Does the center of mass change on a plank?

Yes, the center of mass on a plank can change depending on the distribution of its mass. If the mass is evenly distributed, the center of mass will be at the center of the plank. However, if the mass is unevenly distributed, the center of mass will shift accordingly.

## 4. What is the significance of the center of mass on a plank?

The center of mass on a plank is important because it is the point at which the plank is in equilibrium, meaning there are no net forces acting on it. This can be useful in understanding how an object will behave when placed on a plank.

## 5. Can the center of mass be outside of the plank?

No, the center of mass on a plank will always be located somewhere on the plank itself. It cannot be outside of the plank as it is a representation of the average position of all the mass on the plank.

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