Center of mass on plank problem

[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?

 

[.Scott]

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.

 

[reshmaji]

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.

 

[TomHart]

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:
x_cm = (x₁m₁ + x₂m₂ + . . . x₁m_n)/(m₁ + m₂ + . . . m_n)

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.