The diagram below is a model of a water molecule. Where is the center
of mass of this molecule?

Dear gog34-ga,

You did not post a link to a diagram of a water molecule, however, one
water molecule is like any other.  Here is a link to a diagram of the
water molecule with dimensions:

http://upload.wikimedia.org/wikipedia/en/9/9b/Water_molecule_dimensions.png

The distance between the center of the oxygen atom and one of the
hydrogen atoms is 0.9584 Angstroms (1 angstrom = 10^-10 meters =
0.0000000001 meters).
http://liftoff.msfc.nasa.gov/academy/universe/angstrom.html

Call this distance D and the angle formed by the molecule A.  The
general formula for the center of mass of a system can be calculated,
in general, from the equations here:

http://en.wikipedia.org/wiki/Center_of_mass

Because this is a discrete (countable, rather than continuous)
collection of masses (three all together, two hydrogen, one oxygen),
the integrals are replaced by the sum (the last equation above):

X_ave = 1/M * (Sum (m_I * x_I)

What this equation means is this: calculate the position of the center
of mass for x, y, and z (for a three dimensional problem), or, in this
case, just x and y, separately.  Do this by looking at each mass
separately.  For each, multiply the mass times the x position and add
all of the results, dividing by the total of all the masses.  This
will give the x_ave for the x coordinate of the center of mass.  Do
the same for the y coordinates.  Here?s how to do this for the case of
the water molecule:

Set the origin of the coordinate system at the center of the oxygen
molecule to simplify the problem.  Also, orient the coordinate system
so that the x-direction is running towards one of the hydrogen atoms. 
Express the masses of the atoms in atomic mass units (amu) ?
hydrogen=1amu, oxygen=16amu.

Then the x coordinate of the center of mass (COM) is this:

X_com = [m_O * 0 + M_H * D + M_H * D * cos (A)] / (2 * m_H + m_O)

The y coordinate is this:

Y_com = [m_O * 0 + M_H * 0 + M_H * D * sin (A)] / (2 * m_H + m_O)

Substituting the actual numbers from the diagram:

X_com = [0  + 1amu * 0.9584Angstroms +
1amu*0.9584Angstroms*cos(104.45deg)] / (2 * 1amu + 16amu)

     = [0.9584 Angst*amu + 0.9584*(-0.25) Angst*amu] / (18 amu)
     = (0.72 / 18) Angstroms
     = 0.04 Angstroms

y_com = [0 + 0 + 1amu * sin(104.45deg)] Angst * amu / (2 * 1amu + 16amu)

     = 0.968 Angst * amu / 18 amu
     = 0.054 Angstroms

So, the coordinates of the center of mass, with the above coordinate system are
(0.04 Angstroms, 0.054 Angstroms)

This is very near the oxygen atom, which makes sense since it makes up
16/18 = ~90% of the mass of the molecule.

The best way to master these is to do lots of problems and start with
one dimensional problems (masses all in a row).

Here?s a link with some one-dimensional examples:
http://www.sparknotes.com/testprep/books/sat2/physics/chapter9section6.rhtml


Also, please note that a similar question was posed here:
http://www.madsci.org/posts/archives/feb99/918612075.Ph.q.html

With suggested answer here:
http://www.madsci.org/posts/archives/feb99/918612075.Ph.r.html

Unfortunately, while the math used to solve the problem appears to be
without error, the diagram used to set up the problem is incorrect ?
They have attempted to set up the diagram to avoid using trigonometry
(sines, cosines), but in doing so assume that the length which they
call ?D? is the same as the D in the above equations.  The ?D? in the
madsci.org link above should really be
?D? * sin[(180-105)/2] = (0.96 * sin(75/2)
to be correct.  If the angle between the hydrogen atoms were very
small, the answers would agree, however it is actually greater than 90
degrees, so the assumption in the link above breaks down.

I hope this was helpful,

           -welte-ga