Why is the median of a PDF not always equal to the mean?
If you consider the pdf as defining the distribution of mass in a thin rod of infinite length and variable density (and so the "weight per inch" of the rod is different depending on which part of the rod we are talking about), then the mean is the point about which the rod is perfectly balanced (that is, if we set up a fulcrum and place the rod on it, it will be horizontal and balanced perfectly if the fulcrum is at the mean, and will tilt downwards on one side or the other if the fulcrum is anywhere else.

The median, on the other hand, is the point for which we can say that half the mass is to the left of the median and half the mass is to the right of the median. If that is what floats your boat as the one true and the only meaning of "center of mass" and you feel that everyone else who believes otherwise is in a state of mortal sin, so be it. Just be aware that when you refer to the median as the center of mass, you are likely to be misunderstood by those lesser mortals who have not attained your state of enlightenment and who still believe that center of mass is another name for what they call the mean of the pdf or random variable.

With respect to the comments on the main question as to an asymmetric pdf means that the mean and median will be different, this answer over on stats.SE (where this question more properly belongs) gives an example of an asymmetric pdf for which the mean, median and mode all have the same value.