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Center of Mass Formula

  1. Sep 5, 2013 #1

    I'm reading Mathematical Methods in the physical sciences by Mary Boas and in it, she defines the center of mass of a body in 3 dimensions

    [tex] \int \overline {x}dM=\int xdM [/tex]

    [tex] \int \overline {y}dM=\int ydM [/tex]

    [tex] \int \overline {z}dM=\int zdM [/tex]

    In standard undergraduate textbooks, I've always seen it written as

    [tex] \overline {X}=\dfrac {1} {M}\int xdM [/tex]

    I guess I don't understand the reasoning behind defining it the way she did. I know that [tex]\overline {x} [/tex] is constant so you can pull it out and you'd just simply get the [tex] \int dM [/tex], leaving you with the formula that is generally seen in undergraduate texts.

    But why write the formula as she did to begin with. Is there a particular benefit to doing so?

    Any insight would be great, thanks.
  2. jcsd
  3. Sep 5, 2013 #2


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    No, there's no benefit to writing it that way. Different style, I guess.
  4. Sep 5, 2013 #3


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    Her way is more 'mathematical', which makes her book awkward.
  5. Sep 6, 2013 #4
    I think the advantage is that Boas' form gives the center of mass for any volume in a system, rather than only giving the center of mass for the entire system. For example, when considering the earth-moon system, we might want to calculate the center of mass of the moon and not the center of mass of the system--so you take your volume of integration around just the moon subset of the system, and you get the center of mass for just the subsystem. I guess you could do it like the style of Griffiths E&M and call it Menclosed but Boas' definition automatically clears up that ambiguity.
    Last edited: Sep 6, 2013
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