Center of Mass Formula: Understanding the Intricacies

[eprparadox]

Hello,

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

\int \overline {x}dM=\int xdM

\int \overline {y}dM=\int ydM

\int \overline {z}dM=\int zdM

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

\overline {X}=\dfrac {1} {M}\int xdM

I guess I don't understand the reasoning behind defining it the way she did. I know that \overline {x} is constant so you can pull it out and you'd just simply get the \int dM, 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.

 

[Khashishi]

No, there's no benefit to writing it that way. Different style, I guess.

 

[Meir Achuz]

Her way is more 'mathematical', which makes her book awkward.

 

[Jolb]

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 M_enclosed but Boas' definition automatically clears up that ambiguity.