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.