Conceptual/mathematical center of mass question

[quincyboy7]

The sigma notation of the center of mass is

x_com = 1/M * SIGMA m_ix_i. I understand this because you are just taking a "weighted average" of sorts to find the correct x-value. My difficulty in understanding arises when this is expressed in integral form i.e.

x_com=1/M * INT xdm. First of all, doesn't integrating with respect to m suggest that x is dependent on m? Similarly, what does a "dm" mean in terms of an x-value? Couldn't an infinitesimal amount of mass exist anywhere on the x-axis? Wouldn't mdx make a lot more sense in the integral? Any help would be appreciated.

 

[Doc Al]

x_com=1/M * INT xdm. First of all, doesn't integrating with respect to m suggest that x is dependent on m?

You'll need to express "dm" in terms of "dx" in order to integrate.

Similarly, what does a "dm" mean in terms of an x-value?

You are taking a tiny element of mass "dm" and multiplying it by its x-value. This is exactly the same thing that you did in the first formula (with SIGMA instead of INT), except there the pieces were macroscopic with mass "m" instead of "dm".

Couldn't an infinitesimal amount of mass exist anywhere on the x-axis?

Of course.

Wouldn't mdx make a lot more sense in the integral?

No. What would "m" be the mass of? The point with the integral is that the mass is now continuously distributed, so we have to break the total mass into a zillion tiny elements (dm) and integrate. And what would "dx" mean in that expression? We want the position of each tiny piece of mass, which is x, not dx.