Boas Mathematical physics book, definition of center of mass

[fluidistic]

In Boas' book I can read that the definition of center of mass of a body has coordinates $x_{CM}= \int x_{CM}dM= \int x dM$.
Shouldn't it be this same integral but divided by M?!
Also, I didn't find the definition of center of mass for particles or any non continuous bodies.
I'd be grateful if someone could point me what I'm missing.

Edit: I forgot to say it's on page 210 in the 2nd edition.

 

[WiFO215]

$$x_{CM}= \int x_{CM}dM$$

How could this be true? The second equality is correct. Since x_CM is a constant, you can pull it out of your integral. There's your missing M. You'll thus have x_CM = integral(stuff)/M

For discrete objects, it is just a weighted mean of the positions. Should be in there somewhere!

 

[fluidistic]

How could this be true? The second equality is correct. Since x_CM is a constant, you can pull it out of your integral. There's your missing M. You'll thus have x_CM = integral(stuff)/M

For discrete objects, it is just a weighted mean of the positions. Should be in there somewhere!

My bad, I misunderstood the book. The first equality is wrong and the second is right as you said. Now I get it. Thanks a lot.
Yeah I know how to calculate the center of mass of discrete objects. I just wanted to be sure and referred to the book but couldn't find it (still didn't find it).

 

[WiFO215]

If that is so, then simply substitute in the density of the object delta functions for those mass points and the continuous reduces to the discrete ;)

 

[rezeew]

Thank you for pointing out this discrepancy in Boas' definition of center of mass. It is indeed a typographical error and the correct definition should include the division by the total mass M, as you have correctly noted. This can be seen in other sources such as the textbook "Classical Mechanics" by John R. Taylor.

As for the definition of center of mass for particles or non-continuous bodies, it is important to note that the center of mass is a concept that applies to any system of particles or objects, not just continuous bodies. For a system of particles, the center of mass is defined as the weighted average of the positions of all the particles, where the weight is the mass of each particle. This can be expressed as x_{CM} = (m_1x_1 + m_2x_2 + ... + m_nx_n) / (m_1 + m_2 + ... + m_n), where m_i and x_i are the mass and position of each particle, respectively.

For non-continuous bodies, the center of mass can also be defined as the weighted average of the positions of all the constituent particles, where the weight is the mass of each particle. This can be expressed as x_{CM} = (m_1x_1 + m_2x_2 + ... + m_nx_n) / (m_1 + m_2 + ... + m_n), where m_i and x_i are the mass and position of each constituent particle, respectively.

I hope this clarifies any confusion and provides a more comprehensive understanding of the concept of center of mass. Thank you for bringing this up and for your interest in mathematical physics.