Boas Mathematical physics book, definition of center of mass

In summary, the definition of center of mass for a body has coordinates x_{CM}= \int x_{CM}dM= \int x dM. However, the first equality is incorrect and the second one is correct. This is because x_CM is a constant and can be pulled out of the integral. For discrete objects, the center of mass is just a weighted mean of the positions. This can be found in the book on page 210 of the 2nd edition.
  • #1
fluidistic
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In Boas' book I can read that the definition of center of mass of a body has coordinates [itex]x_{CM}= \int x_{CM}dM= \int x dM[/itex].
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.
 
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  • #2
fluidistic said:
[tex]x_{CM}= \int x_{CM}dM[/tex]

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!
 
  • #3
WiFO215 said:
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).
 
  • #4
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 ;)
 
  • #5


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.
 

Related to Boas Mathematical physics book, definition of center of mass

1. What is the definition of center of mass?

The center of mass of an object or system is a point where the entire mass of the object or system can be considered to be concentrated. It is the point at which the object will balance in all directions.

2. How is center of mass related to the study of mathematical physics?

The concept of center of mass is crucial in the study of mathematical physics as it helps in analyzing the motion of systems and predicting their behavior. It is used to determine the overall motion of a system by considering the motion of its individual components.

3. Can center of mass be outside the physical boundaries of an object?

Yes, the center of mass can be located outside the physical boundaries of an object. This is possible when the object has a non-uniform density distribution or when external forces are acting on it.

4. How is the center of mass calculated?

The center of mass is calculated by taking the weighted average of the position of each particle in the object or system. The position of each particle is multiplied by its mass and then divided by the total mass of the object or system.

5. Why is the center of mass an important concept in physics?

The center of mass is an important concept in physics because it simplifies the analysis of complex systems by reducing them to a single point. It also helps in understanding the overall motion of a system and predicting its behavior. Additionally, the concept of center of mass is used in various laws and principles of physics, such as the law of conservation of momentum and the principle of moments.

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