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

[tex] \int \overline {x}dM=\int xdM [/tex]

[tex] \int \overline {y}dM=\int ydM [/tex]

[tex] \int \overline {z}dM=\int zdM [/tex]

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

[tex] \overline {X}=\dfrac {1} {M}\int xdM [/tex]

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

 

[PJC01]

I can understand your confusion and curiosity about the different ways of representing the center of mass formula. The reason for the difference lies in the perspective from which the formula is being approached.

In Boas' definition, the center of mass is being calculated by taking the integral of the position coordinates (x, y, z) multiplied by the differential mass element (dM). This approach is more commonly used in advanced mathematical and physics texts, as it allows for a more abstract and general understanding of the concept. It also allows for the calculation of the center of mass for irregularly shaped bodies, which may have varying mass densities at different points.

On the other hand, the undergraduate formula represents the center of mass as a single point, represented by the vector \overline {X}. This approach is more practical for simple and symmetrical bodies, where the center of mass can be easily identified and calculated without the need for integrals. It also simplifies the calculation for introductory physics courses.

Both approaches are valid and have their own benefits. It ultimately depends on the context and purpose of the calculation. I hope this helps to clarify the intricacies of the center of mass formula for you.

 

[PJC01]

Dear reader,

Thank you for your question about the center of mass formula and its intricacies. I can provide some insight into the reasoning behind the different forms of the formula and their benefits.

First, let's define what the center of mass is. It is a point in a system or body where the mass is evenly distributed, meaning that the body would balance perfectly on that point. This point is important in physics and engineering, as it helps us understand the motion and stability of objects.

Now, onto the different forms of the formula. The first form, as written by Mary Boas, is known as the moment form. This form breaks down the center of mass calculation into its three components: x, y, and z. This can be useful in situations where the mass is not evenly distributed in all directions, and we need to consider each component separately. For example, if we have a rod with a heavier weight on one end, the center of mass will not be in the middle but will be closer to the heavier end. In this case, using the moment form allows us to easily calculate the center of mass along each axis.

On the other hand, the second form, as seen in undergraduate texts, is known as the weighted average form. This form simplifies the calculation by taking the average of the x, y, and z coordinates of all the individual mass elements. This is useful when the mass is evenly distributed, as it is a simpler and quicker calculation.

In summary, both forms have their benefits depending on the situation. The moment form allows for a more detailed analysis of the center of mass, while the weighted average form is a simpler and faster calculation for evenly distributed mass. As a scientist, it is important to understand and be able to use both forms in different scenarios. I hope this explanation helps you better understand the intricacies of the center of mass formula.

Best regards,