# Center of Mass Formula: Understanding the Intricacies

In summary, Mary Boas defines the center of mass of a body in 3 dimensions using three equations, where the integrals of the coordinates are multiplied by the mass and divided by the total mass. This differs from the standard undergraduate textbook formula, which simply divides the integral of the coordinate by the total mass. The advantage of Boas' definition is that it allows for calculating the center of mass for different subsets of a system, rather than just the entire system.

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.

No, there's no benefit to writing it that way. Different style, I guess.

Her way is more 'mathematical', which makes her book awkward.

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 Menclosed but Boas' definition automatically clears up that ambiguity.

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## 1. What is the center of mass formula?

The center of mass formula is a mathematical equation used to calculate the point at which an object's mass is evenly distributed in all directions. It takes into account the mass and position of all the individual particles that make up an object.

## 2. How is the center of mass formula used in physics?

In physics, the center of mass formula is used to determine the overall motion of a system of particles. It is also used in calculations involving torque, rotational motion, and collisions.

## 3. Can the center of mass formula be applied to objects of any shape?

Yes, the center of mass formula can be applied to objects of any shape as long as the mass and position of each individual particle is known. It can even be used for irregularly shaped objects by dividing them into smaller, simpler shapes.

## 4. How does the center of mass formula differ from the centroid formula?

The center of mass formula takes into account the distribution of mass within an object, while the centroid formula only considers the object's geometrical shape. This means that the center of mass can be located outside of the object, while the centroid is always located within the object.

## 5. Why is understanding the intricacies of the center of mass formula important?

Understanding the intricacies of the center of mass formula is important in physics, engineering, and other fields where the motion and stability of objects are critical. It allows for accurate predictions and calculations, and can also help in designing structures that are more stable and efficient.