Does human body have a centre of mass?

According to this site,

The center of mass of the human body depends on the gender and the position of the limbs. In a standing posture, it is typically about 10 cm lower than the navel, near the top of the hip bones.

And in this site,

In the anatomical position, the COG(centre of gravity) lies approximately anterior to the second sacral vertebra. However, since human beings do not remain fixed in the anatomical position, the precise location of the COG changes constantly with every new position of the body and limbs. The bodily proportions of the individual will also affect the location of the COG.

Physics says,

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation.

But, mass of human body is not concentrated in a particular point. It is evenly distributed. All the components of human body(bones, muscles, organs etc.) contribute to mass of human body. So, why there is centre of mass as experimentally measured and calculated?

4 Answers

In every object, there is a unique point called 'center of mass (CM)' around which the object's mass is equally distributed in all directions. In other words, mass is balanced at the CM in all directions. For a human it depends on body posture and positions of the limbs. An example CM is given below, which was obtained by segmenting the body; see the source below for details.

Source: Oregon State University.

Generally, the CM lies around the area of the abdomen, but can even be outside the body as shown below.

I am not 100% sure if I understand your question but this might clarify the concept of center of mass (COM) of an object. It is defined by the following equation $$\int_{\mathbb R^3} \rho(\vec r) \cdot (\vec r- \vec r_{com}) \, \mathrm d V =\vec 0$$

where $\rho(\vec r)$ is the density of the object at a position $\vec r$ from the origin. If you modify this integral you get:

$$\int_{\mathbb R^3} \rho(\vec r) \cdot \vec r \, \mathrm d V = \int_{\mathbb R^3} \rho(\vec r) \cdot \vec r_{com} \, \mathrm d V = \vec r_{com} \int_{\mathbb R^3} \rho(\vec r) \cdot \, \mathrm d V = \vec r_{com }\cdot M$$ $$\implies \vec r_{com} = \frac 1 M \int_{\mathbb R^3} \rho(\vec r) \cdot \vec r \, \mathrm d V$$

where $M=\int_{\mathbb R^3} \rho(\vec r) \, \mathrm d V$ is the total mass of the object This integral tells you to do one thing: Take density-weighed average of the position vectors.

You see however there is one (technical) problem with this definition. The equation tells you if you give me $\rho(\vec r)$ at every point in space. I'll give you $\vec r_{com}$. For a complex object like a human being $\rho(\vec r)$ is a very very complicated function. Your muscles have different density from your bones and your fat tissue etc. There is one more problem and that is you don't always have the same fat tissue or whatever at the same position for every person. Thus calculating that integral is for all practical purposes impossible for a human being. However the nature doesn't care about the complexity of the object. There is still a unique COM at each fraction of time for each person.**

Let's take a very rough model of a human being. I.e. a box with height $h$ and a square base with side length $a$. Let the density constant $\rho(\vec r) = \rho$ and zero outside of the box. Let's choose the origin to be on the bottom corner of the box.

$$M \cdot \vec r_{com} = \int_{\mathbb R^3} \rho(\vec r) \cdot \vec r \, \mathrm d V = \iiint\limits_{\text{Box}} \rho \cdot \vec r \, \mathrm d V = \int_{z=0}^h \int_{y=0}^{a} \int_{x=0}^a \rho \cdot \left( \begin{smallmatrix} x \\ y \\ z\end{smallmatrix} \right) \, \mathrm dx \mathrm dy \mathrm dz$$

$$\implies M \cdot \vec r_{com,\ standing} = \frac \rho 2 \left( \begin{smallmatrix} a^3h \\ a^3h \\ a^2h^2\end{smallmatrix} \right) \implies \vec r_{com} = \frac 1 2 \left( \begin{smallmatrix} a \\ a \\ h\end{smallmatrix} \right)$$

Suppose you have raised your hands above. We can very crudely model this a box wiht a base of a rectangle with sides $a/2$ and $a$ and height $2h$. Note that the volume and thus the mass of the box is the same. If you calculate the integral you'll get

$$\vec r_{com,\ hands\ raised} = \left( \begin{smallmatrix} a/2 \\ a/4 \\ h\end{smallmatrix} \right) \neq \vec r_{com,\ standing}$$

Note that in both cases you have a unique center of mass. However the COM of a person raising hands is clearly different from a person just standing.

What you can do however is to measure the COM experimentally and get a rough idea of where the COM of a human being might be in certain positions.

*In order for this to make physical sense $\rho(\vec r) =0$ for all $\lVert \vec r \rVert > R$ i.e. your object has to end somewhere.

** Your COM is not the same when your stomach is full and empty. That is why you have to take a small fraction of time to make sure that there is nothing funny going on.

Every system of particles has a centre of mass! We just give a definition (given in other answers), which locates a unique point. We just choose to call the point given by that definition as the centre of mass for that system.

We can either find that unique point for any system of particles by just employing the definition. But physics tells us that that point behaves rather nicely in the real physical world. We can exploit that 'nice behaviour' and can determine that point experimentally too.

Since the human body is nothing but a collection of matter particles (a huge number of them, though!), it must have a centre of mass defined. However, the thing to note is that the position of that is not fixed, as the configuration of particles composing human body is not fixed. You eat food, you breathe, you shit. However, the changes caused by these processes are rather small and almost non-noticeable.

I would like to answer this question, from an efficient and experienced standpoint:

I have now been teaching dance for over 13 years and dancing for about 18 and I thought that I can confidently answer this question in the right way:

From my understanding of the human body and how it functions, from experience, your center is your sacrum, which is located inside the body, in front of/behind the tailbone; I think the best way to access your “center”, is to find the point right above your pubic bone and use all of the muscles around there, like your core and abdominals, etc... to squeeze that area and push it toward the spine and lift everything up, along the spine. There is another center, which is considered your levital center and that is where your sternum us. If you would have to pick one center, it should be an imaginary point, inside of the body, between the front of your body, right above the pubic bone and your sacrum. The other center, which is generally responsible for the balance of the upper body, is the sternum. That should answer that question most efficiently.