Physics of the dead man's (or jellyfish) float

Question

This question arised during a discussion with a friend, when I discovered that he cannot do the dead man's (or jellyfish) float, i.e. float horizontally on the water, not even when he was a child. Basically, he sinks by his feet and he floats with just his head, immersed in the water at a 45-60 degrees angle. His point was: "My body is just not made for it". My point was:"You never learned properly, at least as a child you should have been able to!". So, as he argued that the laws of nature were against him, I began to dig a bit into the physics of the dead mans's float.

I think that, a part from the magnitude of the vectors at play, a major factor comes from their point of application. Indeed, while the center of mass, where the weight vector that pulls you down is applied, is located more or less towards your hips, the bouyant vector is applied at the center of bouyancy, located more or less in your lungs. To do the dead man's float the two centers must be aligned. If this is not the case, you will rotate in the water until the two centers are on the same vertical, and only at this point you will reach equilibrium. This is what happens to him, I guess. So he was right.

My conclusion is: To make my friend do the dead man's float, we should move his center of mass towards his center of bouyancy, by making him lift his arms or spread his legs. And be sure that his lungs are full, so that his diaphragm is in a lower position and his center of bouyancy is as towards the center of mass as possible. First question: Did I get this part right?

I have a problem on all of this though. I thought: "well, if I can use physics to make him float, I can make myself unstable so that I rotate in the water using the same principle". So yesterday I went to the sea and tried to test some things. My idea was: if I'm doing the dead man's float, the two centers are aligned. So I will put myself in that condition, and then change a bit the volume of my lungs and the position of my center of mass, so to sink by the head or by the feet. I wanted to test if I got the physics right. My attempts:

1. I tried to empty a bit my lungs, so to move my center of bouyancy from my center of mass, and sink by the feet. But nothing happens.

2. If I completely empty my lungs, I sink. But I don't rotate, I just sink vertically, and I guess my centers are still vertically aligned, but the magnitude of the bouyant force is not enough anymore.

3. If I lift my arms and spread my legs, my center of mass should move upwards, away from my center of bouyancy, and I should rotate by sinking with my head. This doesn't happen. In this step I tried to keep my breathing as steady as possible, and in some occasions I completely filled my lungs and kept the air in, so to fix the position of my center of bouyancy, while moving my arms and legs to move my center of mass. Still, I don't sink.

I approached this problem with "How can I make my friend float?", but now I also have the reverse question "How can I make myself sink??". Is there something that I didn't take into account in the physics of this process? Maybe I oversimplified things? Of course what I did is a bit "empiric", as I cannot precisely control the volume of my lungs. But nevertheless, I was expecting to find a configuration in which I could sink, by the head or feet. But I just always float like a wooden plank, no matter what I do (except if I completely empty my lungs of course). Is there a reason for that? Given what I understood about the physics, for me it doesn't make sense. I will give it another shot in the next days in any case, so if you have some additional suggestions I haven't think of I will gladly take them.

EDIT: As the user Bob D pointed out, I should have used "center of gravity" instead of "center of mass"

Answer 1

I approached this problem with "How can I make my friend float?", but now I also have the reverse question "How can I make myself sink??"

First some introductory remarks.

The only potential control you have over whether or not you sink or float is the amount of air you have in your lungs. That's because, regardless of the bodies orientation, whether a person sinks or floats depends only on the person's average density relative to water. That's the average density of the specific person's body components (water 1 gm/ml; muscle 1.055 gm/ml; fat 0.9 g/ml; bone 0.92-1.39 g/ml) and the amount of air in lungs (air 0.001225 g/ml). The relative percentage and the distribution of these components will vary from person to person.

Since, with the exception of the amount of air in your lungs, you have no control (other than diet an exercise) over the other contributions to your average density, you can only control your average density by the amount of air in your lungs. Ultimately, with the minimum amount of air in the lungs, a person sinks or floats based only on the person's average density. If the person's average density is greater than water, the person sinks. If equal to or less than water, the person either floats completely submerged (if equal density), or floats partially submerged (if lower density).

With regard to orientation at equilibrium, although the location of the center of gravity (COG) of an object is independent of the orientation of the object, the location of buoyant force acts through the center of buoyancy (COB), which is the location of the centroid of the displaced volume of fluid. That location will change with the orientation of the body since the geometry of the volume of water displaced changes.

If the COB and COG are not vertically aligned with the COG below the COB, There will be disequilibrium (or possible unstable equilibrium if vertically aligned with the COG above the COB) causing rotation until the COG and COB align vertically with the COG below the COB..

Now, with respect to your other comments/statements in light of the above:

My conclusion is: To make my friend do the dead man's float, we should move his center of mass towards his center of buoyancy, by making him lift his arms or spread his legs.

Although the COM and COG are the same point in a uniform gravitational field, it's probably better to talk in terms of COG instead of COM since we are interested in the force of gravity (weight) of the person. As discussed above, equilibrium depends on the relative position of the COG and COB and since the COG is fixed, for the dead mans float to be in equilibrium the COB needs to be moved so the two align with the COG below the COB.

It is quite possible that the distribution of your friend's mass gives him a COG so low on his body that he cannot get his COB to align with the COG in the dead mans float position by inflating his lungs or moving his arms and feet.

I have a problem on all of this though. I thought: "well, if I can use physics to make him float, I can make myself unstable so that I rotate in the water using the same principle".

Yes, if your orientation is such that your COB and COG don't align vertically with the COG below the COB, you will have a net moment causing rotation, or unstable equilibrium if the COB an COG align vertically but with the COG on top.

1. I tried to empty a bit my lungs, so to move my center of bouyancy from my center of mass, and sink by the feet. But nothing happens.

It's possible you empty them enough to make a difference.

2. If I completely empty my lungs, I sink. But I don't rotate, I just sink vertically, and I guess my centers are still vertically aligned, but the magnitude of the bouyant force is not enough anymore.

Yes, it appears the alignment of the COG and COB was already there. Fully emptying you lungs appears to have reduced your average density to that below water, thus you sunk but didn't rotate.

3. If I lift my arms and spread my legs, my center of mass should move upwards, away from my center of buoyancy, and I should rotate by sinking with my head. This doesn't happen...

Your COG remains unchanged. And since your movements appear to be vertically symmetric, the vertical alignment of you COG and COB should also be unchanged. And since you also didn't say anything about changing the air in your lungs nothing observable should happen.

Hope this helps.

Answer 2

The diagram with the woman floating has the centers of buoyancy and mass correct for a person with their arms at their sides. With arms overhead, the centers change so they are more aligned, allowing one to float horizontally. A good model is a cork with two metal rods sticking out and pointing in opposite directions. The lungs full of air are like the cork. The downward forces on either side almost balance like a see-saw. The woman trying to float on her back would rotate more horizontally if she put her arms overhead.

Answer 3

If I completely empty my lungs, I sink. But I don't rotate, I just sink vertically, and I guess my centers are still vertically aligned, but the magnitude of the bouyant force is not enough anymore.

If you empty your lungs you are decreasing your mass (mass of expelled air) so why do you sink? Buoyancy is based on density $\frac{mass}{volume}$. When you empty your lungs, your chest and stomach get sucked in which reduces volume. The change in volume dominates the buoyant effect and you sink. Buoyancy can be altered by simply sucking or sticking out your stomach