# Shores of the Dirac Sea

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## Everyday physics: boats and ice float.

January 19, 2009 by dberenstein

As you might have noticed, some objects float and some others don’t. Here below I have a rendition of a boat and a cube of ice floating.

Various floating objects

Today, I will go on a bit about flotation. As a matter of fact, some of you might remember a puzzle with an egg I wrote down a while ago. Of course, most of you have probably heard of Archimedes Principle as describing flotation, so I will explain some aspects of how that principle comes about.

So the first thing you have is a fluid, and secondly you place an object on top of the fluid and you can ask if it will float or not. But this can depend on the shape of the object that is floating. If someone asks wether iron floats on water or not, the answer is that iron sinks to the bottom. Except that we have iron boats (nowadays they are mostly made out of steel). Indeed, Metal boats were a XIX century wonder and many thought they could not possibly float. Nowadays we are so used to it that we don’t stop to see the wonder of it.

In the end, the important concept is pressure. In particular, the liquid exert pressure on every object inserted in it. The pressure is defined as a force per unit area. It is exerted uniformly in all directions. This is a property that liquids have. So if you have an area that encloses some volume, you can expect that there is some pressure exerting force on the corresponding area.

Pressure is not uniform in a liquid because of gravity.

However, the force is perpendicular to the area in question, so if you have some complicated object, force will be exerted in all kinds of directions, because the direction of the surface of the object is changing. So let us stick to a rectangular -shaped like a box- object as it is more simple. We can also make it as small as we like.

On each area of the box there will be some force pushing it. If we look at the vertical direction, there is a force from the bottom pushing it up and there is also a force from the top pushing it down. In the vertical direction these are equal to

$F_z= -P(z+h)A + P(z)A\sim -\partial_z P(z) A h$

Where P is the local pressure at various points, A is the area of the base and h is the height. There will be similar expressions for the horizontal directions. So if we divide the formula by the volume, we find that there is a force density per unit volume equal to the gradient of the pressure. This is

$\vec f = -\nabla P$

In particular, consider a liquid in equilibrium in the presence of gravity. That means that the liquid is not moving and it is not experiencing acceleration. We have two forces acting on the liquid: gravity and the pressure gradients. There should be a balance of forces between these two. We this find that

$-\rho \vec g -\nabla P=0$

where $\rho$ is the mass density of the fluid.

Thus, we can solve for the pressure at various depths. We find that

$P(z) = P(0) -\rho g z$

So if we submerge a box, the top of the cube feels less pressure than the bottom. The total force is

$\vec F= (P(z+h)-P(z))A= \rho g Ah= W$

where we see that the right hand side is the weight of a box of the liquid (the displaced liquid as in Archimedes Principle).

This explains why a metal boat can float. So long as the displaced volume of water weighs more than the boat, the boat will float. This requires one to have a shape that guarantees that the boat is in the end lighter than the water. We do that by adding a lot of empty space filled with air.

Incidentally, since ice floats, the volume of liquid water that it displaces weighs more than the ice itself. This means that ice is less dense than water. So when water cools off below freezing: it expands. This expansion can cause excess pressure on water pipes. After all, the ice will be trying it’s best to expand to full size and may cause the water pipes to break.

There is one more thing to consider. Most objects get a bit smaller when compressed, but water is pretty incompressible. Thus, if you try to keep an object at a fixed depth and it goes a bit further down you will see that it will most probably start sinking, whereas if it goes a bit up it will most probably start to rise. This actually happens to fish and many have developed a swim bladder to help with this. The fish ‘empty’ the swim bladder if they go too high and fill it up if they are sinking too much. Thus, any equilibrium in depth they have is dynamical. Sharks on the other hand rely on hydrodynamic lift instead to accomplish this equilibrium.

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Posted in hydraulics | 13 Comments

### 13 Responses

1. Weight is a manifestation of curvature, nothing else or more. Flat is heavy (steel plates sink), curved is light (steel bowls float). You will be voluntarily buying Curvature Credits, by law, to save the little children.

http://www.mazepath.com/uncleal/flat.htm

2. on January 20, 2009 at 2:18 am carlbrannen

I clicked down hoping to find an explanation for the dip in the water level near the iceberg but no.

3. Waves in the sea……

4. David,

If this is to much myth in perpetuity being supplied then by all means delete it.

I have an ole factory experiment that you might look at, too dispel some mythical superstition I have about weight.

This experiment was done in my youth as well, and again after some years in this factory by men. As stupid as it sounds it has been the motivator of my pursuance for understanding gravity better.

Hulse and Taylor is gravity wave generations?:) Soliton wave generation? Hmmm…..

The story:

The scale in the kitchen is supplanted by a large scale that will hold five men and one chair. One man is sitting and four are standing with their fingers to balance under the chair to lift. Before this lift ensues, each places his hand over top of the man in a hand over hand fashion, and then proceeds to lift.

IN our perception of the lift previously we were astounded to think that a man who weighted so much could be lifted by all, and we not feel this weight appropriately. So we sought to see if this “weight changes when we lifted him.”

AS we watch the scale, it did not move, yet our perception of his weight remained. I know perception is a strange thing( not reliable as a measure) yet, I had seen it go further in my youth. Was part of the incentive to get the men to try it.

Also….

IN thinking about your egg experiment some things came to mind about the “spaces in between the fluid,” versus ice etc…. and I wanted to know about how much water will rise depending on how much volume per tea spoon sugar can be added to the fluid to raise it’s level? The displacement value. So…..even in a liquid form, there is space?

The aluminum gravity bars seemed related in this context when thinking about resonance? Webber.

Best,

5. Just to go a little bit further here and I’ll leave you alone.

Hmmmm…….IN Gems, a refractive index, in water, an illusion of a bent pencil. Ligo operations, interferometry on a large scale, measuring light (speed of light constant in a vacuum)changes of light pinpointed to mirror? Kip Thorne valuation?

Grace satellite changing our understanding of the landscape of earth from our first pristine view of “stepping outside” the boundaries perceived of earth, now, in it’s most spherical shape “quite lumpy” in it’s measured valuation.

Not sure where I am going with this. I think I am basing on it on the assumption you are heading in the direction of explaining gravity waves? Gas in a box under gravity? If not, sorry.

Best,

6. on January 20, 2009 at 11:07 pm carlbrannen

Actually, the old aluminum gravity wave detectors were called Weber bars.

Joe Weber was quite a guy, a bit infamous for repeatedly observing gravity waves in error. I remember his recollections of the sinking of the carrier Lexington, on which he was stationed, in the battle of the Coral Sea, 1942. He said she was incandescent as she sank beneath the surface, but that the closest brush with death he experienced was on the island afterwards, when a monkey nearly nailed him with a coconut.

I took a couple classes from him, general relativity and an advanced statistical mechanics sort of class. I guess I should plug my general relativity simulator, which does test masses around a black hole in Schwarzschild and Painleve coordinates. Which reminds me, I need to update the java code because there’s an error in the Runge-Kutta algorithm and so it’s not as accurate as it could be (guilt).

7. on January 21, 2009 at 12:29 am Plato

Sorry for going off topic David.

Carl:Joe Weber was quite a guy, a bit infamous for repeatedly observing gravity waves in error.

Wow, that’s cool Carl. I mean knowing him and his history. What was his initial insight into using the Aluminum bars. Do you remember off hand?

I assume it was acoustical in nature and relevant thinking based on “sound values” were thought at one time to be held relevant in terms of Acoustic Hawking Radiation.

Renaud Parentani wrote something in this regard, phonon’s versus photons. Unruh showed something in this respect that phonon’s were not sensitive enough.

Do think Joe Webber thought this way in terms of gravitational wave detection? To this day I think people do think this way in terms of gravitational wave signatures, as being sound by nature? Wayne Chu for example.

Similarily, the laws of gravity and light seem totally dissimilar. They obey different physical assumptions and different mathematics. Attempts to splice these two forces have always failed. However, if we add one more dimension, a fifth dimension, to the previous four dimensions of space and time, then equations governing light and gravity appear to merge together like two pieces of a jigsaw puzzle. Light, in fact, can be explained in the fifth dimension. In this way, we see the laws of light and gravity become simpler in five dimensions.Kaku’s preface of Hyperspace, page ix, para 3.

Thinking of Susskind’s Gedanken experiment, and about the elephants, it would seem from the fifth dimensional perspective, to have some value in the photon description, in terms of valuating that Hawking Radiation from the blackhole?

Best,

8. [...] is very similar to the notion of pressure, as described in the previous post on flotation. Indeed, the units of stress and the units of pressure are the same. The main difference, however, [...]

9. on January 22, 2009 at 5:48 am carlbrannen

Plato,

It’s been a very long time, but I seem to recall that Weber’s was trained as an EE. Just a second. Let me check wikipedia for this… Yes, I recall correctly. He got into the US Naval Academy after “his senator visited me to make sure I wasn’t black”, which would be exactly in keeping with the kind of stories I remember him telling, self deprecating. It’s hard for me to imagine a better naval officer.

For an electrical engineer, it is very natural to think in terms of resonances. And so, he examined the resonant conditions of a cylinder. His detectors were only good for gravity waves of a very particular frequency, and their ability to detect also depended on their orientation. More recent resonance based detectors are spheres.

The only book on Weber I know is “Gravity’s Shadow” by Harry Collins, which is really about the sociology of gravity wave detectors. Of course I bought the book and read it because I knew him, but it turns out that it gives fascinating insights on how experimental and theoretical physics is done, from a sociological point of view. These transfer to other branches of physics.

One of the classes I took from him was a 2nd year statistical mechanics class (my favorite subjects were probability and group theory so I took a lot of these sorts of classes). It was mostly about the “fluctuation dissipation theorem”.

He assigned us a homework problem that I couldn’t solve, which was quite rare for me in grad school. I should write it up as a blog post. It was years before I forgave him for this problem.

Take a 75 ohm resistor. Use it to short out a length of 75-ohm coaxial cable. Attach the other end of the cable to a short dipole antenna. Place the antenna in a black body cavity at temperature T. Show that the resistor reaches an equilibrium temperature T.

When it came time to turn in the homework, only the two or three students who had asked him for hints were able to solve it. We were bending up our copies of JD Jackson calculating approximations of absorption cross sections and integrating over solid angles and frequencies, multiplied by the black body radiation curve, etc..

Instead, he was using a fact about antennas well known to electrical engineers but pretty much unknown to physics grad students, certainly nothing that had ever been discussed in class etc. One might look here for a clue.

25 years on, I’m still pissed over that homework assignment. So to make it even, I’m going to pass on the advice he gave us: “never start working on a calculation unless you already know the answer you will get”, which is a good short hand description of how he got into trouble with gravity waves.

10. Thanks Carl.

11. Wegener proposed that the continents floated somewhat like icebergs in water. Wegener also noted that the continents move up and down to maintain equilibrium in a process called isostasy.

Alfred Wegener

Just thought I would add this for consideration. Grace satellite does a wonderful job of imaging these features into a global perspective?

In the simplest example, isostasy is the principle of buoyancy observed by Archimedes in his bath, where he saw that when an object was immersed, an amount of water equal in volume to that of the object was displaced. On a geological scale, isostasy can be observed where the Earth’s strong lithosphere exerts stress on the weaker asthenosphere which, over geological time flows laterally such that the load of the lithosphere is accommodated by height adjustments.

Isostasy

12. on February 3, 2009 at 8:53 am rufus

hai..a have a project on preliminary design of amphibius vehicle..and i think it is an mazing project..the problem is the reliable calculation on how can i make sure that the car will be float..anyone can help?..all your opinion will be useful for me..tq

13. Dear Rufus:

Make a scaled toy model and submerge it in a container of water to the desired level. If you check how much does the water level rise, that tells you an estimate of the volume. After that, calculate the weight (with typical load included). If the weight divided by the displaced volume is too high, it will sink. Otherwise, there is a chance that it will work.

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