## Tuesday, August 14, 2012

### Crowns,Boats, Subs, and other things that float in water...

Abstarct:
A short story about a friend,
and things that go "ping" in the dark.

It has been some time since last I wrote something, but life, the multiverse and everything, got in the way.

By the way, if the answer to life universe and everything is 42, then for the multiverse it should be what? a Vector? a Tensor? and what are the entries? in which linear basis?
I guess I could get cute and say that its a $$42^n$$ Tensor, but that's just my regular idiotic nonsense at play...

So yes, in layman terms I was simply very otherwise occupied, but today, as my wife is away, and after almost a week of sleep deprivation and other torture methods, I found the will and the way to actually start writing stuff.

Anyway the idea for this post came from a rather dubious experience, and I might have a follow-up post on the physics of water-closets, otherwise known as restrooms, bathrooms or the unsavory "latrines".

What's the connection?

And Seeing as approximately a quarter of my life was spent on boats or in the context thereof, I deem it fit to dedicate a post to the physics of these floating wonders.

### What makes things float?

Most of us at one point or the other have had a chance to hear that quite famous cry "EUREKA" but I daresay less of us actually know the origin of this cry. so here goes...

Some 2500 years ago in the golden age of the great kingdom of Crete, there lived a king and his queen, or, more probably a queen and her lackey of a king.

Now that queen was little capricious, not unlike the queen of hearts from Lewis Carroll's Alice's adventures in wonderland, and at some point in time decided that the crown allotted to her, was not fancy enough. Thus, she ordered a new crown to be made, a crown of solid gold.

The word spread, and a huge congregation of goldsmiths, jewelers, and their apprentices, flocked to the isle of Crete, in hopes of gaining the queen's grace and be chosen to forge the queen's new crown.

But as is often the case with tyrannical rulers, this queen was a very suspicious being, I could theorize she had a bad experience with  goldsmiths, or might be she was simply a bitch.

In any event, she was absolutely terrified by the prospect of being swindled and as a result she wanted to make sure the crown was actually made of solid gold.
But here's the pickle... suppose she WASN'T swindled and she melted the crown to check if it's solid gold, she's now left with a pot of molten gold, while still having to pay the goldsmith.
Thus a method had to be devised to check the crown without damaging it.

Enter stage left: Archimedes.

Archie, our friend happened to be in the vicinity and having a reputation of the genius he was, was charged with the daunting task of finding that method.

Well, Archie thought hard, maybe losing some weight (why can't I?) and some hair (why am I?) at the prospect of failing the queen and subsequently failing to breath, and at the end of 3 excruciating days, his wife decided she would have none of it anymore, "You stink!" shrilled the shrew "Go have a bath or it's the couch tonight, for you!".

Archimedes, being the thoughtful husband that he was, climbed into a warm bath, and noticed, that when he submerged more of his body, the water level rose and spilled over the sides of the tub.

## "EUREKA"

he shouted then, followed by his wife's "Shut up already you git! you'll wake the baby and then you'll have to deal with it!!"

Anyway, I will leave you wondering as for how the story ends, did old Archie indeed wake the baby, how long was spent in the dog-house, and whether or not a goldsmith found his premature demise.

Now for the physics:

Suppose an object with a volume V is partially submerged in water - the elevation force is due to pressure differences, the partially exposed part of the object experiences just the atmospheric pressure, but the underside experiences the upward pressure from the water, so let's see what that pressure is:
Let's consider a column of water and a thin strip $$\Delta Z$$ thick
$\Sigma F=0 \Rightarrow s\cdot\left(P(z+\Delta z)-P(z)\right)-s\cdot\Delta z \cdot \rho_{water} g \\ \text{or in other words} \frac{\partial P}{\partial z}=\rho_{water}\cdot g\Rightarrow P(z)=\rho_{water}\cdot g\cdot z+ P_{atm}$

Where z is the depth of water.

So, we have to consider $mg=s\cdot\rho_{water}\cdot g\cdot z \Rightarrow h\cdot\rho_{object}=\rho_{water}\cdot z$

In other words, the depth of immersion is given by the height and relative density of the object and the fluid.

Now, there's an easy way to see that this dynamic is correct, simply take a piece of wood , and see that it submerges deeper when you hold it length up, than when it's laying flat on the water.

So that takes care of boats, we just have to make sure the average density of the boat is lesser than the density of the water, mind you we're talking the average density of the space occupied by the boat meaning also the air inside the boat, unless you start to take on water, and then guess what? your downward bound.

That also might provide a hint why we sometime encounter "unflushables"...

That also might provide an insight as to how submarines stay submerged at a constant depth:
When the sub is at "bubble up" state, basically the density of the sub is lower than the surrounding water thus the sub tends to float up. to hasten the process the sub might or might not apply it's propeller or other means of propulsion.

When the sub is at "bubble down" state, the average density of the sub is higher than that of the surrounding water making the sub "heavier" and thus sinks down, again applying propulsion or not is at the captain's discretion.

By the way, subs mostly have compressed air tanks, which they discharge into  ballast sections, to change the average density of the vessel, and then use compressors to re-compress said air to the tanks, evacuate the ballast air ballast sections to increase the average density (water then flood the ballast sections).

So it seems fairly simple right? WRONG, we actually took the water's density to be constant where it really isn't, cold water is denser than hot water, and deep water is a tad denser than shallow, so what's the deal?

Well the physics for this is fairly complicated in terms of the math involved, but the IDEA is fairly simple, water density is a product of the mutual forces between water molecules, that are essentially electric in nature, and so external pressure is somewhat involved in this, but even more so, temperature.

So with pressure $$\rho_{water}$$ rises linearly at first but pretty quick stabilizes to a constant.

Like so:
 Courtesy Windows To the Universe (NESTA)
With temperature the change is much more pronounced, but still pretty much the same applies, A linear rise in density when temperature drops, and then exponential decay to a constant.
I suspect somewhere in the middle there's actually a point where it all turns to ice...

That remind me of the cool Thermometer where there are different glass bells with different nifty colored liquids in a glass water tube, and when the water in the tube is in thermal equilibrium with the area, some bells float up, some sink down, and the one left in the middle shows the right temperature on it.... pretty cool if you ask me...

 Told you water density changes with temperature!
So anyway making the calculations needed to predict the density at a certain depth and temperature is a pretty nasty undertaking thus usually subs employ feedback loop mechanisms to apply the right density. either that or they do it by hand and eye i.e. "bubble up"\"bubble down" mechanism.
By the way, remember the couple of pictures in the beginning? well there you go:
 Baby Ruth is swimming pool - not quite what you think...
I could go on and on about this, about weighing ships, (as opposed to sheep), and using partially submerged sonar buoys, Thermocline, and using different water densities to mislead enemy vessels as to your true location etc. etc. But I'm pretty sure if you read Clancy's "The hunt for Red October" you'd learn all this and have great time doing so...