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...

Physics of marriage and hypoxia

Abstract:

Got married, took a hike,

got high, got dehydrated and almost suffocated.

oh and a gerbil.

I lied.
I said the next post would deal with how being fat can save your life, and there are actually several ways that can happen. but that will have to wait for another instance.

A couple of years ago I got married.
Yes. I, like loads of better men than I, donned the world's smallest handcuffs, and walked the "green mile" as it were of single men, to happy cheers of "dead man marrying".

his: +5 ring vs. reason&accountability.
hers: one ring to rule them all and in the darkness bind them.

Well, actually I count myself lucky. most men get fucked on the deal, but as for myself, I got myself a ninja for a wife and a Jedi knight to boot. You should see her performing Jedi mind tricks on me or using The Voice on people - damn she's good!

Anyway after about a year of marriage we decided it's time to go for our much overdue honeymoon.
so somehow we came up with the obscene amount of money it now costs to fly to the USA, and we went to visit grandma, the family and take a hike. no really. we traveled California's national parks and hiked some...
at some point we even had oxygen shortage due to height sickness.

What had happened was, we trekked from Camp White Wolf to Ten lakes in Yosemite national park, and back, in one day.
Now Camp White Wolf is at \(\approx 2,400 m\), and the mountain overlooking Ten lakes is at \(\approx 3000 m \). so all in all it doesn't seem like a distinctive height difference right?
BUT (and it's a big butt for sure!) being overzealous and not so experienced hikers we spent the night near Briceburg which is at \(\approx 350 m\).

Now for a bit of physics:
Humans usually breath air which is comprised among other things of roughly 21% \(O_2\).
Assuming air is an ideal gas we get :
\[PV=NRT\]
Which means in a cubicle meter at sea level pressure, at \(298^\circ K\), we get \(\approx 40.9 {} moles \) of air which translates roughly to \(8.6\text{ moles of }O_2\).

Now we will consider a thin spherical layer \(\Delta r\) of gas:
\[ A(r)\cdot P(r)-P(r+\Delta r) = V\cdot\rho (r) \frac{GM_{\oplus}}{r^2} \]
\[\Rightarrow -4\pi r^2 \Delta r \frac{\partial P}{\partial r}=4\pi r^2 \Delta r \rho (r) \frac{GM_{\oplus}}{r^2} \Rightarrow -\frac{\partial P}{\partial r}= \rho (r) \frac{GM_{\oplus}}{r^2}\]

For all intents and purposes the gravitational acceleration on the earth's surface and at 3 km differs by a factor of \(1 \pm 10^{-4}\) so really there's no need to take into account the gravitational change in that small a difference in distance.
Thus we will treat the gravitational potential as \(\Phi_N=gr\), and thus the force per kg is directed downwards and is given by \(F_g=g\hat{r}\).
\[\rho = \frac{N\cdot m}{V} \Rightarrow \rho=\frac{Pm}{RT}\]
and so we quickly get:
\[\frac{dP}{P}= -\frac{gm}{RT}dr\]
Which assuming a constant temperature (which is a WRONG assumption as we will see immediately) we get:
\[ P=P_0\,e^{-\frac{gmr}{RT}} \]
Where \(m\) denotes the molar mass of air, \(R\) the ideal gas constant, and \(T\) the temperature.

Now using this formula we get that the air pressure at 3000 m above sea level is \(\approx 52595\) pa. which means the amount of moles in a cubic meter in 3000 meters of height is 21.2 moles. keeping in mind that \(O_2\) is only slightly heavier the \(N_2\) we allow ourselves the assumption that the \(N_2\sim O_2\) ratio in 3000 meters is similar to that of about ground level we get that the air up there has only ~11% oxygen in the same volume.
Just for the sake of good form - at 15-19% you get decreased ability to work strenuously.  impaired coordination may occur and there is a chance of induced early symptoms with individuals that have coronary, pulmonary, or circulatory problems. at 12-15% respiration and pulse increases; impaired coordination, perception, and judgment occurs. , at 10-12% respiration further increases in rate and depth; poor judgment and bluish lips occur. At 8-10% symptoms include mental failure, fainting, unconsciousness, an ash-colored-face, blue lips, nausea, and vomiting.

you get the picture right?

So 11% is quite close to being knee-deep in shit.

Let's have a more realistic estimate though, because we know the temperature decreases with height (at least in the atmosphere).

So basically the temperature is a physical function and we like to think of those as analytical so the first approximation of \(T\) around \(T_0\) is given generally by:
\[T(r)\approx T_0 - ar\]
The minus sign owing to our understanding of temperature decreasing with height.
and so the above differential equation becomes:
\[\frac{dP}{P}= -\frac{a\,gm\,dr}{a\,RT_0\left(1-\frac{ar}{T_0}\right)} \Rightarrow P=P_0\left(1-\frac{ar}{T_0}\right)^{\frac{gm}{aR}}\]
And so using the corrected formula with \(a\) denoting the linear-approximate rate of temperature decline in the atmosphere which is given by \(a=6.5 \cdot 10^{-3}\) we get the pressure up there was about 70970 pa which is much likelier, whereas the effective air percentage there is almost 15%.

Meaning that we we're only in an ankle-high depository.

Just as an additional reference, in normal circumstances the oxygen percentage in our exhaled breath is about 16%, and in some circumstances could go as low as 14%.
So imagine yourselves in a hot, non-ventilated room, stuffed with cheap foreign workforce, and imagine the wicked headache you'll get. Now multiply it, as we were climbing up at a quick pace thus taking even shallower and quicker breaths.

At some point we actually sat there at the top of the mountain, breaking for lunch, not realizing we are only making matters worse...

The payoff was worth it though - it's such a beautiful view up there - it's AMAZING!

Ten Lakes view, headache and dehydration included.

Anyway, on our way back we also came to the undeniable conclusion we were idiots, as we ran out of water, we had a pounding headache, and we had to virtually run all the way back to Camp White Wolf before dark, since we were really afraid to walk in the woods at night, with no light, no fire, no reception, and loads of wolves howling in the distance.

We got there in the nick of time,  the sun was already setting as we hit the marked trail back to camp, about 20 minutes later, in full dark, we got there.
Sore-foot, dead-tired and otherwise completely wretched, we decided that we are NOT making camp to sleep on cold bare ground. I drove us to a nearby motel outside the national park, and we took a shower, and slept the deepest sleep achievable by mankind...


oh, and a gerbil:

Gerbil

Next time: Lenses and idiots.