I don't think there is anything that would fundamentally forbid that, it just couldn't happen in the universe we live in.
Isn't it kind of an open question whether our universe is open or closed?
If it's open then it would have required for the initial singularity to have a certain topology (because you obviously can't get an open, infinite universe by expanding a point singularity). Don't ask me what this topology would be, though, because that goes well beyond my grasp.
Isn't it kind of an open question whether our universe is open or closed?
Whether or not at least one of our spatial dimensions is infinite is related to the topology of the universe, though I think that having an open universe doesn't necessarily require any dimension to be infinite. I'm not even sure if infinite dimensions are compatible with the big bang model - I can understand a small universe growing, but how would its size increase from a finite value to infinity?
But even assuming such a spacetime, the construction of an infinite singularity is still a problem. Postulating "it just magically pops into existence" or "it has always been there" doesn't count.
Obviously I don't have any more answers than you do. Waiting for a pro to pop in :)
Whether or not at least one of our spatial dimensions is infinite is related to the topology of the universe, though I think that having an open universe doesn't necessarily require any dimension to be infinite. I'm not even sure if infinite dimensions are compatible with the big bang model - I can understand a small universe growing, but how would its size increase from a finite value to infinity?
I'm not a physicist, nor even a mathematician, but I would imagine that it's mathematically not impossible to have a zero-volume singularity that nevertheless is infinite in some manner (such as being an infinite line or plane) and which then expands. And that's just thinking about it in a cartesian manner. When we go to topology things can get really weird. You don't even need a singularity extending to infinity in order for it to have some infinite dimension.
But even assuming such a spacetime, the construction of an infinite singularity is still a problem. Postulating "it just magically pops into existence" or "it has always been there" doesn't count.
Well, the whole question of how the initial singularity became, where it came from and what its original characteristics were is completely open, so I wouldn't dare to postulate anything about it (eg. that it must have been a point, for instance.)
To be honest, most physicists do not think singularities ever occur at all, but that they are unnatural artifacts that appear when extrapolating general relativity too far outside its region of validity. For example, standard cosmology predicts that the density and curvature should approach infinity at every point as we move back in time (the big bang singularity), but the only thing we know for sure is that they were very high in the past, not that they were infinite. We need a theory of quantum gravity to fill in the parts of the picture where general relativity gives up and produces singularities.
PS. It is no problem for an infinitely large universe to expand. Expansion can be defined locally as every small volume element being replaced by a slightly larger volume element. The expansion of the universe does *not* refer to it expanding into some void outside it, which is the mental model that I think is responsible for most of the confusion about infinite universes expanding.
I know there are some really bright physicists here, so I hope this problem is a piece of cake for you. I got into an argument about it this morning with a fellow grad student.
Why does water shoot higher/faster when you place your thumb partially over the open end of a garden hose?
(Think carefully...)
One could roughly state that the garden hose emits a certain volume of water per second, regardless of the diameter of the hose. (In practice it's not so precise, but serves as a rough approximation.) I think you can intuitively deduce what that means when the hole is smaller.
One could roughly state that the garden hose emits a certain volume of water per second, regardless of the diameter of the hose. (In practice it's not so precise, but serves as a rough approximation.) I think you can intuitively deduce what that means when the hole is smaller.
That's a very common answer on the internet (all the top links of a Google search for "hose thumb physics" offer it as a solution).
It's also completely wrong.
How can it be completely wrong? My google search revealed the following:
"The answer centers on the idea of matter conservation. If a certain amount of water flows into the hose from the spigot, that water must either flow out of the hose at the other end, or accumulate inside the hose itself. Since a garden hose is made of stiff plastic (generally), it won't expand to let more water accumulate, so the water must eventually flow out of the end of the hose. When you put your finger over the tip of the hose, you decrease the amount of space the water has to flow through. Since the same amount of water has to flow out of the hose before and after you place you finger over the end, the water must shoot out faster (to keep the amount of water flowing out a constant). And, thus, since the water shoots out faster, it travels farther."
http://www.physlink.com/education/askexperts/ae185.cfm
How can it be completely wrong? My google search revealed the following:
"The answer centers on the idea of matter conservation. If a certain amount of water flows into the hose from the spigot, that water must either flow out of the hose at the other end, or accumulate inside the hose itself. Since a garden hose is made of stiff plastic (generally), it won't expand to let more water accumulate, so the water must eventually flow out of the end of the hose. When you put your finger over the tip of the hose, you decrease the amount of space the water has to flow through. Since the same amount of water has to flow out of the hose before and after you place you finger over the end, the water must shoot out faster (to keep the amount of water flowing out a constant). And, thus, since the water shoots out faster, it travels farther."
http://www.physlink.com/education/askexperts/ae185.cfm
What physical principle indicates a certain (constant) amount of water must flow out the hose regardless of any constrictions at the mouth? In fact, there is a relevant physical principle that states that the amount of water flowing out is decidedly non-constant. (I won't divulge the physical principle just yet, but you can probably guess it...)
But that's just theory. Limiting cases in experiment also tell you that the amount of outflowing water is highly dependent on constrictions in the system: shut off the valve to the hose. Your cross-sectional area goes to zero and the flow rate goes to zero. When you open the valve just a crack, water comes out at just a trickle, far less than when it is opened entirely. I'd say that's persuasive evidence that the flow rate is not constant.
Keep thinking...
Let's look at the simple case of a horizontal hose connected to a faucet on one side, and open to the air on the other side. The water provider ensures that the faucet end has high pressure, while the open end has normal air pressure, which is much lower. So inside the pipe, the pressure is changing from high on one side, to low on the other. This pressure gradient accelerates the water towards the low-pressure end.
For a constant cross-section hose, it doesn't really work to have water move at different speeds at different points: Water would have to pile up or be stretched thin in this case, and water is almost impossible to compress. Hence, the water can only accelerate or decelerate if the cross-section of the hose changes. It follows that the pressure gradient is zero for sections of constant cross section, and nonzero when the cross-section changes. So for a constant cross-section hose, the whole pressure drop happens at its exit.
If the cross-section in parts of a hose shrinks, this will obstruct some of the water, letting less through the constricted section. This cases the pressure on the exit side of the constriction to fall, and this pressure gradient accelerates the water, causing it to move more quickly in the constricted section. The opposite applies to expanded sections of the hose.
When you hold you thumb in front of the exit of the hose, the water accelerates for this reason. Your thumb reduces the cross-section of the water flow, which lets less water through, which causes pressure to fall on the other side of your thumb, which sets up a pressure gradient, which accelerates the water until the same flow rate is achieved.
As long as we ignore friction, this will always happen. The equilibrium flow speed in volume/s ends up being constant no matter how the cross-section changes along the way, because the flow ultimately is driven by the pressure change between the two ends, and changes in the radius of the tube only change where that pressure change happens, not how large it is in total.
As I said, though, this is only as long as friction is ignored. For a very long pipe, or for a severe constriction, friction against the walls will be considerable. If you pinched the exit of your hose to a millimeter in radius, the water should ideally be shooting through it at more than 10000 km/h, but long before it gets that fast, it is slowed to a trickle by friction. This is why the water stops getting faster once you close the opening too much.
This is reminding me of the waterguns I used to make as a kid -- take a section of PVC pipe, glue an endcap on one end (or use threaded caps and a threaded pipe), drill a small hole in the endcap, and then use a plunger (made from a dowel and a piece of leather) to suck water in / shoot it out. It was pretty clear to me that smaller holes got you more speed, but at some point you had to start working a lot harder to get a decent distance.
You can buy commercial waterguns that work on the same principle but what fun is that?
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
What physical principle indicates a certain (constant) amount of water must flow out the hose regardless of any constrictions at the mouth?
We are not talking about every possible system where water flows from point A to point B. We are talking about a garden hose connected to a typical water faucet, which is a very specific situation.
Naturally there are other, quite different situations, where partially obstructing the way will make water flow more slowly. A typical example is a bottle of water. The difference is that there isn't much pressure forcing the water out of of the bottle, and therefore no "n liters of water per second" effect.
A water faucet typically pushes water at a certain rate because the pressure behind it is much more constant. (Of course it's not fully constant because then the water faucet would explode when it's closed, but it's a good-enough approximation.)
Actually another common situation with water flow in pipes is when you have mineral buildup in the pipes, which creates basically a maze of near-blockage that the water has to thread its way through to reach the faucet. This will greatly reduce pressure and flow, even though the water is not actually blocked; presumably due to friction as amaurea says.
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
You guys need a little help with this. Both amaurea and Derakon have elements of the right answer, but your answers are either incomplete or take things in the wrong direction.
Consider Bernoulli's principle. Suppose there is a large water tower supplying water to the hose. It can be shown that the water will leave the mouth of the hose with the same speed, regardless of how wide it is. This is in direct contradiction with your explanation that the volume rate of water leaving the hose is constant (or nearly so). How do you reconcile the two theories? Which one (if either of them) is correct?
It can be shown that the water will leave the mouth of the hose with the same speed, regardless of how wide it is.
You mean, like, contrary to observation and evidence? How do you "show" it?
Pick location 1 to be at the water's level in the tower. Pick location 2 to be just outside the mouth of the hose. Take the ambient pressure to be atmospheric and all heights relative to the mouth of the hose.
Bernoulli's equation states that
P1 + pgz1 + 1/2 pv12 = P2 + pgz2 + 1/2 pv22
where P is the ambient pressure and p is the density (the other variables should be clear from context). We know that both P terms are one atmosphere, so they cancel. We know that z2 is zero and, as long as the water tower has a very large cross-sectional area, v1 is negligible. This leaves
pgz1 = 1/2 pv22
Solving for v2, we find
v2 = sqrt(2gz1).
This is an expression for the speed at which the water leaves the hose. It is independent of the cross-sectional area at the mouth of the hose. This strongly implies that the volume flow rate is not constant-- instead, it appears that the flow rate increases directly proportional to the area of the open end. So which of the two analyses is (more) correct? How can the flow rate be constant when it's completely contradicted by Bernoulli's equation? If Bernoulli's equation applies, why does the water leave the hose faster when you place your thumb over it? (You're right, Warp: experiment is the ultimate arbiter here. I'd be very interested in whether the flow rate indeed stays constant as you change the aperture size. I don't have a hose and nozzle handy, but I encourage you to time how long it takes to fill up a bucket when the nozzle setting is changed. If the time is not nearly constant, it would seem to contradict the prevailing theory here.)
Note that this analysis doesn't change substantially if the pressure is instead maintained through, say, mechanical means as opposed to gravitation. All that's important is that the reservoir's pressure is independent of what goes on at the mouth of the hose (which I'd say is a very safe assumption).
New question: what's going on in this video? The wood is dry, as is the climate; there's no oil involved; that's a random orbit sander being used. My best guess is that vibrations in the board being sanded are creating local stable nodes where the sawdust naturally clumps together, but that doesn't explain why the "droplets" flow (and per my dad, who made the video, they do flow with gravity).
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
Okay, how about answers that can't apply to any question asked ever?
What exactly is it that you want?
Do you understand that any chaotic system like that is extremely difficult to model or give any kind of function of, because there are so many small things affecting the whole? It's the same thing as with weather forecasting: There are enormous supercomputers calculating weather patterns day and night using really complex formulas and algorithms, and even they get it only half-right.
With weather, often recognizable semi-regular patterns form, such as hurricanes and tornados. The same is true with many other chaotic systems.
There is clearly some force creating a degree of order here, because you wouldn't in a million years see this kind of patterning spontaneously form out of what is more or less a Maxwell distribution. Of course I don't expect a set of equations that can predict where each particle will be; I want a general sense of how this system is possible. We're not talking about forecasting the weather here; we're talking about describing the precipitation cycle.
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
I'd guess it's about the same as water droplets running down a window: the vibrating of the wood reduces friction, allowing for the droplets to flow, and the air flow around the droplets (note: I'm just guessing here, but I'd say some sort of air current has to emerge from that sort of rapid vibrating) causes them to stick together.
Disclaimer: any statement in this post is made at 1:44 AM after plenty whiskey and does not in any way, form or shape reflect the regular thinking capacity of the poster.
There's still plenty of water in both the wood and the air, though I consider a vibration-based effect more likely, but the video stops too early to determine the cause.
What happens if you stop the sander? Do the "droplets" stick together or do they fall apart when you shake the board a little? Are they wet when touched?
What happens if you obstruct the airflow from the sander over the board, only allowing the vibrations to affect the dust? Or when moving the sander away, so that only the air flow is left, but no vibrations?
Derakon wrote:
(and per my dad, who made the video, they do flow with gravity).
No they don't. If you watch closely, at the end you'll see the bigger "droplets" move downwards, while a couple of smaller ones move upwards.
Let me return to the "why does water shoot faster out of a hose when you cover it partially with your thumb" question.
I think I owe you all an apology, but please read this carefully. I offer the following summary:
1) I was too antagonistic when I posted it here.
2) There is a reason that the volume flow rate should be approximately constant.
3) That reason still needs to be justified. The problem can be explained using either the Bernoulli equation or the continuity equation, but you must do so carefully in both cases.
For the first part, yes, I dove into this topic a little too headstrong. I am more familiar with the math problems topic, where people typically post fabricated and solved math problems and then drop hints until the solution is guessed. I was trying to do that here but, looking back, I see that this topic tends to work more with unsolved problems. Anyway, I was trying to guide discussion toward the solution, but I did a poor job of recognizing I was pissing you all off. I'm sorry.
My explanation using Bernoulli's principle is as follows: According to Bernoulli's equation, the fluid should reach the same height (the height of the water level in the water tower), regardless of the nozzle size. That isn't what happens, though. Because Bernoulli's principle follows from conservation of energy, energy must not be conserved. It is the frictional losses in the pipes/hose that cause the spray to be so comparatively feeble. Place your thumb over the mouth of the hose and you slow down the water. We know that viscous forces are (to first order approximation) linear with the fluid's shear velocity differential, so by slowing down the water, you decrease those viscous forces. As a result, less energy is lost in the pipes, more energy reaches the mouth of the hose, there is more pressure on your thumb, and the water shoots out much farther.
But I know that answer is unsatisfactory to many of you, some of my peers and professors included. You all want to use the continuity equation to solve it and, upon further reflection, it turns out you can if you justify it carefully.
My professor guided me in this direction: Imagine an analogous situation in circuits. Bernoulli's equation follows from conservation of energy and is therefore an approach that analogously assumes a given voltage. The continuity equation is a statement about the volume flow rate, which is analogous to current. If the continuity equation applies in this situation, the current at the mouth of the hose should be roughly constant. My professor prompted me by asking how a constant current source could be constructed. My answer was that you begin with a large, constant voltage source (which is easy to come by), connect a very large resistor to it, then hook your system (which has a much lower resistance) up in series to the large resistor. Applying Ohm's law to the whole system, we get
I=V/(Rlarge + Rsystem) ~ V/Rlarge = const.
From there, I realized how it could be applied to this problem-- we have a very similar situation. The water tower is a large source of potential. Even though it should creep very slowly, the water loses a lot of its energy to viscous forces in the pipes throughout the city leading up to your house. Finally, it loses comparatively little energy along the hose. Putting this all together in analogy to the circuit problem, the volume rate of flow should be nearly constant. Place your thumb over the mouth of the hose and the water speeds up.
So that, I believe, is the complete answer. Yes, the volume rate of flow is roughly constant, but you still can't blindly apply the continuity equation. In both derivations, you must consider the viscous forces that sap energy from the water as it flows down the pipes. I still prefer the explanation with the Bernoulli equation because it is so straightforward and you can quickly say after the fact that if the water doesn't reach the level of the water tower, it must have lost energy along the way (to viscosity). In contrast, you need to consider viscosity as a first step and carefully analyze the system before using the continuity equation.
