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38

Neither of those statements are true. It's an easy approximation to make: a neutron star has all of that 'space' removed from between nucleons --- so we just need to know how big a neutron star of mass equal to the solar system would be. Well, the only significant mass is the sun (jupiter is about 1% the mass of the sun---negligible). If the sun were ...


10

If you could compress the mass into that small a space, it would collapse into a black hole, at which point the notion of "size" becomes harder to define, with space-time being so warped. The "event horizon" radius would be about 3 km, if I get the formula correctly. The idea of "there's a lot of space in atoms" comes from computations which state that the ...


10

This is perhaps similar to what mbq meant, but I will elaborate. The T-p phase diagram of water tells us, for a given temperature and pressure, what phase we will get if we have a bunch of that substance. If I apply different pressures to a bottle of water, I am moving around in the p-direction of the T-p plane. I am not changing the pressure of the triple ...


9

Generally taking note of these relationships is a precursor to either (a) applying an approximation or (b) using a purturbative or series solution. In case (a) what qualifies is completely a matter of your sensitivity to error. If you are going to throw out terms $\mathcal{O}(C^{\pm 2})$ and require a 1% approximation then $C$ had better differ from 1 by a ...


9

This is a complement to dmckee's answer by way of an example. Suppose that I want to determine the acceleration due to gravity of an object near the surface of the earth. Let $h$ be the height of the object above the surface, then according to Newton's Law of Gravitation, I get \begin{align} a = \frac{GM}{(R+h)^2} = ...


9

For forces that can be expressed in terms of a quantum field theory, you can compare the size of the coupling constants (which are dimensionless). In short this means that perturbative expansions of weaker forces are well represented by a small number of leading terms because the series converges quickly, while those of stronger forces require more terms (or ...


8

There is a great answer (with references) to this at http://answers.google.com/answers/threadview/id/539329.html which I'll summarize as follows: From http://www.hawaii.edu/suremath/jsand.html the estimate for the grains of sand is 7.5 x 10^18 or 7.5 billion billion. From http://www.faqs.org/faqs/astronomy/faq/part8/section-3.html the estimate for the ...


8

Live on earth is protected from solar wind by the earth's magnetic field. Charged particles from the sun (mostly) penetrate the earth's atmosphere with great velocity. These particles can be trapped by a magnetic field to follow circular path's around the magnetic field lines, thereby losing their energy due to collisions or bremstrahlung. From first ...


7

The metal needs to be specified as a heavy metal, for Lithium, the answer is the opposite. The heuristic is that every atom gives you 3R per mole specific heat (3k per atom). The reason is the equipartition theorem, which is reasonably accurate for solids at room temperature. The amount of energy at temperature T is $.5kT$ for each quadratic term in the ...


7

a great topic. First, ten gigatesla is only the magnetic field near a magnetar - a special type of neutron star. They were discussed e.g. in this Scientific American article in 2003: http://solomon.as.utexas.edu/~duncan/sciam.pdf Ordinary neutron stars have magnetic fields that are 1000 times weaker than that. It is true that in the magnetar stars, atoms ...


7

The "removing the space" and "atoms are mostly empty" memes for atomic nuclei are interesting, but I do grit my teeth every time I hear this. A description that fits better with me might be "remove the electromagnetic force". Concepts of size and space of particles are based on how they interact using forces. There is no evidence that fundamental particles ...


6

One notable class of exceptions are what are called "hierarchy problems" in particle physics. For example, if you identify the Planck mass as a fundamental energy scale, you end up with huge dimensionless numbers which don't have an obvious explanation (i.e. ratio of Planck to electroweak scale, etc.). Explaining these large (or small depending on how you ...


6

Hmmm...some back-of-the-envelope calculations: The depth of the air column at sea level is $14\text{ lbs/in}^2 = 2 \times 10^5\text{ g/cm}^2$, so neglecting space-charge effects and assuming minimum ionization the whole way we get about $4 \times 10^5\text{ MeV} = 0.4\text{ TeV}$ energy loss. We are actually above minimum ionization, so we can multiply that ...


6

Let's make some assumptions. First, assume the fish is rigid. Second, let's assume he's not flapping. Third, I guess let's assume it's a male fish since I said "he." We'll also assume this is 2D because we're looking for an approximation. I would approximate the fish as an airfoil. NACA airfoils are a pretty good choice because they are analytically defined ...


6

$$ 110 \text{ hp} = 82 \text{ kW} $$ This is 1000x what the laptop draws. You won't notice. To put that change in perspective, you would see a similar increase in the power (85 W) used by the car if your speed changed from 65.00 mph to 65.02 mph, since $P \propto v^3$ (at high speed, the power goes as the velocity cubed) as per this answer, so $ ...


5

Here are two examples of where dimensional estimates fail: Divergent expressions I have a laser pointer pointed at a wall directly facing me, 1 meter away. I turn the laser pointer by 90 degrees over the course of 1 second. What is the average speed of the laser spot during the turn? The dimensional analysis argument is 1meter/1second, which would be ...


5

Molecules vibrate with frequencies in the range 10$^{12}$ to 10$^{14}$Hz. Although I don't know of any strict definition, I would take the view that a molecule must hold together for a few vibrations otherwise what you have is a collision not a molecule. That means the lifetime must be greater than 10$^{-14}$ to 10$^{-12}$ seconds, depending on the molecule. ...


5

According to http://ga.water.usgs.gov/edu/earthhowmuch.html the total volume of water on the earth is $1.386\times 10^9$km$^3$, which is about $1.4 \times 10^{21}$kg (I'm rounding because I don't know the average temperature and therefore density of the water). According to http://en.wikipedia.org/wiki/Biomass_(ecology)#Global_rate_of_production the annual ...


4

Consider a spherical drop of water, initial temp 40C, radius 3mm, mass 0.1g To get it down to 0C, you need to remove 4.18 (J/gK) * 0.1 g * 40 K = 17 J then, to freeze it solid, you need to remove latent heat of fusion 333 (J/g) * 0.1 g = 33 J for a total of 50 J. The heat conductivity equation is $H=\frac{\Delta Q}{\Delta t} = k A\frac{\Delta T}{x}$ ...


4

Well, of course you have to pick the quantities in your dimensional analysis right. Example: Use dimensional analysis to estimate the potential energy of a star, hold together only by gravitation. Solution: Newtons gravitational constant $G$ better show up somewhere. This requires us to include something with units $kg^2 / m$. We can get this by inserting ...


4

In this case it is not the size of the force, but where it stands in relaion to the other forces acting on the same two students. They are subject to: the gravitational attraction of the Earth the normal force from the floor or chair the reaction of the things around them due to thier movements both voluntary and involuntary tiny pushes due to circulation ...


3

While the question is interesting, I guess it should be qualified somewhat, otherwise the answer is not quite right. Ron Maimon already pointed out that lithium has higher specific heat than wood. What's worse, some metals react with water, so technically they would melt much more ice than wood due to the chemical reaction with water. Lithium and sodium are ...


3

In physics we distinguish between the physics of "atoms and molecules" and nuclei. Atoms and molecules are described by the same theory, thus I will ignore those molecules here completely and only consider the difference between nuclei and atoms. I suppose you recognize that an atom is a bound system, so is a nucleus a bound system. Maybe you have seen how ...


3

The reason the Planck mass is big is the same reason that the Planck length is small--- we are living on a scale which is enormous in Planck units. So everything around us is made from enormous atoms which have tiny, tiny masses, and you need a large number of atoms to make 1 Planck mass, just as you need a large number of Planck lengths to make 1 meter. The ...


3

I suppose the OP is looking for some general rule to be used when you want to say "A is N orders of magnitude bigger (smaller) than B". In that case, consider $$N = || \log_b(A/B)) ||$$ (where $ || \dots || $ is taken to mean round to the nearest integer, and negative values just mean chose "A is smaller than B", but the magnitude retains the same ...


3

This is really just a footnote to alemi's answer. The electricity for your car is supplied by the alternator, and the torque required to turn the alternator depends on the current it's supplying. As you draw more current more torque is required to rotate the alternator and the car has to use more power to do it. So yes, plugging in your laptop will increase ...


3

These things don't have to be 'smallest' or 'largest'. They are simply (what especially high-energy physicists would agree to be) the most natural units in which to carry out calculations when doing fundamental research. The crux is realizing that things like a 'second' and a 'meter' or a 'kilogram' are purely invented because they are convenient in everyday ...



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