Hot answers tagged equilibrium
15
To make it fall you need a torque. This torque is provided by the weight force acting on the center of mass of the object and by the offset between the center of mass and the edge of the object.
Imagine your domino standing upright then tilt it. You are moving the center of mass. When the center of mass (blue) is on the right of the edge (red) then you have ...
15
Arnold Neumaier's comment about statistical mechanics is correct, but here's how you can prove it using just thermodynamics. Let's imagine two bodies at different temperatures in contact with one another. Let's say that body 1 transfers a small amount of heat $Q$ to body 2. Body 1's entropy changes by $-Q/T_1$, and body 2's entropy changes by $Q/T_2$, so ...
15
Nature has no preferences, and therefore entropy tends to increase.
Sounds paradoxical?
The point is that each microscopic state (describing the exact position and velocity of each atom in the system) is equally likely. However, what we typically observe is not a micro state, but a course-grained description corresponding to incredibly many micro states. ...
12
From a fundamental (i.e., statistical mechanics) point of view, the physically relevant parameter is coldness = inverse temperature $\beta=1/k_BT$. This changes continuously. If it passes from a positive value through zero to a negative value, the temperature changes from very large positive to infinite (with indefinite sign) to very large negative. ...
10
There was some doubt about Lubos' answer (which I've accepted), so this is just a verification.
I copied the method Lubos described and found the potential difference for an ellipsoid with different eccentricities. Sure enough, for an oblate spheroid, if you make the center-equator distance a fraction $e$ larger than the center-pole distance, the ...
10
If you made the most perfect cone possible, so that its tip was a single atom, and stood it on the most perfect surface possible (a perfectly smooth, perfectly hard sheet of atoms), and completely removed all forces other than gravity, it would still topple. This is because those atoms are all jiggling around due to thermal motion. This effect fundamentally ...
5
It takes a lengthy proof, but Lyman Spitzer shows in the second chapter of Physical Processes in the Interstellar Medium (the standard text in interstellar matter studies) that the velocity distribution of interstellar gas particles (which is what forms nebulae) is very nearly Maxwellian - the deviation is less than 1%.
Other larger systems, probably not so ...
4
Start with the unperturbed gravitational potential for a uniform sphere of mass M and radius R, interior and exterior:
$$ \phi^0_\mathrm{in} = {-3M \over 2R} + {M\over 2R^3} (x^2 + y^2 + z^2) $$
$$ \phi^0_\mathrm{out} = {- M\over r} $$
Add a quadrupole perturbation, you get
$$ \phi_\mathrm{in} = \phi^0_\mathrm{in} + {\epsilon M\over R^3} D $$
$$ ...
4
Think of the layer of skin on the finger tip which is in contact with the object that has a lower temperature than the body. If the object has a high heat capacity and a high thermal conductivity (like metal), then the skin of the finger will come to an equilibrium temperature that is lower than if it is in contact with an object that has a lower heat ...
4
Some engineering texts use "moment" and "couple" to talk about forces that tend to rotate an assembly (what physicist mean when they say "torque", but the engineers sometimes have a slightly different meaning for that word).
A roughly translation guide is...
A "couple" is a pair of opposite forces whose points of action are not co-linear. A couple is ...
4
You gave an expression for the force, whereas Prathyush's answer treats it as the potential. If that expression really is the force, then the answer is different.
You don't really need to do any calculation in this case to see what is going on. Clearly $F(0) = 0$, so this is indeed a point of equilibrium (the question should probably read something like ...
3
You may demonstrate numerical friction by applying a Runge-Kutta method (or even the Euler method if RK is too advanced) to a conservative system (such as Sun - Jupiter - Saturn) and notice the dissipative effect of stochastic perturbances (aka discretization errors): Ultimately, the planets will fall into the sun, though this takes many revolutions. ...
3
This is well outside my area of expertise but I believe ou can apply Maxwell–Boltzmann statistics, at least loosely to clusters of galaxies, clusters of stars and in many cases the gas molecules in nebulae. For clusters this is known as the Virial Theroem and Andrew described it quite well in this question Stellar Viscosity in Galaxies.
For the nebular ...
3
Since at higher altitudes, the air pressure is lower, the boiling point of water decreases, since it's easier for the energy insde the water to get free.
When A liquid starts to boil, you reach a critical point where the liquid loses a lot of heat, much more than when not boiling, thus requiring much more energy for the same increase in temperature, and ...
3
Strictly speaking there are no reversible processes in Nature; it is an idealization that enables one to get bounds on efficiency of nonequilibrium processes by using techniques of equilibrium thermodynamics only.
A reversible process is therefore primarily a theoretical concept for discussing what would happen in a process if dissipation were absent. It ...
3
I look at two models of a "fat earth":
a spherically symmetric interior with an aspherical surface layer in hydrostatic equilibrium. This analysis generalizes from the constant density assumed in other answers and thereby exhibits the sensitivity of the flattening to the surface density. I compare the result to those of various other answers.
To estimate ...
3
In this answer, I will present a framework to use, and then I will frame the prior answers within that framework. Let me sum up the values we have here. I'll use the same notation (as best as possible) as everyone else and Wikipedia for an oblate spheroid where $a$ is the large, equatorial, radius.
Mark1, method in the question, $2 (a-b) = 21.6 km$
...
2
Here I would like to numerically check the theoretical prediction of a factor $\frac{2}{5}$ in difference from Mark Eichenlaub's original monopole argument. In practice, this means calculating the difference in the gravitational potential between the North pole and the Equator, and divide by the corresponding difference in monopole potentials. For numerical ...
2
You have 3 obvious weak points in the system.
Where the (longer) arm is attached to the existing bracket
Where the bracket attaches to the pole
Where the pole is attached to the ground / base.
Notation I'll use:
$W$ = weight of monitor
$L$ = maximum arm length
$=> M = WL$ = moment at point of pole attachement
$x$ = distance between attachment ...
2
The direction that the domino falls is determined by the location its center of mass. It falls to the left or right depending on whether the center of mass is to the left or right of the bottom-most edge. If the center of mass is precisely above the edge, then it balances on that edge in an unstable equilibrium.
2
I've started to take Jaynes' wacky point on this subject more seriously --- the key is experimental reproducibility, of which equilibration is a helpful but neither necessary nor sufficient condition. The point is that you know you have enough "macroscopic" degrees of description when you find that it is sufficient to reproduce the phenomena you are ...
2
Throughout the outer body shell (ectoderm), encompassing skin and subcutaneous fat tissue, thermoceptors are embedded on free nerve fibres (so, these types of receptors are not special cell types, in contrast to photoreceptors for example). There are two types of fibres: group-III afferences (cold-sensitive) and group-IV afferences (heat-sensitive).
In the ...
2
Chemical equilibrium is a subset of thermodynamic equilibrium. So there are no examples of systems in thermodynamic equilibrium that are not also in chemical equilibrium.
Here's an example of a hypothetical system in chemical equilibrium but not in thermodynamic equilibrium. A system containing two phases, one a condensed phase containing two equilibrating ...
2
Superfluids can "climb walls" and whatnot.
http://en.wikipedia.org/wiki/Superfluid
Superfluids have zero viscosity, but may have surface tension. Those with surface tension creep up walls in a capillary-like fashion (except here, they can creep up a single wall, whereas capillary action requires a tube).
I don't know about the immiscible liquids, though. ...
2
This problem of three phases in contact was recently studied by the elder Widom (the one from Cornell University), whose stuff is very deep. Here is a link to something related: http://128.84.158.119/abs/1111.2884v1
The quick answers are:
yes
not entirely (no line tension, but zero line-tension is ok physically, just not negative line tension)
surface ...
2
Ok, I dug out our old stat mech/thermo textbook. YES, Maxwell-Boltzmann statistics definitely apply to stars in a globular cluster or galaxy, but you have have to pare back the results to the absolute most general.
Sears and Salinger go through an excellent derivation of Maxwell-Boltzmann statistics as well as the Maxwell-Boltzmann distribution function. ...
2
First, if ${\rm d}S\neq 0$, then the entropy will change, and because something is changing, it's obviously not an equilibrium.
If the physical system doesn't maximize the entropy and it's composed of many parts that may interact with each other, directly or indirectly (it's not disconnected), then any path that allows the entropy to be increased (given ...
2
The obvious example would be diffusion. For example, you could show a simulation of a gas, consisting of rigid 2d circles, some of which are one colour and some of which are another. Start of with all of one colour on the left and all of the other colour on the right, and watch as they gradually become mixed. You can explain that the system could go from the ...
2
This will run indefinitely, and it is the basis of just about every water fountain produced commercially. In a given time, the pump will move a certain volume of water from IN to OUT. That lowers the height of the column at IN, and increases the headspace over that column. Both of those reduce the pressure that column of water provides at the interface ...
1
For the simulation part, I've played some time ago with the phun 2D physics engine, where you can simulate molecules like rigid circles. You should be able to simulate the box on a floor scenario you discussed above quite quickly.
If you don't have the time to make your own, you can also find some nice demonstrations on Youtube, like
A demonstration of ...
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