Planck mass: Difference between revisions

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factor is not added, it's multiplied
Significance: an extended object (which is a subjective description in any case) can have ANY mass - if it's big enough (so density remains low) it won't collapse into a black hole. c.f. Himiko (Lyman Alpha Blob) - possibly 40 billion solar masses
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{{Refimprove|date=August 2010}}
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In [[fluid dynamics]], the '''drag equation''' is a formula used to calculate the force of [[drag (physics)|drag]] experienced by an object due to movement through a fully enclosing [[fluid]]. The formula is accurate only under certain conditions: the objects must have a blunt form factor and the fluid must have a large enough [[Reynolds number]] to produce [[turbulence]] behind the object. The equation is
:<math>F_D\, =\, \tfrac12\, \rho\, v^2\, C_D\, A</math>
where
:''F<sub>D</sub>'' is the [[drag force]], which is by definition the force component in the direction of the flow velocity,<ref>See [[lift force]] and [[vortex induced vibration]] for a possible force components transverse to the flow direction.</ref>
:''ρ'' is the [[mass density]] of the fluid, <ref>Note that for the [[Earth's atmosphere]], the air density can be found using the [[barometric formula]]. Air is 1.293 kg/m<sup>3</sup> at 0°C and 1  [[atmosphere (unit)|atmosphere]]</ref>
:''v'' is the [[velocity]] of the object relative to the fluid,
:''A'' is the reference [[area]], and
:''C<sub>D</sub>'' is the [[drag coefficient]] – a [[dimensionless number|dimensionless]] [[coefficient]] related to the object's geometry and taking into account both [[skin friction]] and [[form drag]].
 
The equation is attributed to [[Lord Rayleigh]], who originally used ''L''<sup>2</sup> in place of ''A'' (with ''L'' being some linear dimension).<ref>See Section 7 of Book 2 of Newton's [[Principia Mathematica]]; in particular Proposition 37.</ref>
 
The reference area ''A'' is typically defined as the area of the [[orthographic projection]] of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as a sphere, this is exactly the same as a [[cross section (geometry)|cross sectional]] area. For other objects (for instance, a rolling tube or the body of a cyclist), ''A'' may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion. [[Airfoils]] use the square of the [[chord_(aircraft)|chord length]] as the reference area; since airfoil chords are usually defined with a length of 1, the reference area is also 1. Aircraft use the wing area (or rotor-blade area) as the reference area, which makes for an easy comparison to [[lift_(force)|lift]]. [[airship|Airships]] and [[Solid_of_revolution|bodies of revolution]] use the volumetric coefficient of drag, in which the reference area is the square of the cube root of the airship's volume. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.
 
For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the [[Reynolds number]] is greater than 1000.<ref>[http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Dynamics/Forces/DragForce.html Drag Force<!-- Bot generated title -->]</ref> For smooth bodies, like a circular cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 10<sup>7</sup> (ten million).<ref>See Batchelor (1967), p. 341.</ref>
 
== Discussion ==
The equation is based on an idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up [[Pressure#Stagnation pressure|stagnation pressure]] over the whole area. No real object exactly corresponds to this behavior. ''C<sub>D</sub>'' is the ratio of drag for any real object to that of the ideal object. In practice a rough unstreamlined body (a bluff body) will have a ''C<sub>D</sub>'' around 1, more or less. Smoother objects can have much lower values of ''C<sub>D</sub>''. The equation is precise – it simply provides the definition of ''C<sub>D</sub>'' ([[drag coefficient]]), which varies with the [[Reynolds number]] and is found by experiment.
 
Of particular importance is the <math>u^2</math> dependence on velocity, meaning that fluid drag increases with the square of velocity. When velocity is doubled, for example, not only does the fluid strike with twice the velocity, but twice the [[mass]] of fluid strikes per second. Therefore the change of [[momentum]] per second is multiplied by four. [[Force]] is equivalent to the change of momentum divided by time. This is in contrast with solid-on-solid [[friction]], which generally has very little velocity dependence.
 
== Derivation == <!-- [[Drag (physics)]] links here -->
The '''drag equation''' may be derived to within a multiplicative constant by the method of [[dimensional analysis]]. If a moving fluid meets an object, it exerts a force on the object. Suppose that the variables involved – under some conditions – are the:
* speed ''u'',  
* fluid density ''ρ'',
* viscosity ''ν'' of the fluid,
* size of the body, expressed in terms of its frontal area ''A'', and  
* drag force ''F<sub>D</sub>''.
Using the algorithm of the [[Buckingham π theorem]], these five variables can be reduced to two dimensionless parameters:
* [[drag coefficient]] ''C<sub>D</sub>'' and
* [[Reynolds number]] ''R''<sub>e</sub>.
 
Alternatively, the dimensionless parameters via direct manipulation of the underlying differential equations.
 
That this is so becomes apparent when the drag force ''F<sub>D</sub>'' is expressed as part of a function of the other variables in the problem:
 
:<math>
  f_a(F_D,\, u,\, A,\, \rho,\, \nu)\, =\, 0. \,
</math>
 
This rather odd form of expression is used because it does not assume a one-to-one relationship.  Here, ''f<sub>a</sub>'' is some (as-yet-unknown) function that takes five arguments. Now the right-hand side is zero in any system of units; so it should  be possible to express the relationship described by ''f<sub>a</sub>'' in terms of only dimensionless groups.
 
There are many ways of combining the five arguments of ''f<sub>a</sub>'' to form dimensionless groups, but the [[Buckingham π theorem]] states that there will be two such groups.  The most appropriate are the Reynolds number, given by
 
:<math>
  \mathrm{Re}\, =\, \frac{u\,\sqrt{A}}{\nu}
</math>
 
and the drag coefficient, given by
 
:<math>
  C_D\, =\, \frac{F_D}{\frac12\, \rho\, A\, u^2}.
</math>
 
Thus the function of five variables may be replaced by another function of only two variables:
 
:<math>
  f_b\left( \frac{F_D}{\frac12\, \rho\, A\, u^2},\, \frac{u\, \sqrt{A}}{\nu} \right)\, =\, 0.
</math>
 
where ''f<sub>b</sub>'' is some function of two arguments.
The original law is then reduced to a law involving only these two numbers.
 
Because the only unknown in the above equation is the drag force ''F<sub>D</sub>'', it is possible to express it as
 
:<math>
  \frac{F_D}{\frac12\, \rho\, A\, u^2}\, =\, f_c\left( \frac{u\, \sqrt{A}}{\nu} \right)
</math>
 
or
 
:<math>
  F_D\, =\, \tfrac12\, \rho\, A\, u^2\, f_c(R_e), \,
</math> &nbsp; &nbsp; and with &nbsp; &nbsp; <math> C_D\, =\, f_c(R_e).</math>
 
Thus the force is simply ½ ''ρ'' ''A'' ''u<sup>2</sup>'' times some (as-yet-unknown) function ''f<sub>c</sub>'' of the Reynolds number ''R''<sub>e</sub> – a considerably simpler system than the original five-argument function given above. 
 
Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.
 
The analysis also gives other information for free, so to speak.  The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.
 
To empirically determine the Reynolds number dependence, instead of experimenting on huge bodies with fast-flowing fluids (such as real-size airplanes in wind-tunnels), one may just as well experiment on small models with more viscous and higher velocity fluids, because these two systems are [[Similitude (model)|similar]].
 
== See also ==
* [[Aerodynamic drag]]
* [[Angle of attack]]
* [[Morison equation]]
* [[Stall (flight)]]
* [[Terminal velocity]]
 
==Notes==
{{Reflist}}
 
== References ==
*{{cite book
| first=G.K.
| last=Batchelor
| authorlink=George Batchelor
| title=An Introduction to Fluid Dynamics
| year=1967
| publisher=Cambridge University Press
| isbn=0-521-66396-2 }}
*{{cite book
| last = Huntley | first = H. E.
| year = 1967
| title = Dimensional Analysis
| publisher = Dover
| id = LOC 67-17978
}}
 
[[Category:Aerodynamics]]
[[Category:Equations of fluid dynamics]]
[[Category:Aircraft wing design]]

Revision as of 23:38, 8 February 2014

"Why does my computer keep freezing up?" I was asked by a great deal of individuals the cause of their pc freeze problems. And I am fed up with spending much time inside answering the query time and time again. This article is to tell you the real cause of the PC Freezes.

You can find which there are registry products which are free plus those that you'll have to pay a nominal sum for. Some registry cleaners provide a bare bones program for free with all the way of upgrading to a more advanced, efficient version of the same program.

It doesn't matter whether you are not pretty obvious about what rundll32.exe is. But remember which it plays an important character inside maintaining the stability of the computers and the integrity of the system. Whenever certain software or hardware couldn't respond normally to a system procedure, comes the rundll32 exe error, that may be caused by corrupted files or missing information in registry. Usually, error content might shows up at booting or the beginning of running a program.

Paid registry cleaners on the alternative hand, I have found, are usually inexpensive. They supply normal, free updates or at least cheap changes. This follows considering the software maker must guarantee their product is best in staying ahead of its competitors.

Another thing we should check is whether or not the fix it utilities program you are considering has the ability to detect files plus programs which are wise. One of the registry cleaner programs we might try is RegCure. It is helpful for speeding up and cleaning up issues on the computer.

2)Fix your Windows registry to speed up PC- The registry is a complex section of the computer which holds different types of information within the aspects we do on a computer every day. Coincidentally, over time the registry may become cluttered with information and/or will receive several sort of virus. This is pretty critical plus we MUST get this issue fixed right away, otherwise you run the risk of the computer being permanently damage and/or a sensitive info (passwords, etc.) is stolen.

Maybe you may be asking why these windows XP error messages appear. Well, for we to be able to know the fix, you need to initially recognize where those mistakes come from. There is this software called registry. A registry is software that shops everything on a PC from a usual configuration, setting, information, and logs of escapades from installing to UN-installing, saving to deleting, and a lot more alterations we do inside the system pass through it and gets 'tagged' plus saved as a simple file for recovery reasons. Imagine it as a big recorder, a registrar, of all your records inside the PC.

Registry products have been designed to fix all the broken files inside the system, permitting the computer to read any file it wants, when it wants. They work by scanning from the registry and checking each registry file. If the cleaner sees that it is corrupt, then it can replace it automatically.