Szemerédi's theorem: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>David Eppstein
m refactor lead sentence to avoid awkward sandwiching of the main idea by prepositional phrases, decapitalize link, and add sections
en>Fullbarre
m Fixed typo
 
Line 1: Line 1:
{{Unreferenced|date=December 2009}}
Another day I woke up and realised - I've been single for some time now and after much intimidation from friends I now locate myself signed up for web dating. They promised me that there are lots of normal, sweet and enjoyable individuals to fulfill, therefore the pitch is gone by here!<br>I endeavor to stay as physically fit as potential staying at the fitness center many times per week. [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=I+enjoy&gs_l=news I enjoy] my sports and attempt to perform or view while many a possible. I shall regularly at Hawthorn matches being wintertime. Notice: I have noticed the carnage of   [http://lukebryantickets.iczmpbangladesh.org cheap luke bryan tickets] wrestling suits at stocktake sales, If you will contemplated shopping an athletics I don't mind.<br>My household and friends are amazing and [https://www.Vocabulary.com/dictionary/hanging hanging] out with them at bar gigabytes or meals is consistently a necessity. As I find that you can do not have a good conversation using the noise I  the today show luke bryan ([http://lukebryantickets.pyhgy.com more info here]) have never been into clubs. In addition, I have 2 unquestionably cheeky and quite cunning canines that are invariably enthusiastic to meet up new folks.<br><br>Feel free to visit my web site :: [http://lukebryantickets.lazintechnologies.com luke bryan schedule 2014]
In [[physics]], the '''stretch rule''' (sometimes referred to as '''Routh's rule''') states that the [[moment of inertia]] of a [[rigid object]] is unchanged when the object is stretched parallel to an axis of rotation that is a [[Principal_axis_(mechanics)#Principal_axes|principal axis]], provided that the distribution of mass remains unchanged except in the direction parallel to the axis.<ref>{{cite web |url= https://ccrma.stanford.edu/~jos/pasp/Stretch_Rule.html|title=Stretch Rule |author=Smith, J.O. |date=2010 |work=Physical Audio Signal Processing |publisher=W3K Publishing |accessdate=26 November 2012}}</ref> This operation leaves [[cylinder (geometry)|cylinder]]s oriented parallel to the axis unchanged in radius.
 
This rule can be applied with the [[parallel axes theorem]] and the [[perpendicular axes rule]] to find moments of inertia for a variety of shapes.
 
==Derivation==
The (scalar) moment of inertia of a rigid body around the z-axis is given by:
 
:<math> I_z = \int_V d^3 r \, \rho(\mathbf{r})\,r^2</math>
 
Where <math>r</math> is the distance of a point from the z-axis. We can expand as follows, since we are dealing with stretching over the z-axis only:
 
:<math> I_z = \int_{0}^{L} dz \int_{x,y} dx dy \, \rho(x, y, z)\,r^2 </math>
 
Here, <math>L</math> is the body's height. Stretching the object by a factor of <math>a</math> along the z-axis is equivalent to dividing the mass density by <math>a</math> (meaning <math>\rho'(x, y, z) = \rho(x, y, z/a)/a</math>), as well as integrating over new limits <math>0</math> and <math>aL</math> (the new height of the object), thus leaving the total mass unchanged. This means the new moment of inertia will be:
 
:<math> I_{z}' = \int_{0}^{aL} dz \int_{x,y} dx dy \, \rho'(x, y, z) \,r^2 </math>
::<math> = \int_{0}^{L} adz' \int_{x,y} dx dy \, \frac{\rho(x, y, z/a)}{a} \,r^2 </math>
::<math> = \int_{0}^{L} dz' \int_{x,y} dx dy \, \rho(x, y, z') \,r^2 = I_z</math>
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Stretch Rule}}
[[Category:Classical mechanics]]

Latest revision as of 08:44, 30 November 2014

Another day I woke up and realised - I've been single for some time now and after much intimidation from friends I now locate myself signed up for web dating. They promised me that there are lots of normal, sweet and enjoyable individuals to fulfill, therefore the pitch is gone by here!
I endeavor to stay as physically fit as potential staying at the fitness center many times per week. I enjoy my sports and attempt to perform or view while many a possible. I shall regularly at Hawthorn matches being wintertime. Notice: I have noticed the carnage of cheap luke bryan tickets wrestling suits at stocktake sales, If you will contemplated shopping an athletics I don't mind.
My household and friends are amazing and hanging out with them at bar gigabytes or meals is consistently a necessity. As I find that you can do not have a good conversation using the noise I the today show luke bryan (more info here) have never been into clubs. In addition, I have 2 unquestionably cheeky and quite cunning canines that are invariably enthusiastic to meet up new folks.

Feel free to visit my web site :: luke bryan schedule 2014

Retrieved from "https://en.formulasearchengine.com/index.php?title=Szemerédi%27s_theorem&oldid=232464"