Szemerédi's theorem: Difference between revisions

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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 07:44, 30 November 2014

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