Re: Is there an equation which links mass per unit length to the poisson's rati

Thank you, Daniel, for such an interesting question!  My first thought
was that the two were completely unrelated.  Actually, they are, it's just
that we aren't normally used to dealing with mass per unit length.  Once we
figure out what that's all about, relating it to poisson's ratio turns
out to be a problem that you've probably already solved.


Let's start by taking a look at the first part of your question.  What
exactly is the mass per unit length?   Density is mass per unit volume, so
if we are to measure the volume and mass of a particular object, we can
calculate its density by

	(density) = m / V

where m = mass, V = volume.


Your question was in terms of mass per unit length, and that really only 
makes sense if you are talking about rods or beams.   So I'm going to make
the assumption that you are dealing with rubber bands, which are
essentially long slender beams. These beams have a cross sectional area 'A'
and a length 'L'.   The total volume is then

	V = A*L.  

and the density then becomes

	(density) = m / (A*L)

The mass per unit length (m/L) is then given by multiplying the density by
the cross sectional area, giving the mass per unit length:

	(density)*A = m/L.

If we assume that the density remains constant, then the change in the 
mass per unit length of a material is directly proportional to the change
in cross sectional area.   This is what initially confused me... although
the density can usually be considered constant under elastic deformation, 
the cross sectional area  will change as a function of poisson's ratio, 
the overall stiffness of the material, and the load applied.  Therefore the 
_linear_ density (mass per unit length) will change, but not the overall 
density.   

Now it is relevant to ask 

"Is there an equation that links the cross sectional area of a rubber beam
to its Poisson ratio when put under known tensions?"


The answer is yes:  Hooke's Law!   Hooke's law relates the deformation
of a body to the applied loads.  Since you are asking a question about 
poisson's ratio, I will assume that you are familiar with this equation.  
Using Hooke's law, you can calculate the area of a particular cross section 
as a function of applied loads, and use that area to calculate the linear
density.

Additionally, an important thing to keep in mind when dealing with rubber 
is that Young's Modulus and Poisson's ratio are not constant:  they change
depending on how 'stretched' the material is.  You can see this for
yourself by stretching a rubber band.   The more you stretch, the harder
you must pull to stretch the rubber band further.  The rubber band stiffens
as it is stretched.  You will also note that the rubber band doesn't 
'thin out' as much when it is stretched.   Make sure to account for this 
non-linear behavior!


I hope this was helpful in getting you started, Good Luck!


Michael Pantiuk       mpanti1@umbc.edu
---------------
Explaining the unknown by means of the unobservable is a perilous task.