Monday, December 29, 2014

Great elasticity demonstration

I've linked to Dustin Sandlin before.  He does a fantastic series of YouTube videos that are readily accessible by a lay audience and show why science is fun.  Here is his look, using ultrahighspeed video, at why dry spaghetti tends to break in at least three pieces when flexed from the ends.  As he says, this is something that Feynman himself couldn't readily unravel.  Watch the video before you scroll down and read my spoiler description of the mechanism.



This video shows some great elasticity concepts that we generally don't teach in the undergrad physics curriculum.  Flexing the noodle puts the top in tension and the bottom in compression - if you assume simple elasticity (e.g., stress = (Young's modulus)(strain)), and you consider the resulting forces and torques on a little segment of the noodle, you can calculate the shape (e.g., vertical deflection and tilt angle as a function of position along the noodle) of the bent spaghetti, though you have to assume certain boundary conditions (what happens to the displacement and tilt at the ends).  As the flexing is increased, at some point along its length (more on this in a minute) the noodle fractures, because the local strain has exceeded the material strength of the pasta.  One way to think about this is that the boundary condition on one end of each piece of the noodle has now changed abruptly.  Each piece of noodle now starts changing shape, since the previous strained configuration isn't statically stable anymore.  That shape change, the elastic information that the boundary condition has changed, propagates away from the fracture point at the speed of transverse sound in spaghetti (probably a couple of km/s).  The result is a propagating kink in the noodle, the severity of the kink depending on the local curvature of the pre-fracture shape.  If that local strain again exceeds the critical threshold, the noodle will fracture again.  The fact that we need really high speed photography to see the order of breaking shouldn't be that surprising - the time interval between fractures should be the size of the noodle fragment (around 3 cm) divided by the speed of sound in the pasta (say 2000 m/s), or around 15 microseconds!  (If I was really serious, I'd go the other way and use the video record of the propagating kink to measure the speed of transverse sound in pasta.)

This problem is actually somewhat related to another mechanics question: why do falling chimneys tend to break into three pieces?  Again, treating the chimney as some kind of elastic beam clamped at the bottom but free at the top, one can find the (quasi static, because the time it takes sound to propagate in the chimney material is much shorter than the time for the chimney to fall) shape of the flexing chimney.  There are two local maxima in the strain, and that's where the chimney tends to break.  Note that the chimney case is quasi static, while the spaghetti case really involves the dynamics of the flexing noodle after fracture. 

The bottom line:  I want one of those cameras.


Unknown said...

Hello Douglas,I have just watched the video. About Great elasticity demonstration
And its quiet interesting about spaghetti breaking and function of compression and tension. Richard Feynman absolutely a genius for behavior of subatomic particles. It’s interesting explanation. Thanks a lot.
voyance nuit

Tobias said...

The nice this is that these kind of cameras are ever dropping in price - I bought a much less fancier model to do particle tracking for less than $3000, and it handles (I think) 20k frames at full resolution, and >100k for smaller ROIs.
As you can tell from his high-speed videos, the illumination really becomes an issue, however.

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