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Simply Harmonic Jello – Fun Physics for Thanksgiving November 23, 2010

Posted by admin in : Acoustics (ACOU), Condensed Matter and Materials Physics (CMMP), History, Policy and Education (HPE), Physics Education Research (PER) , trackback Bookmark and Share

Jello is fun and delicious any time of year, and everyone has seen it “wiggling” and “jiggling”.  With a simple stopwatch and counting the frequency of the wiggles, serving jello brings up a special opportunity to work a physics experiment into your snack and dinner menu.

Those wiggles and jiggles can be described as simple harmonic motion, i.e., the force causing the displacement (motion) is proportional to the displacement itself,  F = -kx .

Consider a square block of wiggling jello on a flat plate.  If the jello is set into vibrating motion by a shear force that acts on the top of the jello, static friction will keep the bottom of the jello fixed in place on the plate.   The displacement (or deformation) of the top of the jello due to the shear force is some distance,  x . This displacement divided by the original dimension is called the shear strain.

From Giancoli, Physics for Scientists and Engineers

If you measure the wiggling rate, i.e., count the number of back and forth excursions per unit time, this frequency can be related to the a physical property of the jello called the shear modulus.

The shear modulus,  G relates the shear force,  F , and shear strain,  \frac{x}{h}   by  

 G = \frac{Fh}{Ax}    or F = \frac{GAx}{h}

where   A is the area of the top of the block.

Because the center of mass oscillates with half the displacement of the top,

 F=\frac{1}{2} k_e x ,

and the effective force constant is given by

 k_{e} = 2\frac{ F}{x} = \frac{2GA}{h} .

The frequency of the vibrations for any simple harmonic oscillator is

 f =\frac{1}{2 \pi} \sqrt{\frac{k_e}{m}}

where  m is the  mass oscillating object, in this case the piece of jello.  The piece of jello can be weighed directly (converting from weight to mass) or given by the density of the jello multiplied by its volume  m= \rho Ah .

So the wiggling frequency of jello is        \frac{1}{2 \pi}  \sqrt{\frac{\frac{2GA}{h}}{\rho Ah}} or  \frac{1}{2 \pi h}{\sqrt{ \frac{2G}{\rho}} .

Thus the shear modulus of jello can be determined from the measured vibrational frequency by  G= 2 \rho ( \pi  f h)^2  .

You can try this experiment at home and even study how the shear modulus changes with how you make the jello, i.e., with water, vinegar, juice, soda, or alcohol. And you can investigate how temperature changes the shear modulus.

Post your results here as a comment.   Check back for updates and useful data.

Updates

Units? When doing any calculation in science it is important to keep in mind the units of the factors in used in the equations.  The units have to be consistent throughout, and the final derived units of your calculation should be consistent with quantity that you are trying to calculate.  It is easy to mix up units if you make length measurements using English units, and mass measurements in the metric system for example.   Even when using the metric system throughout, one could easily make the mistake of mixing CGS units with MKS units.  Always check your units.

The density of jello? Understanding what jello is and how it is made is an interesting lesson in biochemistry, particularly protein structure and function.

The more general name for jello is gelatin.  (Jell-0 is a brand name for the foodstuff – edible gelatin – that has become synonymous with the food itself.) Gelatin is made from the connective tissue proteins of cows or pigs. It is made first by breaking down the cellular structure of the connective tissues.  Then collagen proteins from these tissues are isolated, denatured and subsequently rendered to a powdered form.  Sweeteners, flavoring agents, dyes and other additives are added to this powder to make the familiar gelatin dessert.  To make jello you have to add boiling water to the powder which dis-aggregates the proteins.   Cooling the mixture re-aggregates the proteins.   The final jello mold will be a complex solid mixture of proteins, water, air, and chemical additives.

This leads us to consider the density of jello, which like the biological tissue from which it comes, is mostly water.

Water’s density is  1 \frac{g}{cm^3} = 1000 \frac{kg}{m^3} .  So the density has to be close to water.  But the various additives result in partial molar volumes that contract or expand the total volume.   The final volume depends on the thermodynamic nature of the additives and their relative concentrations.  So while it is easy to think that in any given volume of jello there are constituents that are heavier than water, and that the density should be greater than  1 g/cm^3 , the complex mixture of additives could result in the overall density being less than  1 g/cm^3 .  The most prudent thing to do is to take a well measured cube of jello, calculate its volume (or use volume displacement), weigh it, then calculate its density.

Reported densities for  jello have ranged from  0.98 - 1.3 g/cm^3 (with sugar-free variants being on the low end), while for scientific gelatin (without all the food additives) the density has been reported to be  1.3 g/cm^3 .

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