MadSci Network: Physics

Re: How do you find the 'efficiency of a rubber band' ?

Date: Thu Dec 13 07:25:15 2001
Posted By: Steve Nelson, Grad student, nuclear astrophysics Ph.D. program, Nuclear Lab, Duke University
Area of science: Physics
ID: 1007325821.Ph

You would measure the force "given out" in the same way you would measure 
the force applied to stretch it, position by position with a "Newton meter" 
(by that I assume you mean a simple spring scale).  I've never heard of 
anyone calculating the efficiency of a rubber band before.  A heat engine, 
yes, a rubber band, no.  I suppose it's a valid question, though.  The 
problem with the calculation is that the force is a function of distance 
that you stretch the rubber band, you have to stretch it in small steps to 
measure this force.  Then you'll have to let it contract in equal steps to 
measure the force of contraction at each point.  If you're using a 
spring scale as a force meter, then you'll have to be aware that you 
must measure the distance the rubber band stretched and not the 
distance that the rubber band and scale (which will also stretch) 
stretched.  You're not interested in the force meter, just the rubber 
band.  Stretch the rubber band 10 cm in half cm steps or something 
like that, measuring the force it exerts at each point.  Then, you'll know 
both the energy you used to stretch it and the work the rubber band did to 
contract.  Just multiply the distance of your stretching step (half a cm 
is just a random number I guessed at) by the force at each point and add 
up all the little bits of energy to get a total.  Throwing 
this into the standard efficiency formula for a heat engine (i.e. dividing 
the work out by the work in) will give an efficiency.

This would be a horrible way to do the experiment, in practice, because the 
differences would probably be incredibly difficult to measure.  Error in 
the measurement would be large compared to the actual measured difference.  
I would do the actual experiment (if this is your plan) in the following 
way:  First take several measurements of the work it took to stretch the 
rubber band by the aforementioned method (measuring forces at each point, 
adding them up and multiplying by the incremental distance you stretched) 
to get a relatively accurate number for that energy.   Then let the rubber 
band float in a calorimeter for a few hours to let everything come to 
thermal equilibrium (a calorimeter with as small an amount of water as 
possible to maximize the temperature change). Next take the rubber band 
out, quickly stretch it a few dozen times (wearing gloves or using two 
sticks to hold it in order to prevent heat transfer from your hands) and 
then drop it back in the calorimeter (a coffee thermos and ordinary 
thermometer might work, if the thermometer is accurate enough and you want 
to do this at home).  The rise in temperature should be more quantifiable, 
and it will be the result of energy lost in the stretching and unstretching 
process.  Another key to the success of this experimental method will be to 
make sure that when you are stretching the rubber band you make the 
distance you stretch it as close as possible to the full distance over 
which all the forces were measured.

To figure out what your measurement means you will have to use some more 
physics.   A calorimeter measures energy deposited in it by using the 
specific heat of water.  Assuming most of the energy from the heated rubber 
band goes into the water, just take the rise in temperature of the water in 
Celcius, multiply by 4.186 (that's the number of Joules it takes to heat 1g 
of water 1 degree C, defined as one calorie), multiply again by the number 
of grams of water in your calorimeter, and divide by the number of times 
you stretched the rubber band.  That gives you the energy that was lost 
heating the band per stretch in Joules.  This number should be small 
compared to the actual energy of the stretch (a rubber band will have an 
efficiency close to 1), so it's good to measure it separately.  Subtract 
this amount of energy from the total energy of a single stretch (to get the 
energy returned by the rubber band) and divide by that same total energy.  
This will give you a good approximation of the efficiency of the rubber 
band which eliminates most of the error associated with measuring a small 
change in a large number.  It would be an impressive experiment if you get 
it to work either way.

It's important to remember that the efficiency of a rubber band will 
probably not be a nice constant.  It will be different for every rubber 
band you measure, and the same rubber band will have different efficiencies 
for different stretching distances.  As it ages, of course, these numbers 
will change.  It's kind of a messy system, despite its apparent simplicity, 
heat engines are much cleaner to analyze.

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