Efficiency of an elastic band. Hooke's Law

[Atomicbomb22]

1. The problem statement, all variables and given/known data
I am doing a lab experiment and the objective is to find the efficiency of an elastic band.

Distance to ceiling=2.9m
Mass of elastic band= 1.85g
Force 2.6N (found using Newton meter)
Extension: 6.4cm


2. Relevant equations

Well I am not looking for a solution, I just need the way to setup the formulas in order to find efficiency.

F1=-kx (Hooke's Law)
F2=ma


3. The attempt at a solution

I think that I should compare the force stored in the elastic band as mechanical energy to the force required to make the rubber band hit the ceiling.

so I'd do this:

-kx=ma

Problem is that I don't know if acceleration should be taken to be gravity. Once I find acceleration, I should be able to find the initial velocity. Then, couldn't I use the work-energy theorem to compare the theoretical initial velocity to the actual velocity? Meaning, I compare the initial velocity I get when I use mgh=1/2mv^2 to the velocity I get when using acceleration.

Sorry maybe what I said above is completely wrong. Please help, I was missing from class when the teacher went over it.

[Atomicbomb22]

Actually I think I figured it out. Should I do the following?

Potential Energy of Spring=1/2kx^2
Potential Energy at cieling= mgh

so I just compare the P.E of the spring and the PE of the elastic band at the cieling. The PE of the Spring> than PE at cieling because some energy is lost, and 1-%age loss is efficiency.

Does that seem right? This lab is very important so your help is appreciated.

[denverdoc]

efficienciency seems like a strange word to use, normally we talk about non-ideal behavior or non-Hookean behavior in situations like this. Both eqns are valid F=ma=-kx as well as mgh=1/2kx^2 but not sure exactly whats being described with your setup.

[Atomicbomb22]

Well my teacher said "efficiency" is:

Energy in/Energy out

Basically, I need to find out how much (as a %) the elastic band deviates from the ideal elastic band due to energy loss.


My experiment is basically like this:

m=0.00185kg
g= -10ms^-2
h=2.93m
k= -40.0
x= 0.064

Now if this was an ideal band then the following would be true:

mgh=0.5kx^2

mgh=(0.00185)(-10)(2.93)=-0.0542
0.5kx^2=(0.5*-40.0)(0.064)^2=0.0819

So, 0.0819-0.0542=0.0277J has been removed from the system

So, 100[1-(0.0277/0.0819)]= 66.18% efficient

Does that seem correct?

[denverdoc]

Here's where I get confused, the k should be derived from static experiment with various masses. Then I believe what you descibe makes very good sense, since that would more accurately reflect energy lost vs purely non-linear elongation of band.

[Atomicbomb22]

Yea i derived k from experiment, I just used 40 as an example. But the problem is that I only have the height attained (h) for one experiment. You think I need to redo the experiment or can I somehow derive height (h) with the the variables I know?

Thanks for all your help so far.

[denverdoc]

hey if h of 2.93 is what you measured, thats the number to use. For completeness since you gave the mass of the band itself, that should be included if possible. and your calcs for effieciency show 40, so use whatever you determined. One other way to have done this is to watch over several cycles how the amplitude dimished. If you were to redo, I'd think about trying that.

[Atomicbomb22]

Well, I varied the extension of the rubber band for each time I repeated the experiment. So it obviously won't go as high (if with 200J it goes 2.93m, with 150J it won't go 2.93m), and I didn't measure the new height. I think thats what you mean by amplitude. Guess I made a stupid mistake, I'll do like you said, luckily I still have the rubber band.

Its difficult to explain, but after the first experiment, instead of measuring the height, I simply measured the force and the extension (using a newton meter and ruler). Seems like I really did do a silly mistake


Thanks a lot

[denverdoc]

Right, amplitude is height in this case. Your approach ov varying the extension is a good idea. then you get efficiency measurements which vary over a range, and may show some interesting trends.