Showing/proving a physical relationship

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I derived a relationship between frequency and tension of a string, accounting for tension's effect in the linear density of the string.

So in a nutshell, the equation is more complicated and is in the form of

f^2=aT^2+bT (f is frequency, T is tension, ab are constants involving the control variables which I would know from another experiment).

If I graph T against f^2 with collected data, I would get a quadratic line of best fit. I am so used to modifying the data to show a linear relationship in my high school physics class, but this data is impossible to modify to be linear. I want to obtain the percent error of the constants a and b to suggest the validity of the equation. Is this a reasonable scientific argument?
 

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  • #2
anorlunda
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Good job so far. But I'm confused about your actual question. The title suggests you seem to ask how you know if your equation is valid. In the body, you seem to ask how you do error analysis on your data. They are very different things.

As to the validity, where did you begin? If you started with what we call first principles (meaning things like Newton's Laws of Motion, and conservation of mass/energy/momentum) then things you derive from that are valid if you made no mistakes.

So, tell us more. Where did you begin? How did you get from the beginning to your equation?
 
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  • #3
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Good job so far. But I'm confused about your actual question. The title suggests you seem to ask how you know if your equation is valid. In the body, you seem to ask how you do error analysis on your data. They are very different things.

As to the validity, where did you begin? If you started with what we call first principles (meaning things like Newton's Laws of Motion, and conservation of mass/energy/momentum) then things you derive from that are valid if you made no mistakes.

So, tell us more. Where did you begin? How did you get from the beginning to your equation?
Alright, so the Mersenne's laws can be used to describe the relationship between tension and frequency: f=sqrt (T/u)/2L
The relationship states that frequency is directly proportional to the square of tension.
What I want to show in my research essay is that this is not necessarily the case, especially in substances like a rubber band (used in my experiment), where the linear density—another variable in the law— changes significantly by applied tension.

So we use the Hooke's law T=kx -> x= T/k
The linear density u=m/L will really be u=m/(L+T/k)
This changes the equation above to f= sqrt [T*(L+T/k)/m] /2L

If we square both sides:
f^2= T(L+T/k)/(4mL^2)
f^2= (1/(4mL^2k))T^2 +(1/(4mL^2))T
So my theory indicates that if I use my data to graph the f^2 against T, there will be a quadratic relationship.

In my physics class, we usually prove a law by making the data linear For instance, we would graph pressure against 1/Volume for the equation PV=nRT, and show the accuracy of the data by comparing the slope to nRT) But in this case, the data cannot be modified so that I display a linear relationship; a quadratic was the simplest I could derive. So I wish to obtain the percent errors of the constants (1/(4mL^2k)) and (1/(4mL^2)) from the theoretically calculated values to the constants of the line of best fit to show the validity of the equation I derived.

So my question is this:
Is this, especially the part in bold, a valid scientific method to argue for my theory?
 
  • #4
ZapperZ
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Wait... you have a string with a tension, but then you used Hooke's law that is for a spring to solve it?

This is very puzzling.

Please note that you have neglected to describe the full scenario.

Zz.
 
  • #5
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Wait... you have a string with a tension, but then you used Hooke's law that is for a spring to solve it?

This is very puzzling.

Please note that you have neglected to describe the full scenario.

Zz.
The definition of the Hooke's law states: a law stating that the strain in a solid is proportional to the applied stress within the elastic limit of that solid.
The Hooke's law is not limited only to springs, but strings including a rubber band also follow the law
 
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ZapperZ
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The definition of the Hooke's law states: a law stating that the strain in a solid is proportional to the applied stress within the elastic limit of that solid.
The Hooke's law is not limited only to springs, but strings including a rubber band also follow the law
A "rubber band" is a rubber band, not a "string". You started the thread with a "string", which is often massless and non-extensible!

The mistake here, and you're still making it, is that you have not describe the proper scenario. How about you start from the very beginning. What exactly are you doing? A sketch (which is often required in all physics classes) will help!

BTW, you may assume that I know Hooke's law very well!

Zz.
 
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  • #7
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A "rubber band" is a rubber band, not a "string". You started the thread with a "string", which is often massless and non-extensible!

The mistake here, and you're still making it, is that you have not describe the proper scenario. How about you start from the very beginning. What exactly are you doing? A sketch (which is often required in all physics classes) will help!

BTW, you may assume that I know Hooke's law very well!

Zz.
I have an extended essay for the IB diploma in physics, where my research question is: How does the tension of a rubber band affect the frequency of the standing wave when plucked?

The relationship/theory I want to explore is stated above. It starts off with the Mersenne's law, but I try to extend the theory so that it is more suitable for an extended essay (it is a 4000-word research paper after all), by accounting for the change in linear density with applied.

I now understand your point. The rubber band does not exactly follow the Hooke's law, which is a flaw in my experiment. But using a string (e.g. violin string) makes it hard for me to investigate f= sqrt [T*(L+T/k)/m] /2L, as the string stretches minimally. It's hard. Now I don't know what direction to head...
 
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ZapperZ
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OK, NOW I'm seeing what you are trying to do.

So is the problem boils down to you not knowing how to do a quadratic curve fit? Have you tried online curve fitting apps?

https://mycurvefit.com/

Zz.
 
  • #9
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OK, NOW I'm seeing what you are trying to do.

So is the problem boils down to you not knowing how to do a quadratic curve fit? Have you tried online curve fitting apps?

https://mycurvefit.com/

Zz.
Thank you, but that's not really the problem now. Since the rubber band does not exactly follow the Hooke's law, I'm not necessarily sure of how valid my theory is. I did some research and found out a method to make the rubber band behave closely to a spring. Since this is not a major part of my essay, I wish to mention it as a limitation, but still use the equation above assuming the band followed the Hooke's law. What do you think of this?

source: https://www.wired.com/2012/08/do-rubber-bands-act-like-springs/
 
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anorlunda
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I love PF. @ZapperZ made you think a bit more an soon you found a fundamental error. It is always a mistake to rush to the advanced analysis until we are 100% sure of our fundamentals.

You might get some hints here. (Wikipedia alert ZZ :rolleyes:) https://en.wikipedia.org/wiki/Long-string_instrument

With short strings, stretch is negligible. With a long string, stretch becomes more significant. A consequence of that can be longitudinal waves racing up and down the string that (according to Wikipedia) make a sound like that of a laser blaster in the Star Wars movie.

I think it would make an very good high school paper if you could derive Mersenne's laws from first principles. If you could do that, then perhaps you could extend it to the case with two joined sections of string with different mass per unit length in each section. It may not be necessary to go all the way to continuously variable mass density.
 
  • #11
ZapperZ
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Thank you, but that's not really the problem now. Since the rubber band does not exactly follow the Hooke's law, I'm not necessarily sure of how valid my theory is. I did some research and found out a method to make the rubber band behave closely to a spring. Since this is not a major part of my essay, I wish to mention it as a limitation, but still use the equation above assuming the band followed the Hooke's law. What do you think of this?

source: https://www.wired.com/2012/08/do-rubber-bands-act-like-springs/
So now this is a different issue (gosh, I wish some people can stick to one thing at a time).

First of all, did you do an experiment on this to get your data? If you did, why didn't you also do a Hooke's Law-type experiment on the rubber band? In other words, figure how the relationship between the tension and the extension. It will not be as simple as the linear relationship we got from Hooke's Law, but at least you can make a reasonable fit to your data and get some analytical form. Then use THAT form in your derivation instead of using Hooke's law relationship. Maybe that can be your paper.

Zz.
 
  • #12
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So in a nutshell, the equation is more complicated and is in the form of

f^2=aT^2+bT (f is frequency, T is tension, ab are constants involving the control variables which I would know from another experiment).

If I graph T against f^2 with collected data, I would get a quadratic line of best fit. I am so used to modifying the data to show a linear relationship in my high school physics class, but this data is impossible to modify to be linear. I want to obtain the percent error of the constants a and b to suggest the validity of the equation. Is this a reasonable scientific argument?
An answer to your actual question: You can plot ## \frac{ f^2}{T} ## versus T.
If your data "agrees" with the proposed relationship you will get a straight line. The slope will be your "a". ( and the intercept will be "b")
 

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