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        riddles >> putnam exam (pure math) >> guitar string
        
(Message started by: Larissa_Preedy on Jun 8^th, 2005, 12:11am)

Title: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 12:11am

A guitar string makes a higher note the tighter the tension in the string. Suppose

* the guitar has length L;
* the string tension is T; and
* the mass of the string is M kg per meter.

what is the order of magnitude of the expression for the frequency of vibration of the string?

i can get it down to 1/s²..but i need 1/s

L = metres
T = kgm/s²
M = kg/m

T/ML³

gives me 1/s²

how do i get that down to 1/s?

or am i going on the wrong tracK?

Title: Re: guitar string
Post by Sir Col on Jun 8^th, 2005, 12:56am

Don't forget that you're working with a sinusoidal wave, so when you derive the wave velocity you would obtain an expression involving a square root (v=sqrt(T/r), where r is mass per unit).

Then using the result that the fundamental frequency of a vibrating string is given by v/(2L), we get f=sqrt(T/r)/(2L), where T is tension (kgm/s²), r is mass per unit length (kg/m) and L is length of string (m).

U(T/r)=(kgm/s²)/(kg/m))=m²/s²
U(sqrt(T/r))=m/s
So U(f)=((m/s)/m)=1/s.

I hope that helps.

Title: Re: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 1:35am

hm, i dnt think it's meant to be so complex...

i think we're only meant to use the items given :S

Title: Re: guitar string
Post by Sir Col on Jun 8^th, 2005, 1:48am

Sadly the word "fundamental" rarely means simple in mathematics and physics, rather it means of significant importance.

Simply put, the fundamental frequency of a vibrating string is of the order sqrt(T/M)/L, using the symbolic conventions you were given.

Title: Re: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 2:00am

that's my probelm though, i cannot have

sqrts

otherwise i would have put

sqrt (T/ML^3)

but i can't do that :(

that's why i'm stuck

Title: Re: guitar string
Post by towr on Jun 8^th, 2005, 2:56am

What is meant by "order of magnitude of the expression"?

Title: Re: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 3:01am

it's dimensional analysis.

i have to basically work out a formula from the given 3 identities, and get the answer in hertz..

L = m
T = kgm/s²
M = kg/m

i got to "cancel" the things down, to end up with 1/s or hertz

as i said, T/ML³ gives 1/s²

This is done by

kgm/s²*m/kg*m^-2

the kg's cancel, the m's cancel, leaving 1/s²

however, i need to cancel down to get 1/s

the only method i can think off is square rooting the final answer, but that is not allowed :(

Title: Re: guitar string
Post by towr on Jun 8^th, 2005, 4:01am

Is it alowed to use ^(1/2) (to the power 1/2) instead of sqrt? (Even though it's the same thing)

Title: Re: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 6:48am

omg, i'm such a fruitcake

i think i can do that..i'll try now

Title: Re: guitar string
Post by Larissa_Preedy on Jun 8^th, 2005, 6:55am

yeh that worked :-[ :-[ :-[ :-[

so many silly mistakes i make :(

Title: Re: guitar string
Post by towr on Jun 8^th, 2005, 9:14am

on 06/08/05 at 06:55:53, Larissa_Preedy wrote:

so many silly mistakes i make :(

Well, this is the best time to make them, better than during an exam. As long as you learn from them.
If you make every possible mistake now, you'll be able to recognize them when it counts ;)

Title: Re: guitar string
Post by Icarus on Jun 8^th, 2005, 6:29pm

on 06/08/05 at 02:56:20, towr wrote:

What is meant by "order of magnitude of the expression"?

From Larissa's statements, I think what it really means is: determine the formula up to a multiplicative constant.

E.g., E = mv² would be acceptable answer instead of the actual formula of E = (1/2)mv².

I think the point of the questions is to show students that much of the information they need is available in the units (and hopefully instill in them the realization that carrying the units through the calculation can be helpful, instead of dropping the units at the start, then appending the expected units on the final answer as is their wont).

It is a bit risky determining formula by figuring out how to make the dimensions cancel out correctly. There may be contributors that you miss, or others that you included that should not have been included. (For instance, if their was some time quantity around with this problem, you might have been tempted to multiply your formula by it to get rid of one of the 1/sec factors, instead of taking the square root - thus resulting in an incorrect formula). However, this method does provide a good feel for what the actual formula should look like, which can be helpful in deriving it.