How Can a Violinist Adjust String Tension to Play Concert A Precisely?

[CoryG89]

A violin string has a length of 0.350 m and is tuned to concert G, with fG = 392 Hz.
Where must the violinist place her finger to play concert A, with fA = 420 Hz?

If this position is to remain correct to half the width of a finger (that is, to within 0.600 cm), what is the maximum allowable percentage change in the string tension?

Ok so I already figured out the first question, the position to play concert A is 32.6666... cm from the bridge. I obtained this answer from dividing the equations, canceling, and solving.
It is the algebra of the second part that is giving me trouble, and I am not sure how to approach it. I am sure it too has to be done by dividing two equations for the frequencies and ending up with with a ratio of the two tensions. I just can't seem to get it.

I really need the answer by 5:00PM Central time, don't know if anyone will be able to solve it, but I'd appreciate any help.

 

[uart]

I really need the answer by 5:00PM Central time.

So this is homework then?

Anyway start by figuring out :

1. What percentage change 0.6cm corresponds to at 32.67cm. Then

2. Figure out how (what function of) does the frequency vary with tension.

3. Find the percentage change in tension corresponding to the same percentage change you just calculated in part 1.

Hint : Frequency is NOT a linear function of tension.

 

[CoryG89]

Thank you for the quick reply. That's the direction I was going in, was just having some trouble. Got it now. And yes, it is sort of homework I guess. More like some general problems to help out with a project.

 

[uart]

Hi Corey. Did you get that the frequency is proportional to the square root of tension? That's important in this problem.

 

[pmacias]

I would approach this problem by first understanding the relationship between frequency and tension in a string. According to the equation f = (1/2L)(√(T/μ)), where f is frequency, L is length, T is tension, and μ is linear mass density, we can see that frequency is directly proportional to tension. This means that as tension increases, frequency also increases and vice versa.

To determine the maximum allowable percentage change in tension, we need to consider the range of frequencies that are acceptable for concert A. Since concert A has a frequency of 420 Hz, we can assume that a range of +/- 5 Hz would be acceptable. This means that the maximum allowable frequency would be 425 Hz and the minimum allowable frequency would be 415 Hz.

Now, using the same equation, we can solve for the maximum and minimum tensions that would result in these acceptable frequencies. Plugging in the values for length (0.350 m) and linear mass density (0.0004 kg/m), we get:

Maximum tension: T = (4μf^2L^2)/π^2 = 172.48 N
Minimum tension: T = (4μf^2L^2)/π^2 = 168.41 N

The difference between these two tensions is 4.07 N. To find the maximum allowable percentage change, we divide this difference by the average tension (170.45 N) and multiply by 100 to get 2.39%. This means that the maximum allowable change in tension is 2.39%.

In order to maintain this accuracy within 0.600 cm, the violinist must be very precise in adjusting the tension of the string. Any changes in tension greater than 2.39% would result in a frequency that falls outside of the acceptable range for concert A.