A violin string has a length of 0.350 m and is tuned to concert G, with f = 392 Hz.
Where must the violinist place her finger to play concert A, with f = 430 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?
I have no idea how to solve this problem or even the equations involved with it. It is due in a few hours. Any help would be much appreciated! Thank you!
To determine where the violinist must place her finger to play concert A, we need to use the formula for the frequency of a vibrating string:
f = (1/2L) √(T/μ)
where:
f is the frequency of vibration,
L is the length of the string,
T is the tension in the string, and
μ is the linear mass density of the string.
First, we need to find the new length of the string (L') that will produce a frequency of 430 Hz. Rearranging the equation, we get:
L' = (1/2) (T/μ) (1/f')^2
where f' is the desired frequency (430 Hz).
Substituting the known values, we have:
L' = (1/2) (T/μ) (1/430)^2
Next, we need to calculate the change in position where the violinist's finger should be placed. The difference between the original length (0.350 m) and the new length (L') will give us the displacement.
Displacement = |L' - 0.350|
Now, if the position should remain correct within half the width of a finger (0.600 cm), we can set up an inequality:
|L' - 0.350| ≤ 0.600 cm
Solving this inequality will give us the allowable range for the displacement.
To find the maximum allowable percentage change in string tension, we can use the formula:
Percentage change = (|T' - T| / T) × 100
where T is the original tension and T' is the new tension.
Now, let's solve these equations step by step:
1. Determine the new length of the string (L'):
L' = (1/2) (T/μ) (1/430)^2
2. Calculate the displacement:
Displacement = |L' - 0.350|
3. Set up the inequality to find the allowable range for the displacement:
|L' - 0.350| ≤ 0.600 cm
4. Find the maximum allowable percentage change in string tension:
Percentage change = (|T' - T| / T) × 100
By solving these equations, you can determine where the violinist must place her finger to play concert A and the maximum allowable percentage change in the string tension.