Fundamental Frequency of a String

In summary, a string with a length of 2.5m has adjacent resonances at frequencies 112 Hz and 140 Hz, and the fundamental frequency of the string is determined to be 28 Hz. This can be found by dividing the difference between the two resonances by each of the given solutions and choosing the one that results in a whole number, or by using the formula (n+1)*F - n*F = 1*F = F. Both methods lead to the same answer of 28 Hz.
  • #1
NeRdHeRd
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0

Homework Statement



A string with a length of 2.5m has two adjacent resonances at frequencies 112 Hz and 140 Hz. Determine the fundamental frequency of the string?

A. 14 Hz
B. 28 Hz
C. 42 Hz
D. 56 Hz
E. 70 Hz


2. The attempt at a solution

Since I am not sure how to begin solving this I divided the two Resonances by each of the given solutions to the problem and based on that I came up with 28 Hz as being correct.

112/28 = 4 and 140/28 = 5 since these are adjacent and the others were not I chose 28 Hz

I'm sure there is some type of formula or a method of solving this. If anyone can help me with solving this the correct way it would be appreciated. Thank you.
 
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  • #2
I think the method you solved it is right. One shortcut you can take is 140Hz-112Hz = 28Hz.

So the answer must be 28Hz. (reason being (n+1)*F - n*F = 1*F = F... so if 140 and 112 are adjacent frequencies, F must be 28).

But you should also make sure that 140/28 gives an integer and 112/28 gives an integer... which they both do...
 
  • #3


I would approach this problem by using the equation for calculating the fundamental frequency of a string, which is f = (1/2L) * √(T/μ), where L is the length of the string, T is the tension on the string, and μ is the mass per unit length of the string.

Using the given information, we can set up two equations:

112 = (1/2 * 2.5) * √(T/μ)
140 = (1/2 * 2.5) * √(T/μ)

Solving for T/μ in both equations, we get:

112 = (5/2) * √(T/μ)
140 = (7/2) * √(T/μ)

Dividing the second equation by the first, we get:

140/112 = (7/2) * √(T/μ) / (5/2) * √(T/μ)

Simplifying, we get:

5/4 = √(T/μ) / √(T/μ)

Therefore, we can conclude that the tension and mass per unit length of the string are the same for both resonances.

Now, we can use the fundamental frequency equation to solve for the fundamental frequency:

f = (1/2 * 2.5) * √(T/μ)

Substituting in the values for the tension and mass per unit length that we found earlier, we get:

f = (1/2 * 2.5) * √(T/μ) = (1/2 * 2.5) * √(140/μ) = 28 Hz

Therefore, the fundamental frequency of the string is 28 Hz.
 

What is the fundamental frequency of a string?

The fundamental frequency of a string is the lowest frequency at which a string can vibrate. It is also known as the first harmonic or the first overtone.

How is the fundamental frequency of a string calculated?

The fundamental frequency of a string can be calculated using the equation f = 1/2L * √(T/μ), where L is the length of the string, T is the tension in the string, and μ is the linear density of the string.

What factors affect the fundamental frequency of a string?

The fundamental frequency of a string is affected by its length, tension, and linear density. Other factors such as temperature, material, and stiffness of the string can also have an impact.

What is the relationship between the fundamental frequency and the harmonics of a string?

The fundamental frequency is the first harmonic of a string. The harmonics of a string are integer multiples of the fundamental frequency. For example, the second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental frequency, and so on.

How does the fundamental frequency of a string change with the length of the string?

The fundamental frequency of a string is inversely proportional to the length of the string. This means that as the length of the string increases, the fundamental frequency decreases, and vice versa.

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