How do I find the normal modes of massless string w/ masses?

In summary, the problem involves finding the frequencies of the three smallest modes of transverse motion for a string with length L, mass M, and tension T. This is then compared to a massless string with the same tension and length, but with three masses of M/3 equally spaced. The correct solution involves using the equation ωq=2ω0|sin(q/2)| with ω02 = T/((M/3)(L/4)), resulting in frequencies of .84ν1, 1.55ν1, and 2.04ν1.
  • #1
ChickenTarm
2
0

Homework Statement


So, a string with length L and a mass of M is given tension T. Find the frequencies of the smallest three modes of transverse motion. Then compare with a massless string with the same tension and length, but there are 3 masses of M/3 equally spaced. So this is problem #1
http://www.physics.purdue.edu/~jones105/phys42200_Spring2013/Assignment_5_Spring2013.pdf

Homework Equations


ν * λ = velocity
velocity = sqrt(T * L / M)
νn = nν1 n = 1, 2, 3
ν1 = √(T/(4ML))

The Attempt at a Solution


I tried using coupled oscillators and the equation for finding the frequencies.
ωq=2ω0|sin(q/2)|
q = nπ/(N+1) where n is the index and N is the number of particles
This does not give me the correct answer.
The correct answer is: .84ν1, 1.55ν1, and 2.04ν1.
 
Last edited:
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  • #2
Hello. Welcome to PF!

What did you use for ωo?
 
  • #3
TSny said:
Hello. Welcome to PF!

What did you use for ωo?
ω02 = T/((M/3)(L/4))
 
  • #4
OK. That should give the correct answer. You'll have to show your detailed calculations in order to find the mistake.
 

Related to How do I find the normal modes of massless string w/ masses?

1. How do I determine the normal modes of a massless string with masses?

The normal modes of a massless string with masses can be determined using the wave equation, which describes the motion of a string. The equation takes into account the tension, length, and mass of the string and can be solved using mathematical methods such as Fourier series or separation of variables.

2. What is the significance of finding the normal modes of a massless string with masses?

The normal modes of a massless string with masses represent the different ways in which the string can vibrate. These vibrations are important in understanding the behavior of musical instruments, such as guitars and violins, and can also be applied in other areas such as structural engineering.

3. How do the masses of the string affect the normal modes?

The masses of the string affect the normal modes by changing the tension and the frequency at which the string vibrates. In general, heavier masses will result in lower frequencies and longer wavelengths, while lighter masses will result in higher frequencies and shorter wavelengths.

4. Can the normal modes of a massless string with masses be observed?

Yes, the normal modes of a massless string with masses can be observed through visualizing the string's vibrations or by using special equipment such as a strobe light. These modes can also be heard as different pitches or tones in musical instruments.

5. Are there any real-life applications of understanding the normal modes of a massless string with masses?

Yes, understanding the normal modes of a massless string with masses has many real-life applications. It is used in the design and tuning of musical instruments, in studying the behavior of structures such as bridges and buildings, and in the development of technologies such as sensors and actuators.

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