Calculating Energy of Vibrating String Problem

In summary, the problem involves calculating the energy of vibrations in a string that is pulled up at the center to a height of h, with a constant tension T and length L. Two methods of calculation are suggested, one based on the work required to stretch the string and the other based on the frequency and linear density of the string. The latter method is found to be easier, but a correction is needed in the former method to account for the change in tension along the string.
  • #1
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Problem. A string fixed at two ends (which are a length L apart) is pulled up at the center to a height of h. Assuming that the tension T remains constant, calculate the energy of the vibrations of the string when it is released. [Hint: What work does it take to strech the string up?]

The work to pull the string is

[tex]\int_0^h \frac{y}{c} \, T \, dy[/tex]

where

[tex]c = \sqrt{y^2 + (L/2)^2}[/tex]

right? And if I were to calculate the energy directly, I would need to know the frequency of vibration and the linear density of the string right?
 
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  • #2
Nevermind. One may find the frequency given the linear density using the fact that a standing wave is produced when the string is released so:

[tex]f = \frac{v}{\lambda} = \frac{v}{2L} = \frac{\sqrt{T/\mu}}{2L}[/tex]

The energy is then given by:

[tex]E = \int_0^L 2 \pi^2 f^2 D(x)^2 \mu \, dx[/tex]

where

[tex]D(x) = h \sin (\pi x / L)[/tex]

right? I should, in theory, get the same answer using this method and the method in the first post.
 
  • #3
I ask because the latter integral is much easier to calculate (at least for me) than the former one. For the latter one, I get [itex]\pi T h / L[/itex] as the answer.
 
  • #4
It just dawned on me that

[tex]\frac{d}{dy} \sqrt{y^2 + (L/2)^2} = \frac{y}{\sqrt{y^2 + (L/2)^2}}[/tex]

Duh! So the integral in the first post becomes [itex]T(\sqrt{h^2 + L^2/4} - L/2)[/itex]. This doesn't agree with what I posted earlier. (After a quick dimensional analysis, I realize that the energy I calculated in post #3 is wrong.) Hmm...
 
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1. What is the "Energy of a Plucked String"?

The "Energy of a Plucked String" is a term used to describe the amount of potential energy stored in a string when it is plucked. This energy is initially in the form of potential energy, which is then converted into kinetic energy as the string vibrates.

2. How is the energy of a plucked string calculated?

The energy of a plucked string can be calculated using the formula E = 0.5 * k * x^2, where E is the energy, k is the string's stiffness coefficient, and x is the displacement of the string from its equilibrium position.

3. What factors affect the energy of a plucked string?

The energy of a plucked string is affected by several factors, including the string's tension, length, and thickness, as well as the amplitude and frequency of the plucking motion.

4. How does the energy of a plucked string relate to the sound it produces?

The energy of a plucked string is directly related to the sound it produces. As the string vibrates, it produces sound waves that travel through the air. The greater the energy of the string, the louder and more intense the sound will be.

5. Can the energy of a plucked string be changed?

Yes, the energy of a plucked string can be changed by altering the factors that affect it, such as the tension, length, and thickness of the string. Additionally, different plucking techniques can also change the energy of the string and therefore the sound it produces.

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