Find the Minimum Length of a Vibrating String

In summary, the equation of a stationary wave on a string with fixed ends is given by y= [0.6sin(pi/10)x]cos(600pi)t, and the smallest length of the string can be found by dividing the wavelength by 2. This is because the string must have 2 nodes on each end, and the distance between nodes is equal to half the wavelength. This can be further researched through the concept of string vibration modes.
  • #1
vijayramakrishnan
90
0

Homework Statement


the equation of a stationary wave produced on a string whose both ends are fixed is given by
y= [0.6sin(pi/10)x]cos(600pi)t
what could be the smallest length of string?

Homework Equations


k=2pi/wavelength

The Attempt at a Solution


i got the wavelength to be 20 but don't know how to proceed,in the solution it is given that
"since the string is fixed at both ends the minimum length should be half of the wavelength"i couldn't understand this statement why should it be half?please explain
 
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  • #2
Okay Vijay
But what exactly are you supposed to find?
 
  • #3
Suraj M said:
Okay Vijay
But what exactly are you supposed to find?
sir we are supposed to find the minimum length of the string.it has been given in the question sir.
 
  • #4
As the string is attached on both sides, it's should contain 2 nodes on each end is what I think, so in which case would there be the least number (or fraction ) of both nodes and internodes
Try drawing all the cases you can think of.
I hope I'm helpful
 
Last edited:
  • #5
Suraj M said:
As the string is attached on both sides, it's should contain 2 nodes on each end is what I think, so in which case would there be the least number (or fraction ) of both nodes and internodes
Try drawing all the case you can think of.
I hope I'm helpful
thank you,since the two ends must be at rest so they must be nodes and by finding the distance between two consecutive nodes we get the answer.i think this the explanation
 
  • #6
So, the answer?
 
  • #7
vijayramakrishnan said:
i got the wavelength to be 20 but don't know how to proceed,in the solution it is given that
"since the string is fixed at both ends the minimum length should be half of the wavelength"i couldn't understand this statement why should it be half?please explain
You should be able to do a bit of research to find an explanation. Did you try a web search on "string vibration modes"? Do so and take a look at the Hyperphysics web page (Standing Waves on a String) that turns up in the search results.
 
  • #8
Suraj M said:
So, the answer?
it is half of wavelength that is 10.
 

1. What is the purpose of finding the minimum length of a vibrating string?

The purpose of finding the minimum length of a vibrating string is to determine the shortest possible length at which the string can vibrate and produce a given frequency. This is important in various fields such as acoustics, music, and engineering.

2. How is the minimum length of a vibrating string calculated?

The minimum length of a vibrating string is calculated using the formula L = λ/2, where L is the length of the string and λ (lambda) is the wavelength of the sound wave produced by the string. The wavelength is inversely proportional to the frequency, so as the frequency increases, the wavelength decreases and thus the minimum length of the string also decreases.

3. What factors affect the minimum length of a vibrating string?

The minimum length of a vibrating string is affected by the tension of the string, its mass per unit length, and the speed of the wave traveling through the string. These factors can be manipulated to change the frequency produced by the string and thus the minimum length at which it can vibrate.

4. Can the minimum length of a vibrating string be longer than the string itself?

No, the minimum length of a vibrating string cannot be longer than the string itself. This is because the string needs to have a certain length in order to produce a sound wave with a given frequency. If the string is shorter than the minimum length, it will not be able to produce the desired frequency.

5. How is the minimum length of a vibrating string used in real-world applications?

The minimum length of a vibrating string is used in various real-world applications such as musical instruments, where the length of the string is adjusted to produce different notes or pitches. It is also used in engineering to design and optimize structures that produce sound, such as speakers and musical instruments. Additionally, the minimum length of a vibrating string is important in acoustic research and the study of sound waves.

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