Vibrations problem - Transverse in a string under tension

In summary, the equation for the natural frequencies of a cord under tension with one end fixed and the other attached to a spring-mass system is given by tan(wL/c) = -(T/kL)((wL/c)/(1-(w/w_n)^2)), where w_n^2 = k/m and c^2 = T/(pS). This can be simplified to f=v/lambda, where v^2=c^2 = T/(pS) and lambda is a half-integer multiple of the cord's length.
  • #1
alexisonsmith
29
0

Homework Statement


A cord of length L density p and cross-sectional area S is under tension T with the left end fixed and the right end attached to a spring-mass sstem. Show that the equation for the natural frequencies is given by:

tan(wL/c) = -(T/kL)((wL/c)/(1-(w/w_n)^2))


Homework Equations



w_n^2 = k/m and c^2 = T/(pS)

The Attempt at a Solution



I have tried this but I am really having problems starting this problem out, please help
 
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  • #2
Are you sure that equation is right? The natural frequencies are given very simply by f=v/lambda, where v^2=c^2 = T/(pS) and lambda is a half-integer multiple of the pipe's length.
 
  • #3
Yes it is Q.2 on the sheet which I have attached.
 

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FAQ: Vibrations problem - Transverse in a string under tension

What is a transverse vibration problem in a string under tension?

A transverse vibration problem in a string under tension refers to the study of how a string behaves when it is stretched between two points and then disturbed by a force perpendicular to the direction of the string. This type of vibration is characterized by the string moving up and down, or side to side, as opposed to vibrating in a longitudinal direction.

What factors affect the transverse vibration of a string under tension?

The transverse vibration of a string under tension is affected by several factors, including the tension of the string, the length of the string, and the density or thickness of the string. Additionally, the type of material the string is made of and the frequency of the disturbance also play a role in the vibration.

What is the equation used to calculate the frequency of vibration in a string under tension?

The equation used to calculate the frequency of vibration in a string under tension is known as the wave equation. It is expressed as f = 1/2L * sqrt(T/μ), where f is the frequency, L is the length of the string, T is the tension in the string, and μ is the mass per unit length of the string.

How does tension affect the frequency of vibration in a string?

Tension plays a significant role in determining the frequency of vibration in a string. As the tension in the string increases, the frequency of vibration also increases. This means that a tighter string will vibrate at a higher frequency than a looser string, assuming all other factors remain constant.

What is the relationship between the wavelength and frequency of vibration in a string under tension?

The wavelength and frequency of vibration in a string under tension are inversely proportional to each other. This means that as the frequency increases, the wavelength decreases, and vice versa. This relationship is expressed as λ = v/f, where λ is the wavelength, v is the velocity of the wave, and f is the frequency of vibration.

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