Calculating Total Energy of Vibration for a String

In summary, the total energy of vibration of a string of length L, fixed at both ends, oscillating in its nth characteristic mode with an amplitude A is (A^2)(n^2)(pi^2)T/4L. To calculate the total energy of vibration of the same string if it is vibrating in the following superposition of normal modes, we use the equation y(x,t)=A1sin( xpi/L)cos(w1t) + A3sin(3xpi/L)cos(w3t- pi/4) and the formula U=1/2 mu (dy/dt)^2 + T/2 (dy/dx)^2. By substituting the values for A1
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JB91
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Homework Statement


a.) Find the total energy of vibration of a string of length L, fixed at both ends, oscillating in its nth characteristic mode with an amplitude A. The tension in the string is T and its total mass is M. (HINT: consider the integrated kinetic energy at the instant when the string is straight so that it has no stored potential energy over and above what it would have when not vibrating at all.)

b.) Calculate the total energy of vibration of the same string is it is vibrating in the following superposition of normal modes:
y(x,t)=A1sin( xpi/L)cos(w1t) + A3sin(3xpi/L)cos(w3t- pi/4)
(You should be able to verify that it is the sum of the energies of the two modes separately.)

Answers:
a.) (A^2)(n^2)(pi^2)T/4L
b.) (A1^2 + 9A3^2)(pi^2)(T)/4L[/B]

Homework Equations


y(x,t)=A1sin( xpi/L)cos(w1t) + A3sin(3xpi/L)cos(w3t- pi/4)
U=1/2 mu (dy/dt)^2 + T/2 (dy/dx)^2[/B]

The Attempt at a Solution


https://ca.answers.yahoo.com/question/index?qid=20141120113724AAqz07h
https://ca.answers.yahoo.com/question/index?qid=20141120113815AANH7NE

Thats my attempt, final answer is off ( i put the answer from textbook at the bottom of the last page), sorry couldn't figure out how to post . Any help would be greatly appreciated!

also i just tried to get rid of the omegas using : w(n)=npi/L (T/mu)^(1/2) where mu=M/L
[/B]
 
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  • #2
Oh sorry forgot to mention I figured out A, I'm only trying to solve part b now!
 

1. What is the definition of Total Energy of Vibration?

The Total Energy of Vibration refers to the sum of all forms of energy associated with the motion of a vibrating object. This includes kinetic energy, potential energy, and any other forms of energy that may be present in the system.

2. How is Total Energy of Vibration calculated?

The Total Energy of Vibration can be calculated using the equation E = 1/2kA^2, where E is the total energy, k is the spring constant, and A is the amplitude of vibration. This equation is derived from the conservation of energy principle, which states that energy cannot be created or destroyed, only transferred or transformed.

3. What factors affect the Total Energy of Vibration?

The Total Energy of Vibration is affected by several factors, including the mass of the vibrating object, the amplitude and frequency of the vibration, and the stiffness of the material the object is vibrating in. Changes in any of these factors will result in a change in the total energy of vibration.

4. Why is the Total Energy of Vibration important to understand?

Understanding the Total Energy of Vibration is important in many fields, such as engineering, physics, and chemistry. It allows scientists and engineers to predict the behavior of vibrating objects and design structures that can withstand vibrations. Additionally, it plays a crucial role in fields such as acoustics and seismology, where the study of vibrations is essential.

5. How can the Total Energy of Vibration be controlled or manipulated?

The Total Energy of Vibration can be controlled or manipulated by adjusting the factors that affect it, such as mass, amplitude, and frequency. For example, increasing the stiffness of a material can decrease the amplitude of vibration and thus decrease the total energy. In practical applications, devices such as dampers and shock absorbers can also be used to reduce the total energy of vibration in a system.

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