PartA Learning Goal: To understand the concept of normal modes of oscillation an
ID: 3308630 • Letter: P
Question
PartA Learning Goal: To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct? View Available Hint(s) A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency f, and associated pattem of oscillation. The wave is traveling in the +x direction. The wave is traveling in the -x direction. Consider an example of a system with normal modes: a string of length L held fixed at both ends, located at x = 0 and x-L. Assume that waves on this string propagate with speed v. The string extends in the x direction, and the waves are transverse with displacement along the y direction. The wave will satisfy the given boundary conditions for any arbitrary wavelength The wavelength The wave does not satisfy the boundary condition yi (0,t) = can have only certain specific values if the boundary conditions are to be satisfied. O Submit In this problem, you will investigate the shape of the normal modes and then their frequency The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is Part B Which of the following statements are true? The system can resonate at only certain resonance frequencies f, and the wavelength must be such that Vi (x, t)-Ai sin (2m . ) sin(27/t) y, (0,t)-y, (L,t) = 0 A; must be chosen so that the wave fits exactly on the string Any one of Ai orAi or fi can be chosen to make the solution a normal mode Submit t AnsExplanation / Answer
According to the given problem,
a)
Ans:The wavelength i can have only certain specific values if the boundary conditions are to be satisfied.
b.)
Ans: The system can resonate at only certain resonance frequencies fi and the wavelength i must be such that yi(0;t)=yi(L;t)=0.
The key factor producing the normal modes is that there are two spatial boundary conditions, yi(0,t)=0 and yi(L,t)=0, that are satisfied only for particular values of i .
c.)
Ans: 2L,L,2L/3
The procedure described here contains the same mathematics that leads toquantization in quantum mechanics.
d.)
Ans:v/i
The frequencies fi are the only frequencies at which the system can oscillate. If the string is excited at one of these resonance frequencies it will respond by oscillating in the pattern given by yi(x,t), that is, with wavelength i associated with the fi at which it is excited. In quantum mechanics these frequencies are called the eigenfrequencies, which are equal to the energy of that mode divided by Planck's constant h.