What is the Amplitude of a Standing Wave on a Guitar String?

[General_Sax]

Homework Statement

A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.386 m. The maximum transverse acceleration of a point at the middle of the segment is 9000 m/s^2 and the maximum transverse velocity is 3.00 m/s.

Homework Equations

y(x,t) = A*sin(kx)*sin(wt)

dy(x,t)/dt = A*w*sin(kx)*sin(wt)

d^2y(x,t)/dt^2 = -A*w^2*sin(kx)*sin(wt)

The Attempt at a Solution

I equated the max velocity with the secound derivative of the stading wave function.

3.00 = A*w*sin(kx)cos(wt)

Quick question: The x in the equation refers to the location on the string, not the
displacement of the string from equilibrium correct?

2nd question: is max velocity still at displacement = 0, and max acceleration at x =
1*amplitude?

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From here I tried to solve for k, first by solving for lambda.

lambda = 2L/n

lambda = 2L (because n = 1, because (sorry) f = fundamental frequency

lambda = 0.772 m

from there I solved for k

k = 2pi/lambda

k = 8.1388 /m

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using this I solved for w (omega) using w = v*k

I used max velocity for this equation

w = 24.417/s

now, does the fact this is MAX velocity effect the caluculation?

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I failed at solving for t, and I don't want to waste your time, so I won't repeat my
attempts here.

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From there I subbed the values into the equation and attempted to solve for A
(amplitude).

 

[General_Sax]

I've thought about this question a little more. If velocity and acceleration are maxed, then the cos/sin values must be equal to one. So, I've done this work:





1: A*w = 3.00
2: A*w^2 = 9000

1: w = 3.00/A

subbing into 2

A*(3.00/A)^2 = 9000

then:

9.00/A = 9000

A = 9.00/9000

A = 0.00100m = 1.00*10^-3 m

Does this look correct to you learned folk? I only ask because there is no answer in the textbook.

 

[ehild]

It is correct, as both the maximum acceleration and maximum speed occurs at the same place, at the middle of the string. Was it the maximum amplitude the question?

ehild

 

[General_Sax]

Oh, yes it was, sorry. Thanks for the help.