A string of length (L) and mass (M) vibrating with a linear frequency (f) and am

Question

A string of length (L) and mass (M) vibrating with a linear frequency (f) and amplitude (A) has a periodic transverse disturbance traveling its length at a speed (v).

L = 2.00 meters = length of string
M = 0.005 kg = mass of string
f = 100 Hz = linear frequency of vibration
A = 0.030 meters = amplitude of the period transverse wave traveling in the string
v = 20 m/s = speed of the transverse wave traveling down the string

Determine the following:
a) Wavelength of the periodic disturbance
b) Wave number
c) Tension in the string
d) Fundamental frequency in the string
e) Maximum transverse speed and maximum transverse acceleration of any small bit of string
f) Displacement of the bit of string at x = L/4 when t = 0.5 seconds
g) Speed of the bit of string at x = 3L/4 when t = (1/80) seconds
h) Average total energy of waves on the string
i) Average rate at which energy is transmitted along the string

Explanation / Answer

b) Wave number
           k = 2/ = 31.4

d) Fundamental frequency in the string
       f = 1/2L * T/

where = m/L = 0.005kg/2.00m = 0.0025kg/m

f = 22.12Hz

   maxium accleration = ^2 * A =  (2f )^2* A = 11.831 * 10^3m/s^2

f) The displacement of the string at x = L/4 = 2/4 = 0.5m when t = 0.5s

           y = A sin ( kx-t )

   = 0.030m sin( 31.4 * 0.5m - 628 * 0.5s)

= 0.026m

(g) speed at x = 3L/4 = 6/4 = 1.5 ,t = ( 1/80s)

v = -A cos(kx-t)

= -(628 ) ( 0.030) ( cos (31.4 * 1.5 - (628 * 1/80)

= -14.58m/s

(h) total energy = E = 1/2 * 2 A2

                     = 0.088J

        = 8.87W

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