A string of length L and mass m vibrating with a linear frequency f and amplitud

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

Determine
1. Wavelength of the periodic disturbance
2. Wave number
3. Tension in the string
4. Fundamental frequency for this string
5. Maximum transverse speed of any small bit of string
6. Maximum transverse acceleration of any small bit of string
7. Displacement of the bit of string at x= L/4 at t = 0.5s
8. Speed of the bit of string at x = 3L/4 at the time t = (1/80)s
9. Average total energy of waves on the string
10. 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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