Question

A certain elastic string has a spring constant of 18n/m, a relaxed length of 3 meters, and a mass of 20 grams. A 500 g mass is hung on a string and given a kinetic energy of 2J. While the mass is in oscillating, the string can be plucked and it will produce a sound. What is the highest and lowest frequency of sound which can be produced by plucking the string in this situation.

Answer 1

frequency of wave produced on string f = V/2L = (1/2L)(Tm/L)^1/2   = (T/4mL)^1/2
                                                            = (k(L-Lo)/4mL )^1/2 .......1
where T is tension in the string, m is mass of string and L is length of string, Lo is relaxed length of string, and k is spring constant of string.
From expression, when L tends to Lo, frequency is minimum and for maximum possible value of L, frequency is maximum.
Equilibrium extension of spring Xo = mg/k = 0.5*9.8/18 = 0.27 m
Extreme positions of string are given by
energy of oscillation = kA²/2   ( A is amplitude of oscillations)
2 = 18*A²/2
A = 0.47 m
As A < Xo, during oscillation at some time L = Lo or tension in string = 0
hence minimum frequency = 0
maximum frequency is when, L -Lo is maximum
L- Lo = xo + A = 0.74
L = 3+0.74 = 3.74
From 1 we get maximum frequency = ( 18*0.74/4*0.02*3.74)^1/2
                                                      = 6.7 Hz