Question

Two strings with linear densities of 5 g/m are stretched over pulleys, adjusted to have vibrating lengths of 50 cm, and attached to hanging blocks. The block attached to string 1 has a mass of 20kg and the block attached to string 2 has a mass M. Listeners hear a beat frequency of 2 Hz when string 1 is excited and its fundamental frequency and string 2 at its third harmonic. a) Was is the velocity of the transverse wave in string 1? b) What are the frequencies of the first 3 harmonics of string 1? c) Make a sketch of the first 3 harmonics. d) Write an expression for the frequency of the 3rd harmonic in string 2 in terms of the hanging mass M? e) Determine the possible values of the mass M and explain why there are 2 possible values?

Answer 1

part a:

for string 1, wave velocity=sqrt(tension/mass per unit length)

=sqrt(20*9.8/0.005)=198 m/s

part b:

fundamental frequency of first string=wave speed/(2*length )

=198/(2*0.5)=198 Hz

first three harmonics are:

1st: 198 Hz

2nd: 198*2=396 Hz

3rd : 594 Hz

part c:

sketch:

fundamental 0.5 0.0005 0.001 econic 0.0025 0.003 0.5 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.5 0.0005 0.001 0.0015 0.002 0.0025 0.003

part d:

tension in the string=weight of the block=M*g

speed of wave=sqrt(tension /mass per unit length)

=sqrt(M*g/0.005)

third harmonic frequency=3*speed/(2*length)

=3*sqrt(M*g/0.005)/(2*0.5)

=3*sqrt(M*g/0.005)

part e:

as beat frequency is 2 per second, frequency in second string is either 198+2=200 Hz or 198-2=196 Hz

then 3*sqrt(M*g/0.005)=200

==>M=2.2675 kg

or 3*sqrt(M*g/0.005)=196

==>M=2.178 kg

so two possible values of M can be 2.2675 kg or 2.178 kg.