A string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N. Resonant frequencies are found at 558 Hz and the next one at 744 Hz. What is the fundamental frequency of the string?
A) 558 Hz
B) 372 Hz
C) 93 Hz
D) 186 Hz
E) None of the above.

Question

Quizplus · Accepted Answer

The resonant frequencies of a fixed string under tension are given by:

f_n = n*v/2L

where n is the integer representing the harmonic (n=1 for fundamental frequency), v is the speed of the wave, and L is the length of the string.

The speed of the wave can be calculated using the formula:

v = sqrt(T/mu)

where T is the tension in the string and mu is the linear mass density.

Substituting the given values, we get:

v = sqrt(290 N / 0.0025 kg/m) = 340 m/s

Using the given resonant frequencies, we can set up two equations:

f_1 = v/2L ---> L = v/2f_1
f_2 = 3v/2L ---> L = 3v/2f_2

Equating the expressions for L, we get:

v/2f_1 = 3v/2f_2

Simplifying, we get:

f_1 = 3/2 * f_2

Substituting the values, we get:

f_1 = 3/2 * 744 Hz = 1116 Hz

Therefore, the fundamental frequency of the string is:

f_1 = 1116 Hz ≈ 186 Hz (rounded to nearest whole number)

The correct answer is D) 186 Hz.

A String with Mass Density Equal to 0