Question

Two cello strings, with the same tension and length, are played simultaneously. Their fundamental frequencies produce audible beats with a frequency of 8 Hz. The string with the lower pitch (frequency) is tuned to an “A” (a frequency of 220 Hz). What is the approximate ratio of the linear mass density μ of the string with the higher pitch to that of the string with the lower pitch?

Answer 1

Explanation:

Let f₁ is the fundamental frequency, $f_1=8\ Hz$

Lower pitch frequency, $f_2=220\ Hz$

Fundamental frequency is, $f_1=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu_1}}$.....(1)

Lower frequency is, $f_2=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu_2}}$..............(2)

Dividing equation (1) and (2) as :

$\dfrac{f_1}{f_2}=\sqrt{\dfrac{\mu_2}{\mu_1}}$

$\dfrac{\mu_2}{\mu_1}=(\dfrac{f_1}{f_2})^2$

$\dfrac{\mu_2}{\mu_1}=(\dfrac{8}{220})^2$

$\dfrac{\mu_2}{\mu_1}=0.00132$

So, the ratio of  linear mass density μ of the string with the higher pitch to that of the string with the lower pitch is 0.00132. Hence, this is the required solution.