Answered

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 :

Explanation:

Let f₁ is the fundamental frequency, [tex]f_1=8\ Hz[/tex]

Lower pitch frequency, [tex]f_2=220\ Hz[/tex]

Fundamental frequency is, [tex]f_1=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu_1}}[/tex].....(1)

Lower frequency is, [tex]f_2=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu_2}}[/tex]..............(2)

Dividing equation (1) and (2) as :

[tex]\dfrac{f_1}{f_2}=\sqrt{\dfrac{\mu_2}{\mu_1}}[/tex]

[tex]\dfrac{\mu_2}{\mu_1}=(\dfrac{f_1}{f_2})^2[/tex]

[tex]\dfrac{\mu_2}{\mu_1}=(\dfrac{8}{220})^2[/tex]

[tex]\dfrac{\mu_2}{\mu_1}=0.00132[/tex]

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.

Other Questions