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

Step-by-step explanation:

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

Lower pitch frequency,
`f_2=220\ Hz`

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

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

Dividing equation (1) and (2) as :

`(f_1)/(f_2)=\sqrt{(\mu_2)/(\mu_1)}`

`(\mu_2)/(\mu_1)=((f_1)/(f_2))^2`

`(\mu_2)/(\mu_1)=((8)/(220))^2`

`(\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.