Answer to Question #163179 in Optics for Ada

Question

A string on a cello vibrates in the first normal mode with a frequency of  220 Hz. The vibration segment is 70 cm long and has a mass of 1.20 grams

(a) find the tension in the string 

(b) Determine the frequency of vibration when the string vibrates in three segments.

Answer 1

(a) We can find the tension in the string from the formula:

$$f_n=\dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}.$$

Since, the string on a cello vibrates in the first normal mode, we get:

$$f_1=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}=\dfrac{1}{2L}\sqrt{\dfrac{LT}{m}},$$ $$T=4Lmf_1^2,$$ $$T=4\cdot0.7\ m\cdot1.2\cdot10^{-3}\ kg\cdot(220\ Hz)^2=163\ N.$$

(b) We can find the frequency of vibration when the string vibrates in three segments from the same formula:

$$f_3=\dfrac{3}{2L}\sqrt{\dfrac{LT}{m}}=\dfrac{3}{2}\sqrt{\dfrac{T}{Lm}},$$ $$f_3=\dfrac{3}{2}\cdot\sqrt{\dfrac{163\ N}{0.7\ m\cdot1.2\cdot10^{-3}\ kg}}=660\ Hz.$$