A violin with string length 32 cm and string density $1.5 \mathrm {~g} / \mathrm { cm }$ resonates in its fundamental with the first overtone of a 2.0-m organ pipe with one end closed and one end open. What is the tension in the string if the speed of sound in air is 344 m/s?
A) 1000 N
B) 110 N
C) 450 N
D) 4100 N
E) 56 N

Question

Quizplus · Accepted Answer

The fundamental frequency of the string is given by:$$f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$where L is the string length, T is the tension, and \mu is the string density.The first overtone of the organ pipe is given by:$$f_2 = \frac{3}{4L} v$$where L is the pipe length and v is the speed of sound in air.Since the string and the organ pipe resonate, we have:$$f_1 = f_2$$$$\frac{1}{2L} \sqrt{\frac{T}{\mu}} = \frac{3}{4L} v$$$$T = \frac{9\mu v^2}{16}$$$$T = \frac{9(1.5 	imes 10^{-3} \mathrm{~kg/cm})(344 \mathrm{~m/s})^2}{16}$$$$T = 1000 \mathrm{~N}$$

A Violin with String Length 32 Cm and String Density 1.5 g/cm1.5 \mathrm {~g} / \mathrm { cm }1.5 g/cm

A Violin with String Length 32 Cm and String Density $1.5 \mathrm {~g} / \mathrm { cm }$