Question

On a cello, the string with the largest linear density (1.56 *10-2 kg/m) is the C string. This string produces afundamental frequency of 65.4 Hz and has a length of .08 m betweenthe two fixed ends.
What is the speed of the wave on the string?
What is the wavelength of the wave on the string?
What is the tension in the string?

Answer 1

1) Wavelength of the wave: 1.6 m

2) Speed of the wave: 104.6 m/s

3) Tension in the string: 170.7 N

Explanation:

1)

For the standing waves on a string, the wavelength of the wave is related to the length of the string by

$\lambda = 2 L$

where

$\lambda$ is the wavelength

L is the length of the string

For the string in this problem.

L = 0.8 m is its length (I assume there is a mistake in the text, since 0.08 m is not a realistic value for the length of the string)

Therefore, the wavelength of the wave on the string is

$\lambda=2(0.8)=1.6 m$

2)

The speed of a wave is calculated through the wave equation:

$v=f\lambda$

where

f is the frequency

$\lambda$ is the wavelength

For the standing wave on this string, the fundamental frequency is

$f=65.4 Hz$

while the wavelength is

$\lambda=1.6 m$

Therefore, the speed of the wave is

$v=(65.4)(1.6)=104.6 m/s$

3)

The speed of the wave is related to the tension in the string by

$v=\sqrt{\frac{T}{\mu}}$

where

v is the speed

T is the tension

$\mu$ is the linear density of the string

For this string,

v = 104.6 m/s

$\mu=1.56\cdot 10^{-2} kg/m$

Therefore, the tension in the string is

$T=\mu v^2 = (1.56\cdot 10^{-2})(104.6)^2=170.7 N$

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