A cello string vibrates in its fundamental mode with a frequency of 168 1/s. The vibrating segment is 86.7 cm long and has a mass of 0.55 g. Find the tension in the string. Answer in units of N.Find the frequency of the string when it vibrates in two segments. Answer in units of 1/s.

Answer :

Answer:

Tension = 0.034 N

Frequency in two segments = 336 1/s

Explanation:

The frequency of a string is given by

[tex]f=\dfrac{k}{2l}\sqrt{\dfrac{T}{\mu}}[/tex]

k represents the mode of vibration; for the fundamental frequency, k = 1

l is the length of the string in metre

T is the tension of the string in newton

[tex]\mu[/tex] is the linear density or mass per unit length in kg/m; it is a measure of the thickness of the string

Making T the subject of the formula,

[tex]T =(2fl\mu)^2[/tex]

f = 168

l = 86.7 cm = 0.867 m

[tex]\mu = \dfrac{0.55 \text{ g}}{86.7\text{ cm}} = \dfrac{0.00055 \text{ kg}}{0.867\text{ m}} = 6.3\times10^{-4} \text{ kg/m}[/tex]

Then

[tex]T =(2\times168\times0.867\times\dfrac{0.00055}{0.867})^2 = 0.034 \text{ N}[/tex]

When the string vibrates in two segments, it is in the second harmonic. This is simply twice the fundamental frequency.

Hence, f = 2 × 168 = 336 1/s

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