Question

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 1

Tension = 0.034 N

Frequency in two segments = 336 1/s

Step-by-step explanation:

The frequency of a string is given by

`f=(k)/(2l)\sqrt{(T)/(\mu)}`

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

`\mu` 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,

`T =(2fl\mu)^2`

f = 168

l = 86.7 cm = 0.867 m

`\mu = \frac{0.55 \text{ g}}{86.7\text{ cm}} = \frac{0.00055 \text{ kg}}{0.867\text{ m}} = 6.3*10^(-4) \text{ kg/m}`

Then

`T =(2*168*0.867*(0.00055)/(0.867))^2 = 0.034 \text{ N}`

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