Question

The A string on a cello vibrates in its first normal mode with a frequency of 220 Hz. The vibrating segment is 70.0 cm long and has a mass of 1.20 g. (a) Find the tension in the string. (b) Determine the frequency of vibration when the string vibrates in three segments

Answer 1

Force is 161.27 N and frequency of vibration when the string vibrates in three segments is 660 Hz .

Step-by-step explanation:

Given :

Frequency , f = 220 Hz .

Length of wire , L = 70 cm = 0.7 m .

Mass of wire ,
`m = 1.20\ g=(1.2)/(1000)=1.2* 10^(-3)\ kg .`

a ) We know, frequency in string is :

`f=(1)/(2L)\sqrt{(F)/(\mu)}`

Therefore ,

`F=4\mu L^2f^2` .... equation 1.

Here ,
`\mu` is mass per unit length .

So,
`\mu=(m)/(L)=(1.2* 10^(-3)\ kg)/(0.7\ m)=1.7* 10^(-3)\ kg/m.`

Putting value of
`\mu` in equation 1.

We get ,
`F=4* 1.7* 10^(-3) * 0.7^2 * 220^2=161.27\ N.`

b) We know , frequency of when n segment are in string :

`f_n=nf_1`

For , n = 3

`f_3=3f_1\\\\f_3=3* 220\\\\f_3=660\ Hz.`

Hence , this is the required solution.