Question

The mass of a string is 5.5 × 10-3 kg, and it is stretched so that the tension in it is 230 N. A transverse wave traveling on this string has a frequency of 160 Hz and a wavelength of 0.66 m. What is the length of the string?

Answer 1

L = 0.275 m

Step-by-step explanation:

velocity of transverse wave in a stretched string is given as

`v =\sqrt (T)/(\mu)`

where T = tension = 230N

μ = linear density

`μ = \frac[m}{L}`

where length L is in meters

Velocity =
`n\lambda`

so we have after equating both value of velocity

`\sqrt (T)/(\mu) = n\lambda`

`(T)/(\mu) =(n\lambda)^(2)`

μ =
`(T)/((n\lambda)^(2))`

μ =
`(230)/((160*0.66)^(2))`

μ = 0.020 kg/m

but μ =
`\frac[m}{L}`

so length of string is

L =
`(5.5*10^(-3))/(0.020)`

L = 0.275 m

Answer 2

The length of the string is 0.266 meters.

Step-by-step explanation:

It is given that,

Mass of the string,
`m=5.5* 10^(-3)\ kg`

Tension in the string, T = 230 N

Frequency of wave, f = 160 Hz

Wavelength of the wave,
`\lambda=0.66\ m`

We need to find the length of the string. Let l is the length of the string. The speed of a transverse wave is given by :

`v=\sqrt{(T)/(M)}`

M is the mass per unit length, M = m/l

`v=\sqrt{(lT)/(m)}`

`l=(v^2m)/(T)`

The velocity of a wave is,
`v=\\u* \lambda`

`l=((\\u* \lambda)^2m)/(T)`

`l=((160\ Hz* 0.66\ m)^2* 5.5* 10^(-3)\ kg)/(230\ N)`

l = 0.266 meters

So, the length of the string is 0.266 meters. Hence, this is the required solution.