Question

A guitar string is clamped at both ends and is under a certain tension as it lies horizontally (along the x - axis, say). When plucked, the vibrating string is found to have a harmonic at a frequency of 560 Hz. When plucked another way, the vibrating string is observed to have a harmonic at a frequency of 700 Hz. You are told these harmonics are consecutive modes of oscillation. The velocity of the travelling waves is 200 m/s. Determine the length of the string.

Answer 1

The length of the string is
`\frac17` m.

Step-by-step explanation:

Data provided in the question:

Velocity, V = 200 m/s

Initial frequency
`\Rightarrow f_1=550 \ Hz,`

Final frequency,
`\quad f _(2)=700\ Hz`

Now, we know

`\Rightarrow \lambda=\frac Vf\\ \Rightarrow \lambda_(1)=(v)/(f_1)\\ =(200)/(560)=(2)/(5 .6)=\frac {2.5}7`

and,

`\qquad \lambda_(2)=\frac V {f_2}=(200)/(700)=(2)/(7)`
`\text { Here } \lambda_(1)=2 \lambda+\frac \lambda {2}=(5 \lambda)/(2)=2.5 \lambda`

also,

`\lambda=l=\frac{2.5}7*\frac 1 {2.5}=(1)/(7)`

Therefore,

The length of the string is
`\frac17` m.