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

Answer:

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

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      

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

and,        

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

also,

$\lambda=l=\frac{2.5}7\times\frac 1 {2.5}=\frac{1}{7}$          

Therefore,        

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