Question

A string of length 0.5m and linear density of 0.0001kgper m kept under a tention 1N. Find first three overtones (n>1 harmonic) of the string when it is plucked at mid points​

Answer 1

The tension given in the string is 490 N The linear density of the string is 0.1 kg/m

Therefore, the velocity produced in the string will be,

v=

T/μ

Substituting the values, gives v = 70 m/s

Now, the fundamental frequency of the string is, the HCF of 2 consecutive frequencies, i.e. 400 Hz and 450 Hz is 50 Hz

Now, the velocity anf fundamental frequency of the string are related by the expression,

v=2fL

Therefore, L = v/2f = 70/100 = 0.7

Answer 2

The velocity of the string is calculated by the formula:

$v=\frac{\sqrt{T} }{\sqrt{\mu} }$

Here, the tension is given as=$1N$

The linear density$\mu= 10^{-4} kg/m$

Length of the string$(l)=0.5m$

Velocity:

$v=\sqrt{1} /\sqrt{10^{-4} } \\v=100m/s$

When the string is plucked at the midpoint=

1st Overtone= $2[\frac{v}{2l} ]$

1st Overtone=$2[\frac{100}{2\times0.5 } ]=200Hz$

2nd Overtone=$3[\frac{v}{2l} ]$

2nd Overtone=$3[\frac{100}{2\times0.5 } ]=300Hz$

3rd Overtone=$4[\frac{v}{2l} ]$

3rd Overtone=$4[\frac{100}{2\times0.5 } ]=400Hz$

Hence, the first three overtones of the strings would be 200Hz, 300Hz, and 400Hz.