Question

Determine the frequencies of the fundamental mode and the next two harmonics that can be set up on a sitar string of length 1.0 m. Take the speed of waves of the string to be 2.8 10 ms .

Answer 1

Fundamental mode and next two harmonics:

1. For $\rm fun{da}mental$ frequency:

$\therefore \: \dfrac{ \lambda}{2} = l$

$\implies \: \lambda = 2l$

Now, frequency be f :

$\therefore \: f = \dfrac{v}{ \lambda}$

$\implies \: f = \dfrac{280}{ 2 \times 1}$

$\implies \: f = 140 \: hz$

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For 2nd harmonic:

$\therefore \: \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} = l$

$\implies \: \lambda= l$

Now, frequency be f2:

$\therefore \: f2 = \dfrac{v}{ \lambda}$

$\implies \: f2 = \dfrac{280}{ 1}$

$\implies \: f2 = 280 \: hz$

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For 3rd harmonic:

$\therefore \: \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} = l$

$\implies \: \lambda = \dfrac{2l}{3}$

Let frequency be f3 :

$\therefore \: f3 = \dfrac{v}{ \lambda}$

$\implies \: f3 = \dfrac{v \times 3}{ 2 \times 1}$

$\implies \: f3 = \dfrac{280 \times 3}{ 2 \times 1}$

$\implies \: f3 = 420 \: hz$

Hope It Helps !