Question

When a string is divided into three segments of lengths l₁, l₂ and l₃, the fundamental frequencies of these three segments are v₁,v₂ and v₃ respectively. The original fundamental frequency (v) of the string is:

\((a)\ \sqrt{v} = \sqrt{v_1} + \sqrt{v_2} + \sqrt{v_3}\) 

\((b)\ v = v_1 + v_2 + v_3\)

\((c)\ \frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}\)

\((d)\ \frac{1}{\sqrt{v}} = \frac{1}{\sqrt{v_1}} + \frac{1}{\sqrt{v_2}} + \frac{1}{\sqrt{v_3}}\)  

Answer 1

Correct option is : \((c)\ \frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}\)

\(\because\ V = \frac{1}{2l}\sqrt{(\frac{T}{m})}\)

∴ n₁l₁ = n₂l₂ = n₃l₃ = k

\(\therefore\ l_1 = \frac{k}{n_1},\ l_2 =\frac{k}{n_2}\ and\ l_3 = \frac{k}{v}\)

Original length, \( l = \frac{k}{v} \) 

Here, l = l₁ + l₂ + l₃

\(\frac{k}{v} = \frac{k}{v_1} + \frac{k}{v_2} +\frac{k}{v_3}\)

\(\therefore\ \frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{k}{n_3}\)