Rankers Physics

Standing Wave in String and Organ Pipe: Practice Problem & Solution

If $n_1, n_2$ and $n_3$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency $n$ of the string is given by: (2014)
$\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$
$\frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_1}} + \frac{1}{\sqrt{n_2}} + \frac{1}{\sqrt{n_3}}$
$\sqrt{n} = \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3}$
$n = n_1 + n_2 + n_3$

Solution Explained:

To solve this problem, we apply the core principles of Standing Wave in String and Organ Pipe. Understanding the underlying formula is key to arriving at the correct answer below:

Total length $L = L_1 + L_2 + L_3$. Since frequency $n \propto 1/L \Rightarrow L \propto 1/n$. Substituting lengths, we get $\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$.

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