Question

If $n_1,n_2,n_3,...$ are frequencies of segments of a stretched string, the frequency $n$ of the string is given by

• A. $n=n_1\times n_2\times n_3\times ...$
• B. $n=\sqrt{n_1+n_2+n_3+...}$
• C. $\frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}+...$
• D. none of the above

Answer 1

Answer: (C)

$\frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}+...$

Frequency of the first segment can be given by:

$n_1=\frac{1}{2l_1}\sqrt{\frac{T}{m}}\Rightarrow l_1=\frac{1}{2n_1}\sqrt{\frac{T}{m}}$

Similarly the frequency of the second segment can be given by:

$n_2=\frac{1}{2l_2}\sqrt{\frac{T}{m}}\Rightarrow l_2=\frac{1}{2n_2}\sqrt{\frac{T}{m}}$ and so on.

The total length of the string can be given by:

$l=l_1+l_2+....=\frac{1}{2n_1}\sqrt{\frac{T}{m}}+\frac{1}{2n_2}\sqrt{\frac{T}{m}}+...=d\frac{1}{2}\sqrt{\frac{T}{m}}[\frac{1}{n_1}+\frac{1}{n_2}+...]$

$\Rightarrow 1=\frac{1}{2l}\sqrt{\frac{T}{m}}[\frac{1}{n_1}+\frac{1}{n_2}+...]=\nu [\frac{1}{n_1}+\frac{1}{n_2}+...]$

$\Rightarrow \frac{1}{n}=[\frac{1}{n_1}+\frac{1}{n_2}+...]$