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String Fixed at Both Ends

Two strings A...

Question

Two strings A and B made of same material, have lengths $l_{A}$ and $l_{B}$ and are connected to two other comparitively denser medium strings having masses $M_{A}$ and $M_{B}$ , the upper ends being supported by rigid supports. If $f_{A}$ and $f_{B}$ are their fundamental frequencies of their vibrations, then $f_{A} : f_{B}$ is

A

\frac{M_{A} l_{A}}{M_{B} l_{B}}

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B

\frac{l_{A}}{l_{B}} \sqrt{\frac{M_{A}}{M_{B}}}

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C

\frac{M_{A}}{M_{B}} \sqrt{\frac{l_{A}}{l_{B}}}

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D

None of these

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Solution

The correct option is D None of these

Both the strings will vibrate as if their both ends are fixed.
At equilibrium,
$T_{A} = M_{A} g$ and $T_{B} = M_{B} g$
Fundamental frequency.
$f = \frac{V}{2 l} = \frac{1}{2 l} \sqrt{\frac{T}{μ}}$
$\Rightarrow \frac{f_{A}}{f_{B}} = \sqrt{\frac{T_{A}}{T_{B}}} \times \frac{l_{B}}{l_{A}}$
$∴ f_{A} : f_{B} = \frac{l_{B}}{l_{A}} \sqrt{\frac{M_{A}}{M_{B}}}$
$\Rightarrow f_{A} : f_{B} = \frac{l_{B}}{l_{A}} \sqrt{\frac{M_{A}}{M_{B}}}$

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