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  4. A string vibrates with a frequency of 200 Hz. When its length is doubled and tension is altered, it begins to vibrate with a frequency of 300 Hz. The ratio of the new tension to the original tension is

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Q. A string vibrates with a frequency of $200 \,Hz$. When its length is doubled and tension is altered, it begins to vibrate with a frequency of $300\, Hz$. The ratio of the new tension to the original tension is

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Solution:

$f =\frac{1}{2 L } \sqrt{\frac{T}{\mu}}$
where $f$ is frequency, $T$ is Tension, $L =$ length and $\mu$ is constant $\therefore f \propto \frac{\sqrt{ T }}{ L }$
$\Rightarrow \frac{ f _{2}}{ f _{1}}=\left(\sqrt{\frac{ T _{2}}{ T _{1}}}\right)\left(\frac{ L _{1}}{ L _{2}}\right)$
Squaring on both sides and rearranging we get $\Rightarrow \frac{ T _{2}}{ T _{1}}=\frac{\left( f _{2} L _{2}\right)^{2}}{\left( f _{1} L _{1}\right)^{2}}$
$L _{2}= 2 L _{ 1 }, f _{ 1 }= 2 0 0 H z , f _{2}= 3 0 0 H z$
Substituting above values we get $\frac{ T _{2}}{ T _{1}}=\frac{ 9 }{ 1 }$