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A string vibrates with a frequency of $200 \mathrm{~Hz}$. When its length is doubled and tension is altered, it begins to vibrate with a frequency of $300 \mathrm{~Hz}$. The ratio of the new tension to the original tension is
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Verified Answer
The correct answer is:
9 : 1
Given,
$\begin{aligned}
& v_1=200 \mathrm{~Hz}, v_2=300 \mathrm{~Hz}, \\
& l_1=l \text { and } l_2=2 l \\
& \qquad \mathrm{v}=\frac{1}{2 l} \sqrt{\frac{T}{m}} \Rightarrow \mathrm{v} \propto \frac{\sqrt{T}}{l} \Rightarrow \sqrt{T} \propto v \cdot l \\
& \Rightarrow \quad \frac{T_2}{T_1}=\left[\frac{v_2}{v_1}\right]^2\left[\frac{l_2}{l_1}\right]^2=\left[\frac{300}{200}\right]^2\left[\frac{2 l}{l}\right]^2=\frac{9}{1}
\end{aligned}$
$\begin{aligned}
& v_1=200 \mathrm{~Hz}, v_2=300 \mathrm{~Hz}, \\
& l_1=l \text { and } l_2=2 l \\
& \qquad \mathrm{v}=\frac{1}{2 l} \sqrt{\frac{T}{m}} \Rightarrow \mathrm{v} \propto \frac{\sqrt{T}}{l} \Rightarrow \sqrt{T} \propto v \cdot l \\
& \Rightarrow \quad \frac{T_2}{T_1}=\left[\frac{v_2}{v_1}\right]^2\left[\frac{l_2}{l_1}\right]^2=\left[\frac{300}{200}\right]^2\left[\frac{2 l}{l}\right]^2=\frac{9}{1}
\end{aligned}$
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