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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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The correct answer is:
$9: 1$
$\quad v=\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$
$\Rightarrow \quad v \propto \frac{\sqrt{\mathrm{T}}}{1}$
$\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}=\left[\frac{\mathrm{v}_{2}}{\mathrm{v}_{1}}\right]^{2}\left[\frac{\mathrm{l}_{2}}{\mathrm{l}_{1}}\right]^{2}=\left[\frac{300}{200}\right]^{2}\left[\frac{21}{\mathrm{l}}\right]^{2}=\frac{9}{1}$
$\Rightarrow \quad v \propto \frac{\sqrt{\mathrm{T}}}{1}$
$\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}=\left[\frac{\mathrm{v}_{2}}{\mathrm{v}_{1}}\right]^{2}\left[\frac{\mathrm{l}_{2}}{\mathrm{l}_{1}}\right]^{2}=\left[\frac{300}{200}\right]^{2}\left[\frac{21}{\mathrm{l}}\right]^{2}=\frac{9}{1}$
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