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  4. If the vibrations of a string are to be increased by a factor of two, the tension in the string should be made

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Q. If the vibrations of a string are to be increased by a factor of two, the tension in the string should be made

MGIMS WardhaMGIMS Wardha 2007

Solution:

The frequency of vibration of a string is given by $ n=\frac{1}{2l}\sqrt{\frac{T}{m}} $ where $ l $ is length, T the tension and m the mass per unit length. Given, $ {{n}_{1}}=n,{{n}_{2}}=2n $ $ \therefore $ $ \frac{{{n}_{1}}}{{{n}_{2}}}=\frac{n}{2n}=\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}} $ $ \Rightarrow $ $ \frac{1}{4}=\frac{{{T}_{1}}}{{{T}_{2}}} $ $ \Rightarrow $ $ {{T}_{2}}=4{{T}_{1}} $ Hence, to increase n by a factor of 2 tension must increase four times.