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Question
A sonometer wire of length l vibrates in fundamental mode when excited by a tuning fork of frequency 416. Hz. If the length is doubled keeping other things same, the string will ______.
Options
vibrate with a frequency of 416 Hz
vibrate with a frequency of 208 Hz
vibrate with a frequency of 832 Hz
stop vibrating.
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Solution
vibrate with a frequency of 208 Hz
According to the relation of the fundamental frequency of a string
\[\nu = \frac{1}{2l}\sqrt{\frac{F}{\mu}}\]
where
l is the length of the string
F is the tension
μ is the linear mass density
We know that ν1 = 416 Hz, l1 = l and l2 = 2l.
\[v_1 \propto \frac{1}{l_1}\]
\[ v_1 l_1 = v_2 l_2 \]
\[\left( 416 \right)l = v_2 \left( 2l \right)\]
\[ v_2 = 208 Hz\]
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