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Question
A one-metre long stretched string having a mass of 40 g is attached to a tuning fork. The fork vibrates at 128 Hz in a direction perpendicular to the string. What should be the tension in the string if it is to vibrate in four loops?
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Solution
Given:
Length of the stretched string (L) = 1.00 m
Mass of the string =40 g
String is attached to the tuning fork that vibrates at the frequency (f) = 128 Hz
Linear mass density (m)
\[= \left( 40 \times {10}^{- 3} \right) kg/m\]
No. of loops formed, (n) = 4
\[L = \frac{n\lambda}{2}\]
\[ \Rightarrow \lambda = \frac{2L}{n} = \frac{2 \times 1}{4}\]
\[ \Rightarrow \lambda = 0 . 5 m\]
Wave speed \[ (v) = f\lambda = 128 \times 0 . 5\]
\[ \Rightarrow v = 64 m/s\]
\[We know: \]
\[ v = \sqrt{\left( \frac{T}{m} \right)}\]
\[ \Rightarrow T = \nu^2 m\]
\[ = \left( 64 \right)^2 \times 40 \times {10}^{- 3} \]
\[ = 163 . 84 \approx 164 N\]
Hence, the tension in the string if it is to vibrate in four loops is 164 N.
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