Physics · Wave motion, longitudinal and transverse waves, speed of the travelling wave, displacement relation for a progressive wave

If the density of materials of two strings of same length, tension and area of c

If the density of materials of two strings of same length, tension and area of cross-section are 2 kgm \( ^{-3} \) and \( 4 k g m^{-3} \) respectively then the ratio of their frequencies will be

  • A. \( 1: \sqrt{2} \) 2 \( : \sqrt{2} \cdot \sqrt{2} \)
  • B. 2: 1
  • C. 1: 2
  • D. \( \sqrt{2}: 1 \)

Step-by-step solution

The frequency of a stretched string is given by \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( \mu = \rho A \). Since length, tension, and area are same, \( f \propto \frac{1}{\sqrt{\rho}} \). Thus \( \frac{f_1}{f_2} = \sqrt{\frac{\rho_2}{\rho_1}} = \sqrt{\frac{4}{2}} = \sqrt{2} \), so the ratio is \( \sqrt{2}:1 \).
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