Using a highly sensitive parabolic sound collector, Frank records the frequency of a tuning fork...

Question:

Using a highly sensitive parabolic sound collector, Frank records the frequency of a tuning fork as it drops into the Grant Canyon. He drops the vibrating tuning fork from rest at t=0. He records the frequency of 1891 Hz at t=4.74 s. What is the natural frequency of the tuning fork? (in Hz) Use {eq}V_{sound}=343 m/s {/eq}.

(A) 332.6

(B) 482.3

(C) 699.3

(D) 1014

(E) 1470.3

(F) 2132

(G) 3091.4

(H) 4482.5

Doppler effect:

Doppler effect is given by Doppler. Any decrease or increase in the frequency of sound, waves or light as the source and observer move towards or away from each other.

Answer and Explanation:

Given data

  • The observed frequency is: {eq}F = 1891.0\;{\rm{Hz}} {/eq}
  • The velocity of sound is: {eq}{v_0} = 343.0\;{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}} {/eq}
  • The initial time is: {eq}{t_1} = 0\;{\rm{s}} {/eq}
  • The final time is: {eq}{t_2} = 4.740\;{\rm{s}} {/eq}


Write equation of motion.

{eq}{v_1} = u + g{t_1} {/eq}

Here, {eq}{v_1}{/eq} is velocity of source and u is initial velocity u = 0

The value of acceleration of gravity {eq}g = 9.81\;{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}. {/eq}

Substitute all the value in above expression.


{eq}\begin{align*} {v_1}& = 0 + 9.81 \times 4.74\\ {v_1}& = 46.5\;{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}} \end{align*} {/eq}


Write the Doppler effect.

{eq}F = \left( {\dfrac{{{v_0}}}{{{v_0} + {v_1}}}} \right)F' {/eq}

Here, {eq}F' {/eq}is natural frequency.

Substitute all the value in above expression.


{eq}\begin{align*} F' &= \dfrac{{1891}}{{\left( {\dfrac{{343}}{{343 + 46.5}}} \right)}}\\ F' &= 2147.36\;{\rm{Hz}} \end{align*} {/eq}

The, natulal frequency is {eq}2147.36\;{\rm{Hz}}. {/eq}. Therefore closest value in given in the option (f). Thus, option (f) is correct.


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