# 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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