Question

A girl swings in a cradle with a period of oscillation of $\dfrac{\pi }{4}{\text{ sec}}$ and amplitude of 2 m. A boy standing in front of her blows a whistle of natural frequency 1000 Hz. Find the minimum frequency heard by the girl. (Velocity of sound in air is $320{\text{m/s}}$ ).
A) $850$ Hz
B) $1000$ Hz
C) $750$ Hz
D) $950$ Hz

Answer 1

The girl swinging in the cradle constitutes simple harmonic motion. The girl is moving back and forth from the sound produced by the boy. Thus, the frequency of the sound heard by the girl as she moves varies. This is termed the Doppler effect.

Formulas used:
-The relation between the linear velocity and angular velocity is given by, $v = r\omega$ where $v$ is the linear velocity, $\omega$ is the angular velocity and $r$ is the radius of the circle.
-The frequency observed when the source is stationary and the observer is moving is given by, $f' = \left( {1 + \dfrac{{{v_o}}}{v}} \right)f$ where, ${v_o}$ is the velocity of the observer, $v$ is the velocity of the sound in the medium and $f$ is the natural frequency of the sound wave.

Complete step by step answer.
Step 1: List the data given in the question.
A girl is swinging in a cradle back and forth and a boy standing in front of her is whistling. The motion of the girl is a simple harmonic motion.
The period of oscillation is given as $T = \dfrac{\pi }{4}{\text{ sec}}$ and the amplitude of oscillation is $A = 2{\text{ m}}$ .
The natural frequency of the sound made by the whistle is $f = 1000{\text{ Hz}}$ and the velocity of sound in air is given as $v = 320{\text{ m/s}}$.
Step 2: Find the velocity of the girl in the extreme positions of the oscillation.
Here, the boy is the source of the sound wave and the girl is the observer. The girl moves back and forth from the source as she swings in the cradle. The girl’s speed will be maximum at the extreme positions i.e., at positions right next to the boy and farthest away from the boy.
The relation between the linear velocity $v$ and angular velocity $\omega$ is given by, $v = r\omega$ , $r$ is the radius of the circle.
Here, the amplitude of the oscillation constitutes the radius because simple harmonic motion can be depicted as a circular motion. So, $r = A = 2{\text{ m}}$ .
The angular velocity can be expressed in terms of the period of oscillation, i.e., $\omega = \dfrac{{2\pi }}{T}$ .
Let the velocity of the girl be ${v_o}$ . Then, ${v_o} = A\omega = A\dfrac{{2\pi }}{T}$
Substituting the values for $A = 2{\text{ m}}$ and $T = \dfrac{\pi }{4}{\text{ sec}}$ in the above equation, we get ${v_o} = \dfrac{{2 \times 2\pi }}{{\left( {\dfrac{\pi }{4}} \right)}} = 16{\text{ m/s}}$
Thus the velocity of the girl in the extreme positions of the oscillation is 16 m/s.
Step 3: Find the minimum frequency of the sound heard by the girl using concepts of the Doppler effect.
Doppler effect refers to the frequency change of a sound wave due to the change in motion of the source, observer or both.
Ours is a case of frequency change due to the observer moving back and forth while the source remains stationary. The minimum frequency heard by the girl will be when she is away from the boy.
By Doppler effect, the frequency heard by the girl is given by, $f' = \left( {1 + \dfrac{{{v_o}}}{v}} \right)f$ -------- (1)
where ${v_o}$ is the velocity of the observer, $v$ is the velocity of the sound in the medium and $f$ is the natural frequency of the sound wave.
Here the medium of sound is air.
Substitute the values for ${v_o} = - 16{\text{ m/s}}$ , $v = 320{\text{ m/s}}$ and $f = 1000{\text{ Hz}}$ in equation (1).
Then we have, $f' = \left( {1 + \dfrac{{16}}{{320}}} \right) \times 1000$ and on simplifying we get $f' = \left( {\dfrac{{320 - 16}}{{320}}} \right) \times 1000 = 950{\text{ Hz}}$

Therefore, the minimum frequency heard by the girl is 950 Hz.

Note: When the observer moves away from the source, the velocity of the observer is taken to be negative and when the observer moves towards the source the velocity of the observer is taken to be positive. Thus, when the girl is moving away from the boy, her velocity is taken as negative i.e., ${v_o} = - 16{\text{ m/s}}$ .