A $1024 \mathrm{~Hz}$ tuning fork is dangled at the end of a string such that its center of mass is $1.2 \mathrm{~m}$ below the point at which the other end of the string is attached to door frame, making a pendulum of sorts. The tuning fork may be treated as a point mass for the purposes of this problem. If the pendulum is brought to an amplitude of $22^{\circ}$ and released, what is the range of frequencies heard by an observer standing in the plane of the pendulum's motion? The speed of sound is $340 \mathrm{~m} / \mathrm{s}$.
A) $1020 \mathrm{~Hz}$ to $1028 \mathrm{~Hz}$
B) $1010 \mathrm{~Hz}$ to $1038 \mathrm{~Hz}$
C) $1012 \mathrm{~Hz}$ to $1036 \mathrm{~Hz}$
D) $1015 \mathrm{~Hz}$ to $1033 \mathrm{~Hz}$

Question

Quizplus · Accepted Answer

The range of frequencies heard by an observer due to the Doppler effect depends on the speed of the tuning fork at its lowest point. The speed can be found using conservation of energy, equating potential energy at the highest point to kinetic energy at the lowest point. The frequency shift is then calculated using the Doppler effect formula, considering the speed of the tuning fork relative to the observer and the speed of sound. The calculated frequencies at the extremes of motion give the range of $1020 \mathrm{~Hz}$ to $1028 \mathrm{~Hz}$.

A 1024 Hz1024 \mathrm{~Hz}1024 Hz Tuning Fork Is Dangled at the End of a String

A $1024 \mathrm{~Hz}$ Tuning Fork Is Dangled at the End of a String