Expectant parents are thrilled to hear their unborn baby's...

Expectant parents are thrilled to hear their unborn baby's heartbeat, revealed by an ultrasonic motion detector. Suppose the fetus's ventricular wall moves in simple harmonic motion with an amplitude of 1.80 $\mathrm{mm}$ and a frequency of 115 per minute. (a) Find the maximum linear speed of the heart wall. Suppose the motion detector in contact with the mother's abdomen produces sound at 2000000.0 $\mathrm{Hz}$ which travels through tissue at 1.50 $\mathrm{km} / \mathrm{s}$ . (b) Find the maximum frequency at which sound arrives at the wall of the baby's heart. (c) Find the maximum frequency at which reflected sound is received by the motion detector. By electronically "Istening" for echoes at a frequency different from the broadcast frequency, the motion detector can produce beeps of audible sound in synchronization with the fetal heartbeat.

Key Concepts

Doppler Effect

The Doppler effect is the change in frequency or wavelength of a wave as observed by an observer moving relative to the source of the wave. It applies to various types of waves, including sound and light, and is crucial for interpreting frequency shifts in systems involving motion, such as in medical ultrasound or radar.

Maximum Linear Speed in SHM

In simple harmonic motion, the maximum speed of the oscillating object occurs when it passes through its equilibrium position, and is given by the product of the amplitude and the angular frequency (v_max = A?). This concept is key to understanding the dynamic constraints of oscillatory systems.

Doppler Shift in Reflected Waves

When a wave reflects off a moving object, it undergoes two Doppler shifts—one when the wave approaches the moving object and a second when the reflected wave returns to the detector. This concept is important in applications like medical imaging and speed detection, where precise frequency changes reveal information about the motion of the object.

Simple Harmonic Motion (SHM)

This concept describes oscillatory motion where an object moves back and forth along a line about an equilibrium position, with its acceleration proportional and opposite to its displacement. In problems involving oscillating systems, such as vibrating walls or pendulums, SHM provides the framework to relate displacement, velocity, and acceleration via sinusoidal functions.

Angular Frequency

Angular frequency is a measure of how many radians an oscillating system sweeps per unit time and is calculated as ? = 2?f, where f is the ordinary frequency. It is essential for linking the temporal aspects of oscillation to spatial properties, such as amplitude, to determine quantities like maximum linear speed.

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