Expectant parents are thrilled to hear their unborn baby\'s heartbeat, revealed

Question

Expectant parents are thrilled to hear their unborn baby's heartbeat, revealed by an ultrasonic detector that produces beeps of audible sound in synchronization with the fetal heartbeat. Suppose the fetus's heart wall moves in simple harmonic motion with amplitude 2.00 mm and a frequency of 115 beats per minute. Calculate the maximum speed of the heart wall. A source is mounted on the detector in contact with the mother's abdomen, and produces sound at 2,000,000.0 Hz. The speed of sound in body tissue is 1500 m/s. If the heart wall is moving at its maximum speed toward the sound waves from the source, calculate the difference in frequency between the reflected sound received back at the detector, and the original signal. Show your work, and carefully explain your method of solution and your choice of signs. (Keep lots of significant figures in your intermediate calculations.) Although this difference in frequency is very small compared to that of the ultrasound, it can be easily detected by looking for the beats between the two signals!

Explanation / Answer

equation for SHM:

x=A*sin(w*t)

where A is amplitude, w is angular frequency

then velocity=dx/dt=A*w*cos(w*t)

maximum velocity=A*w

=A*2*pi*f

=0.002 m *2*pi*(115 /60) Hz=0.0241 m/s

part b:

using doppler effect:

modified frequency = (c+vr)*f0/(c+vs)

where c=velocity of wave in the medium=1500 m/s

vr=speed of the receiver, positive when receiver is moving towards the source

vs=speed of the source, positive if the source is moving away from the receiver

f0=original frequency=2*10^6 Hz

when the wave reaches the heart wall, frequency received can be cqalculated as below:

c=1500 m/s

vr=0.0241 m/s

vs=0

f0=2*10^6 hz

so f=2000032.133 Hz

when this frequency is received by the detector:

c=1500 m/s

vr=0 m/s

vs=-0.0241 m/s

f0=2000032.133 Hz

then frequency received=2000064.268 Hz

then beat frequency=64.268 Hz

Related Questions

Navigate

• Browse (All)
• Browse E
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.