Question

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 amplitude 1.57 mm and frequency 103 beats per minute. The motion detector in contact with the mother's abdomen produces sound at precisely 2.000000 MHz, which travels through tissue at 1.50 km/s.

(a) Find the maximum linear speed of the heart wall.
m/s

(b) Find the maximum frequency at which sound arrives at the wall of the baby's heart. (Enter your answer to at least seven significant figures.)
Hz

(c) Find the maximum frequency at which reflected sound is received by the motion detector. (By electronically "listening" for echoes at a frequency different from the broadcast frequency, the motion detector can produce beeps of audible sound in synchrony with the fetal heartbeat. Enter your answer to at least seven significant figures.)
Hz

Answer 1

Given

amplitude is A = 1.57 mm ,

Beats per minute is 103 , vonverting it into frequency that is  

no of beats per sec is frequency f = 103/60 = 1.716 cycles /s

nwo the maximum linear velocity is V_max = A*W

wehre W is the angular freqeuncy W = 2pi*f = 2pi*1.716 rad/s

V_max = 1.57*10^-3*2pi*1.716 m/s = 0.016927655 m/s

b)

frequency of the sound produced by detector is f = 2 Hz*10^6

speed is v = 1.5 km/s = 1500 m/s

from doppler effect  

f' = f(v+vo/v-vs)

f is frequency of the emitter f = 2000000 Hz

f' = observed frequency  

V speed 1500 m/s

v0 speed of the observer = 0.016927655 m/s

vs is speed of source here vs=0 m/s

f' = f(V+vo/v-vs)

f' = 2000000((1500+0.016927655)/1500) Hz

f' = 2000022.570 Hz

c )

for the reflected sound is  

here f'= 2000022.570 is the frequency and the observer is stationary and

vs = 0.016927655 m/s

f' = f((v+vo) /(v-vs))

f' =2000022.570((1500)/(1500-0.016927655)) Hz

f' = 2000045.1407160843 Hz