Question Part Points Submissions Used 1 2 3 0/2 0/2 0/2 2/2 1/2 1/2 Total 0/6 Ex

Question

Question Part Points Submissions Used 1 2 3 0/2 0/2 0/2 2/2 1/2 1/2 Total 0/6 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 1.52 mm and frequency 106 beats per minute. The motion detector in contact with the mother's abdomen produces sound at precisely 2 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 six 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 six significant figures.) ____Hz

Explanation / Answer

(a)
   from the theory we know that
    = 2 f
       = 2 (106 / min / 60.0 s/ min)  
       = 11.1 rad / s
   the maximum linear speed will be
   vmax = A
          =(11.1) (1.52 x 10-3 m)
           = 0.017 m / s   

(b)
   from the given we can see that the heart wall isa moving observer and the detector is a stationar
   source so we get
   f ' = f ((v + vmax) / v)
      = (2 x 106 Hz) [(1500 +0.017) / (1500)]
      = 200022.5 Hz = 2 x 106 Hz

(c)
   in this case the heart wall is a moving source and thedetector will be a stationary source
   f '' = f ' (v / (v- vmax))
       = 200022.67 Hz
(OR)

from the doppler effwct we can write
   f ' = (v + vo / v - vs) f
   where vo = vmax is the velocity of the observer and vs is the velocity of the source of sound
   if heart wall is assumed as the observer it will be moving and the detector (source of sound) is
   said to be stationary that means vs =0, then by applying the sign convention to the Doppler effect
   according to the given condition we get
   f ' = (v + vo / v - vs) f
   f ' = 2592942.1 Hz

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