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 motion detector. Suppose the fetus's ventricular wall moves in simple harmonic motion with amplitude 1.64 mm and frequency 109 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 = amplitude = 1.64mm

f = observed frequency = 109 beats per min

f0 = emitted frequency = 2MHZ

a) w = angular frequency = 2*pie *f

= 2*3.14*109/60

= 11.408 rad/s

V_max ( maximum linear speed) = Aw

= 1.64*10-3*11.408

= 0.0187091 m/s

b) from doppler's effect

f = f0* ( c+Vr / c+Vs)

c = 3*108m/s = velocity of waves in medium

Vr = velocity of receiver relative to medium

Vs = velocity of source relative to medium

since Vs = 0m/s

f = f0 ( c+Vr / c)

= 2*106 ( 3*108 + 1.50*103 / 3*108)

= 2.00001*106 hz

c) f = f0 ( c+Vr / c) * ( c/ c-Vr)

= f0 * ( c + Vr / c- Vr)

= 2*106 * ( 3*108 + 1.50*103 / 3*108 - 1.50*103)

= 2.00002*106 Hz

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