Question

EXAMPLE 14.11 Sour Notes Goal Apply the beat frequency concept. Problem A certain piano string is supposed to vibrate ning, what beat frequency is observed by the assistant? at a frequency of 4.40 X 102 Hz. To check its frequency, Note: Assume all numbers are accurate to two decimal a tuning fork known to vibrate at a frequency of places, necessary for this last calculation. 4.40 × 102 Hz is sounded at the same time the piano key is struck, and a beat frequency of 4 beats per second is heard. (a) Find the two possible frequencies at which the string could be vibrating. (b) Suppose the piano tuner oce ot sound and occurs it the pno s runs toward the piano, holding the vibrating tuning fork while his assistant plays the note, which is at 436 Hz. At his maximum speed, the piano tuner notices the beat fre- quency drops from 4 Hz to 2 Hz (without going through a beat frequency of zero). How fast is he moving? Use a sound speed of 343 m/s. (c) While the piano tuner is run- Strategy (a) The beat frequency is equal to the abso- lute value of the difference in frequency between the two d and occurs if the piano string is tuned either too high or too low. Solve Equation 14.20 for these two possible frequencies. (b) Moving toward the piano raises the observed piano string frequency. Solve the Doppler shift formula, Equation 14.12, for the speed of the observer. (c) The assistant observes a Doppler shift sources of souno fieuc hich is at 4Alw> the piano tuner is run for the tuning fork. Apply Equation 14.12. Solution (a) Find the two possible frequencies Case l: -i is already positive, so just drop the absolute-value signs: 五-,-五一, 4Hz = f2-4.40 × 102 Hz 444 Hz

Please explain and show how to do the exercise (at the bottom), parts b and c. Explain what you are substituting in the Doppler equations and why. Also please explain what signs you have to use.

Answer 1

a)

Beat frequency is given by

fb=f2-f1

if f2-f1 is positive then

2=f2-440

f2=442 Hz

if f2-f1 is negative then

2=-(f2-f1)=-f2+440

f2=438 Hz

so flower=438 Hz

fhigher=442 Hz

b)

from doppler shift

fo=fs(V+Vo)/(V-Vs)

fo=440*(343+4)/343=445 Hz

beat frequency

fb=445-442= 3 Hz

c)

beat frequency

fb=445-438= 7 Hz