Spring and Microphone (Doppler Effect...?)

[John123]

Homework Statement

A microphone is suspended from a spring attached to the ceiling. Directly below, on the floor, is a source of sound that is emitting at f=440 Hz. The microphone vibrates vertically in simple harmonic motion (SHM) with a period of 2.0 seconds. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz. a

Determine the amplitude of the SHM of the microphone.

b. Determine the maximum speed of the microphone.

c. If the microphone has a mass of 0.120 kg, determine the spring constant k. (Take the speed of sound to be 343 m/s, and ignore all reflections of sound in the room)

Homework Equations

Doppler Effect, various frequency equations

The Attempt at a Solution

[/B]
I am having trouble figuring out what this question is telling me. What does the final sentence about max and min sound frequencies mean?
I understand since the observer is moving towards and away from the source, giving a doppler effect, however, can I use this to find Amplitude.

How do I go about solving this?

Thanks.

 

[haruspex]

Homework Statement

A microphone is suspended from a spring attached to the ceiling. Directly below, on the floor, is a source of sound that is emitting at f=440 Hz. The microphone vibrates vertically in simple harmonic motion (SHM) with a period of 2.0 seconds. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz. a

Determine the amplitude of the SHM of the microphone.

b. Determine the maximum speed of the microphone.

c. If the microphone has a mass of 0.120 kg, determine the spring constant k. (Take the speed of sound to be 343 m/s, and ignore all reflections of sound in the room)

Homework Equations

Doppler Effect, various frequency equations

The Attempt at a Solution

[/B]
I am having trouble figuring out what this question is telling me. What does the final sentence about max and min sound frequencies mean?
I understand since the observer is moving towards and away from the source, giving a doppler effect, however, can I use this to find Amplitude.

How do I go about solving this?

Thanks.

I would start with part b.
If at some instant the microphone is moving down at speed v, what frequency will it register?

 

[John123]

I would start with part b.
If at some instant the microphone is moving down at speed v, what frequency will it register?

Itll register a frequency that is related to the doppler shift, however we don't know the doppler frequency nor do we know the velocity.

How can we solve for one of them?

Edit: wait, I am slightly guessing here, but is the frequency due to the doppler shift 440 +/- 2.1?

442.1 when the microphone is oscillating down

437.9 when it is oscillating up?

 

[haruspex]

Itll register a frequency that is related to the doppler shift

Right, and what equation relates that to the source frequency, the receiver speed relative to the medium and the speed of sound?

442.1 when the microphone is oscillating down. 437.9 when it is oscillating up?

Would that match this information:

The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz.

?

 

[John123]

Right, and what equation relates that to the source frequency, the receiver speed relative to the medium and the speed of sound?

Would that match this information:

?

Ok, I've got the correct answer, could you tell me if its the easiest/best way to do this question.

I set up two equations,

f_1 = 440(1 + Vmicro/343)
f_1 - 2.1 = 440(1 + Vmicro/343)

Made them equal to each other and solved for V, which gave the correct of answer of .82m/s.

Now, for part a). Vmax = 2piA/T

.82 = 2piA/2

A = .26 (also correct!)

c)

T = 2 pi sqr(m/k)
1/pi = sqr .120/k
k=1.2N/m

Thank you :)

 

[haruspex]

f_1 = 440(1 + Vmicro/343)
f_1 - 2.1 = 440(1 + Vmicro/343)

Those two equations are in direct conflict. What did you mean to write? And please define your variables,,, what, exactly, is f₁?

 

[John123]

Those two equations are in direct conflict. What did you mean to write? And please define your variables,,, what, exactly, is f₁?

Oh oops, so I meant to write
f_1 = 440(1 + Vmicro/343) -----> (EQN: Observer moving toward source)
f_1 - 2.1 = 440(1 - Vmicro/343) -----> this equation is for the frequency as the microphone oscillates up (EQN: Observer moving away from source)

f_1 is the frequency that the microphone experiences on the way down (so the larger one)

By making the two equations equal:

440(1+Vmicro/343) = 440(1-Vmicro/343)+2.1
Divide both sides by 440
1 + Vmicro/343 = 1 - Vmicro/343 + 2.1/440
1's cancel
Move a Vmicro/343 to the other side giving 2Vmicro/343
2Vmicro/343 = 2.1/440
2.1/440 * 343 / 2 = Vmicro = .8185m/s ~ .82m/s

Answer key states:
a) 0.26 m
b) 0.82 m/s
c) 1.2 N/m

 

[haruspex]

Yes, that looks good.