Doppler Effect Stationary Source/Observer on a Spring

[jamiewilliams]

Homework Statement

A microphone is attached to a spring that is suspended from the ceiling. Directly below on the floor is a stationary 540-Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of 2.20 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 1.83 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude (in m) of the simple harmonic motion.

Homework Equations

f(obs)= f(source) (1-(v(obs)/v))
ω=2\pi/T
v(max)=Aω

The Attempt at a Solution

When I inquired about help I was told that I need to combine the equations for the max and min frequencies to get the maximum velocity, but I can't figure out how to do that. I am also not sure what to do with the difference of the max and min frequencies that was given in the problem. I have figured the value of ω to be 2.86 rad/s.

I think I would use the equation v(max)=Aω once I had the max velocity to get the amplitude.

I feel like this problem should be easier but I just can't figure it out!

 

[Delphi51]

combine the equations for the max and min frequencies

Write your dopplar equation twice, once with positive Vo and again with negative Vo. One gives the maximum frequency observed, the other the minimum, so the difference between the two is your 1.83 Hz. That's your clue to subtract the two equations. I think you will be able to get Vo out of that.

Regarding the amplitude, I wonder if you have an equation something like
x = A*sin(ωt) for the position as a function of time. And can differentiate it with respect to time to get a similar equation for the velocity. The two of them would constitute a relationship between the amplitude and the maximum velocity.

 

[jamiewilliams]

Thank you so much! I just subtracted the min frequency from the max and set it equal to 1.83 Hz.

I did, however, use the v(max)=Aω to find the amplitude, but versus time the other equation would have worked better.

:)

 

[Delphi51]

Most welcome!
v(max)=Aω comes from differentiating x = A*sin(ωt).

 

[jamiewilliams]

Didn't even notice that, I haven't thought about differentiation (or calculus) in a few semesters :)