You attach a speaker to an air track cart and then attach the cart to one end of

Question

You attach a speaker to an air track cart and then attach the cart to one end of the air track by means of a spring and start the system oscillating by stretching the spring 0.677 m from its equilibrium position and then releasing it. The force constant for the spring is k = 48.8 N/m, the total mass of the speaker and air track cart are 4.00 kg, and the speaker emits sound with a frequency of 475 Hz. If your lab partner Pat stands at the end of the air track, determine the highest and lowest frequencies she hears. The speed of sound in air is 343 m/s. (Enter your answers to at least the nearest Hz.)

Explanation / Answer

frequency head by stationary observer
f' = fo ( V / V +/- Vs) -------1
fo is the natural frequency of source
V is the speed of sound
Vs is speed of source
We use +ve sign when souce is moving away from abserver and -ve when it is moving towards observer.

Hence maximum frequency, heard by observer is corresponding to maximum speed of source toward observer and mimimum frequency is corresponding to maximum speed of source away from observer.
Maximum speed of oscillating mass is, when it's at mean position or it's potential energy is zero. Equating initial potential energy of spring to kinetic energy of mass at mean positon we get
1/2 K Xo2 = 1/2 MVm2 (Vm is maxximum speed of source, Xo is initial extension of spring)
Vm2 = Xo (K/M)1/2 = 2.36 m/s  
Using this in equation 1, we get maximum frequency = 478 Hz
and minimum frequency = 472 Hz

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