Question

Two identical 375-g speakers are mounted on parallel springs, each having a spring constant of 50.0 N/cm. Both speakers face in the same direction and produce a steady tone of 1.00 kHz. Both sounds have an amplitude of 35.0 cm, but they oscillate 180° out of phase with each other. What is the highest frequency of the beat that a person will hear if she stands in front of the speakers?

Answer 1

given
m = 375 g = 0.375 kg
k = 50 N/cm = 5000 N/m
A = 35 cm = 0.35 m

maximum speed of speakers at equilibrium point,

v_max = A*w

= A*sqrt(k/m)

= 0.35*sqrt(5000/0.375)

= 40.4 m/s

f = 1 kHz

= 1000 Hz

v_speaker = v_max

= 40.4 m/s

we know, v_sound = 343 m/s

the two speakers are oscillating 180 degrees out of phase each other.

so, if one speaker moves towards you the other moves away from you.

the apparent frequency heard by you from speaker moving towards you,

f' = f*v_sound/(v_sound - v_speaker)

= 1000*343/(343 - 40.4)

= 1133.5 Hz

the apparent frequency heard by you from speaker moving away from you,

f'' = f*v_sound/(v_sound + v_speaker)

= 1000*343/(343 + 40.4)

= 894.6 Hz

maximum beat frequency heard, f_beat = f' - f''

= 1133.5 - 894.6

= 239 Hz <<<<<<<<<<----------------------Answer