A whistle of frequency 490 Hz moves in a circle of radius 2.00 ft at an angular speed of 15.0 rad/s. What are the lowest and the highest frequencies heard by a listener a long distance away at rest with respect to the center of the circle?
To find the lowest and highest frequencies heard by a listener, we need to consider the effect of the Doppler effect. The Doppler effect is the change in frequency of a wave perceived by an observer when the source of the wave and the observer are in relative motion.
In this case, the source of the sound is the whistle and the observer is the listener. The whistle is moving in a circular path with a certain angular speed, and the listener is at rest.
The formula for the observed frequency of a moving source can be given as:
f' = f * (v + v_o) / (v + v_s)
where f' is the observed frequency, f is the source frequency, v is the speed of sound, v_o is the speed of the observer, and v_s is the speed of the source.
In this situation, the listener is at rest, so v_o = 0. The whistle is moving in a circle, so its speed can be calculated using the formula:
v_s = r * ω
where r is the radius of the circular path and ω is the angular speed.
In this case, r = 2.00 ft and ω = 15.0 rad/s. We convert the radius to meters (1 ft = 0.3048 m) to match the units of the speed of sound, which is approximately 343 m/s.
v_s = 2.00 ft * 0.3048 m/ft * 15.0 rad/s
Now, let's calculate v_s:
v_s = 9.144 m/s
Substituting the values into the Doppler effect formula, we get:
f' = f * (v + 0) / (v + 9.144)
The lowest frequency occurs when the whistle is moving away from the listener, which corresponds to the maximum possible value for f. The highest frequency occurs when the whistle is moving towards the listener, which corresponds to the minimum possible value for f.
To find the maximum frequency, we use the maximum value for f. The highest frequency can be calculated by substituting the values into the formula for f':
f_max = f * (v + 0) / (v - v_s)
To find the minimum frequency, we use the minimum value for f. The lowest frequency can be calculated by substituting the values into the formula for f':
f_min = f * (v + 0) / (v + v_s)
Now we have all the necessary information to compute the lowest and highest frequencies.
plugging in the values for v and v_s:
f_max = 490 Hz * (343 m/s + 0) / (343 m/s - 9.144 m/s)
f_min = 490 Hz * (343 m/s + 0) / (343 m/s + 9.144 m/s)
Calculating the answers:
f_max = 506.7 Hz
f_min = 472.8 Hz
Therefore, the lowest frequency heard by the listener is approximately 472.8 Hz, and the highest frequency heard is approximately 506.7 Hz.