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 determine the lowest and highest frequencies heard by a listener, we need to consider two phenomena: the Doppler effect and the circular motion of the whistle.
The Doppler effect describes how the perceived frequency of a sound changes when the source and the observer are in relative motion. When the source moves towards the observer, the frequency appears higher, and when it moves away, the frequency appears lower.
In this case, the whistle is moving in a circular path, and we need to determine the frequencies heard when the whistle is closest and farthest from the listener.
Let's start by calculating the velocity of the whistle at the closest and farthest points.
The linear velocity of an object moving in a circle can be calculated using the formula:
v = ω * r
Where:
v is the linear velocity
ω (omega) is the angular velocity
r is the radius of the circle
Substituting the given values:
v = 15.0 rad/s * 2.00 ft = 30.0 ft/s
Now, we can use the Doppler effect equation to calculate the observed frequencies at the closest and farthest points:
f' = f * (v + v₀) / (v - vₛ)
Where:
f' is the observed frequency
f is the emitted frequency (490 Hz in this case)
v is the velocity of sound in air (approximately 343 m/s or 1125 ft/s)
v₀ is the velocity of the observer (assumed to be zero for a stationary listener)
vₛ is the velocity of the source (in this case, the linear velocity of the whistle)
At the closest point (when the velocity of the whistle is directed towards the listener):
f'₁ = f * (v + v₀) / (v - vₛ)
= 490 Hz * (1125 ft/s + 0 ft/s) / (1125 ft/s - 30.0 ft/s)
= 490 Hz * (1125 / 1095)
= 504.56 Hz
At the farthest point (when the velocity of the whistle is directed away from the listener):
f'₂ = f * (v + v₀) / (v - vₛ)
= 490 Hz * (1125 ft/s + 0 ft/s) / (1125 ft/s + 30.0 ft/s)
= 490 Hz * (1125 / 1155)
= 478.41 Hz
Therefore, the lowest frequency heard by the listener is approximately 478.41 Hz, and the highest frequency heard is approximately 504.56 Hz.