The Humming Helicopter Blade · AI Generator #GEN-3849B132
Physics
Problem
A helicopter blade of length rotates at angular frequency in still air. Due to the Doppler effect, an observer on the ground hears a distinctive humming sound as the blade passes overhead. The blade can be modeled as a line source of sound waves, with each point along the blade moving in a circle. Consider the moment when the blade is horizontal and perpendicular to the line connecting the helicopter to the observer on the ground directly below. At this instant, different points along the blade have different radial velocities relative to the observer. (a) Find the range of frequencies heard by the observer at this instant, given that the blade emits sound at frequency and the speed of sound is . (b) If the observer has perfect pitch and can distinguish frequency differences as small as , how many distinct frequency components can they potentially identify in the sound?Intuitive explanation
The key insight is that different points along a rotating blade have different velocities toward and away from the observer, so they each produce a slightly different Doppler shift. The blade's tip moves fastest, creating the extreme frequency shifts, while the hub moves slowest (or not at all). You'll need to find the radial velocity of points at different distances from the rotation axis, apply the Doppler formula, and determine the spread of frequencies heard simultaneously.
Intuitive explanation
The key insight is that different points along a rotating blade have different velocities toward and away from the observer, so they each produce a slightly different Doppler shift. The blade's tip moves fastest, creating the extreme frequency shifts, while the hub moves slowest (or not at all). You'll need to find the radial velocity of points at different distances from the rotation axis, apply the Doppler formula, and determine the spread of frequencies heard simultaneously.