A physics professor demonstrates the Doppler effect by tying a 600 Hz sound generator to a 1.0-m-long rope and whirling it around her head in a horizontal circle at 100 rpm. What are the highest and lowest frequencies heard by a student in the classroom?

Verified step by step guidance

1. Calculate the angular velocity (\(\omega\)) of the sound generator. Angular velocity is given by \(\omega = \frac{2\pi \times \text{RPM}}{60}\), where RPM is the revolutions per minute.
2. Determine the linear velocity (\(v\)) of the sound generator. The linear velocity can be calculated using the formula \(v = r \times \omega\), where \(r\) is the radius of the circle (half the length of the rope).
3. Use the Doppler effect formula to find the highest and lowest frequencies heard. The formula for the frequency heard by an observer when the source is moving towards the observer is \(f' = f \times \frac{v + v_0}{v - v_s}\) and when moving away is \(f' = f \times \frac{v - v_0}{v + v_s}\), where \(f\) is the original frequency, \(v\) is the speed of sound in air, \(v_0\) is the velocity of the observer (which is 0 in this case as the observer is stationary), and \(v_s\) is the velocity of the source.
4. Substitute the calculated linear velocity (\(v_s\)) into the Doppler effect formulas to find the highest frequency (when the source is moving towards the observer) and the lowest frequency (when the source is moving away from the observer).
5. Consider the effect of the circular motion, where the source velocity towards or away from the observer continuously changes, which means the frequency will vary between the calculated highest and lowest frequencies.

Key Concepts

Doppler Effect

The Doppler effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. When the source approaches the observer, the frequency increases, resulting in a higher pitch, while it decreases as the source moves away, producing a lower pitch. This phenomenon is commonly experienced with sound waves, such as when a passing siren changes pitch.

Centripetal Motion

Centripetal motion describes the motion of an object moving in a circular path, where a net force acts towards the center of the circle. In this scenario, the sound generator is whirled in a circle, creating a situation where the frequency of the sound waves emitted changes based on the motion of the source relative to the observer. The speed of rotation and the radius of the circle are key factors in determining the effective frequency changes.

Frequency Shift Calculation

To determine the highest and lowest frequencies heard by the observer, one must calculate the frequency shift due to the Doppler effect. This involves using the formula for the observed frequency, which accounts for the speed of sound, the speed of the source, and the direction of motion. The maximum frequency occurs when the source approaches the observer, while the minimum frequency occurs when it moves away.