A vibrating 1000-hertz tuning fork produces sound waves that travel at 340 meters per second in air. Points A and B are some distance from the tuning fork. Point P is 20. meters from the tuning fork.
1. If the waves are in phase at point A and B, what is the minimum distance separating points A and B in terms of wavelength?
2. If the vibrating tuning fork is accelerated toward point P, what happens to the frequency of the sound observed at P?
To answer the first question, we need to consider the concept of phase difference in waves. The phase of a wave refers to the position it occupies in its cycle, which can be measured in terms of wavelength. When two points are in phase, it means they are at the same position in their respective waves.
1. To find the minimum distance separating points A and B in terms of wavelength, we need to calculate the phase difference between them. The phase difference is given by the formula:
Δφ = 2πΔx / λ
Where Δφ is the phase difference, Δx is the distance between the points, and λ (lambda) is the wavelength.
Since the points are in phase, the phase difference Δφ will be a multiple of 2π. We want to find the minimum distance Δx that satisfies this condition.
Δφ = 2πn (where n is an integer)
Substituting this into the formula, we get:
2πn = 2πΔx / λ
Simplifying, we find:
n = Δx / λ
Therefore, the minimum distance separating points A and B in terms of wavelength is:
Δx = nλ
Now let's move on to the second question.
2. When a vibrating tuning fork is accelerated towards point P, the frequency of the sound observed at P will increase. This change in frequency is known as the Doppler effect.
The formula for the Doppler effect can be expressed as:
f' = f(v + vp) / (v + vs)
Where f' is the observed frequency, f is the actual frequency, v is the velocity of sound in air, vp is the velocity of the source (tuning fork), and vs is the velocity of the observer (point P).
In this case, the tuning fork is accelerated towards point P, causing the velocity of the source (vp) to increase. As a result, the observed frequency (f') at point P will also increase.
It's worth noting that if the tuning fork was moving away from point P, the observed frequency would decrease.
So, to summarize, when the vibrating tuning fork is accelerated towards point P, the frequency of the sound observed at P will increase due to the Doppler effect.