Two detectors of sound (D1 and D2) are located on y-axis with co-ordinates (0,3.4√3 km) and (0,5.1√3 km) respectively. A source of sound(S) of natural frequency 100Hz is moving along x− axis with a constant speed of 170m/s. The frequencies observed by D1 and D2 when source is nearest to them is f1 Hz and f2 Hz respectively. Find f2–f1. Take speed of sound = 340 m/s.
Question
Two detectors of sound (D1 and D2) are located on y-axis with co-ordinates (0,3.4√3 km) and (0,5.1√3 km) respectively. A source of sound(S) of natural frequency 100Hz is moving along x− axis with a constant speed of 170m/s. The frequencies observed by D1 and D2 when source is nearest to them is f1 Hz and f2 Hz respectively. Find f2–f1. Take speed of sound = 340 m/s.
Solution
The problem involves the Doppler effect, which describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave.
The formula for the Doppler effect is:
f' = f * (v + vo) / (v + vs)
where:
- f' is the observed frequency,
- f is the source frequency,
- v is the speed of sound,
- vo is the speed of the observer, and
- vs is the speed of the source.
In this case, the source of sound is moving, but the detectors are stationary. Therefore, vo = 0. The speed of the source (vs) is given as 170 m/s, but since the source is moving in the opposite direction to the detectors, we take vs as -170 m/s. The speed of sound (v) is given as 340 m/s.
When the source is nearest to D1, the observed frequency f1 is:
f1 = f * (v + vo) / (v + vs) = 100 Hz * (340 m/s + 0) / (340 m/s - 170 m/s) = 100 Hz * 2 = 200 Hz
When the source is nearest to D2, the observed frequency f2 is:
f2 = f * (v + vo) / (v + vs) = 100 Hz * (340 m/s + 0) / (340 m/s - 170 m/s) = 100 Hz * 2 = 200 Hz
Therefore, f2 - f1 = 200 Hz - 200 Hz = 0 Hz.
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