𝑆1 and 𝑆2 are two identical sound sources of frequency 656 Hz. The source 𝑆1 is located at 𝑂 and 𝑆2 moves anti-clockwise with a uniform speed 4√2 m s −1 on a circular path around 𝑂, as shown in the figure. There are three points 𝑃, 𝑄 and 𝑅 on this path such that 𝑃 and 𝑅 are diametrically opposite while 𝑄 is equidistant from them. A sound detector is placed at point 𝑃. The source 𝑆1 can move along direction 𝑂𝑃. [Given: The speed of sound in air is 324 m s −1 ] Consider both sources emitting sound. When 𝑆2 is at 𝑅 and 𝑆1 approaches the detector with a speed 4 m s −1 , the beat frequency measured by the detector is _______Hz.
To solve this problem, we need to calculate the beat frequency measured by the detector at point
when both sources and are emitting sound. Specifically, is at point , and is moving towards the detector at with a speed of .
Step 1: Understand the Setup
Sources: and emit sound at a frequency of .
Positions:
is at and moves towards at .
is at , which is diametrically opposite to on the circular path.
Detector: Placed at .
Speed of Sound: .
Step 2: Calculate the Frequency Shift Due to
Since is moving towards the detector at , the frequency observed due to will be higher than the emitted frequency due to the Doppler effect.
The Doppler effect formula when the source is moving towards the observer is:
Where:
(speed of sound)
(speed of towards )
(emitted frequency)
Plugging in the values:
Step 3: Calculate the Frequency Shift Due to
Since is at , which is diametrically opposite to , the distance between and is the diameter of the circular path. However, is stationary relative to at this instant because it is moving along the circular path and its velocity is tangential to the circle. Therefore, there is no radial component of velocity towards or away from .
Thus, the frequency observed due to remains the same as the emitted frequency:
Step 4: Calculate the Beat Frequency
The beat frequency is the absolute difference between the frequencies observed from and :
Final Answer
The beat frequency measured by the detector is 8.1 Hz.
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