π1 and π2 are two identical sound sources of frequency 656 Hz. The source π1 is located at π and π2 moves anti-clockwise with a uniform speed of 4\(\sqrt{2}\) ms-1 on a circular path around π, as shown in the figure. There are three points π, π and π
on this path such that π and π
are diametrically opposite while π is equidistant from them. A sound detector is placed at point π. The source π1 can move along the direction ππ. [Given: The speed of sound in air is 324 ms-1].
Consider both sources emitting sound. When π2 is at π
and π1 approaches the detector with a speed 4 m sβ1 , the beat frequency measured by the detector is _______Hz
Consider both sources emitting sound. When π2 is at π and π1 approaches the detector with a speed 4 m sβ1 , the beat frequency measured by the detector is _______Hz
Correct Answer: 8 - 8.4
Approach Solution - 1
f0 = 656 Hz
v = 324 m/s Frequency heard due to movement of (S1)
\(f_1=(\frac{v}{v-u_s})f_0\)
\(f_1=\frac{324}{3520}\times656\)
And frequency heard due to movement of (S2)
f2 = 656 Hz
β΄ Beat frequency \(Ξ f = f_1 β f_2 \)
\(=656(\frac{324}{320}-1)\)
\(Ξ f = 8.2\)
Approach Solution -2
The given values are:
- f0 = 656 Hz (frequency of the source)
- v = 324 m/s (velocity of sound in air)
- us = 4 m/s (velocity of the source S1)
The formula for the frequency heard due to the movement of S1 is:
\(f_1 = \left( \frac{v}{v - u_s} \right) f_0\)
Substituting the given values:
\(f_1 = \left( \frac{324}{320} \right) \times 656 = 664.8 \, \text{Hz}\)
The frequency heard due to the movement of S2 is:
\(f_2 = 656 \, \text{Hz}\)
The beat frequency \( \Delta f \) is the difference between the two frequencies:
\(\Delta f = f_1 - f_2\)
Substituting the values:
\(\Delta f = 656 \left( \frac{324}{320} - 1 \right) = 8.2 \, \text{Hz}\)
Thus, the beat frequency is 8.2 Hz.
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Concepts Used:
Oscillations
Oscillation is a process of repeating variations of any quantity or measure from its equilibrium value in time . Another definition of oscillation is a periodic variation of a matter between two values or about its central value.
The term vibration is used to describe the mechanical oscillations of an object. However, oscillations also occur in dynamic systems or more accurately in every field of science. Even our heartbeats also creates oscillationsβ. Meanwhile, objects that move to and fro from its equilibrium position are known as oscillators.
Read More: Simple Harmonic Motion
Oscillation- Examples
The tides in the sea and the movement of a simple pendulum of the clock are some of the most common examples of oscillations. Some of examples of oscillations are vibrations caused by the guitar strings or the other instruments having strings are also and etc. The movements caused by oscillations are known as oscillating movements. For example, oscillating movements in a sine wave or a spring when it moves up and down.
The maximum distance covered while taking oscillations is known as the amplitude. The time taken to complete one cycle is known as the time period of the oscillation. The number of oscillating cycles completed in one second is referred to as the frequency which is the reciprocal of the time period.