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A source of sound S emitting waves of frequency $100Hz$and an observer O are located at some distance from each other. The source is moving with a speed of $19.4m/s$at an angle of ${60^ \circ }$with the source observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer is (velocity of sound in air is $330m/s$)
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Last updated date: 13th Jun 2024
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Hint:Its asking apparent frequency and source is moving with some velocity you need to apply doppler effect of sound along the line joining observer and source, to do that you need to take the component of velocity of source along the line joining source and observer and then apply doppler effect.

Complete step by step answer:
Here source is moving with certain velocity at some angle from the line joining source and observer. Component of velocity of source along the line joining source and observer will be, $ {v_s}\cos {60^ \circ } = 19.4 \times \dfrac{1}{2} = 9.7m/s$
Now we will apply doppler effect .We know that when source is moving and observer is at rest then,
Apparent frequency $f' = {f_o}(\dfrac{{{v_o}}}{{{v_o} - {v_s}}})$ where, ${f_o}$is original frequency , ${v_o}$is velocity of sound in air and ${v_s}$is velocity of source along line joining source and observer.

Therefore putting values we have,
$\ f' = {f_o}(\dfrac{{{v_o}}}{{{v_o} - {v_s}}}) \\
\Rightarrow f' = 100(\dfrac{{330}}{{330 - 9.7}})\\
\Rightarrow f' = 100 \times \dfrac{{330}}{{320.3}} \\
\therefore f' = 103.03Hz$

Hence apparent frequency in this case will be $103.03Hz$.

Additional information:
Doppler effect is an important phenomenon that is useful in a variety of different scientific disciplines, including planetary science. The Doppler effect or the Doppler shift describes the changes in the frequency of any kind of sound or light wave produced by a moving source with respect to an observer.Doppler effect in physics is defined as the increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move towards (or away from) each other.

Note: In this type of questions taking component of velocity along the line joining the source and observer is crucial because only this component of velocity affects the original frequency or contributes to the apparent frequency and the perpendicular component of velocity will not have any impact because relative separation doesn’t change when one particle moves perpendicular to other