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(D) The Doppler effect formula for a moving observer and a stationary source is given by $n' = n \left( \frac{v \pm v_o}{v} \right)$,where $v$ is the

Vedclass · Accepted Answer

$573$

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(D) The Doppler effect formula for a moving observer and a stationary source is given by $n' = n \left( \frac{v \pm v_o}{v} ight)$,where $v$ is the speed of sound and $v_o$ is the component of the observer's velocity along the line joining the source and the observer.The actual frequency $n$ is heard when the apparent frequency $n' = n$,which occurs when the component of the observer's velocity along the line joining the source and the observer is zero. This happens when the velocity vector of the motorcyclist is perpendicular to the line joining the siren at $Y$ and the motorcyclist.In one complete circular revolution,there are $2$ such points where the velocity vector is perpendicular to the radial line connecting the center of the circle to the source (or more accurately,the line of sight from the source to the observer). Thus,the motorcyclist hears the actual frequency $2$ times per revolution.The time period of one revolution $T$ is given by $T = \frac{2 \pi R}{v_o} = \frac{2 \pi 	imes 50}{25} = 4 \pi \ s$.The number of revolutions in one hour $(3600 \ s)$ is $N = \frac{3600}{T} = \frac{3600}{4 \pi} = \frac{900}{\pi} \approx \frac{900}{3.14159} \approx 286.48$.Since the motorcyclist hears the actual frequency $2$ times per revolution,the total number of times in an hour is $2 	imes 286.48 \approx 572.96$,which is approximately $573$.

$A$ motorcyclist is moving around a circular track of radius $50 \ m$ with a constant speed of $25 \ m/s$. $A$ static siren at point $Y$ emits sound of frequency $n$. How many times (approximately) in an hour will the motorcyclist hear the sound of the actual frequency $n$?

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