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A small source of sound vibrating at frequency $$ 500\,Hz $$ is rotated in a circle of radius $$ 100/ \pi \, cm $$ at a constant angular speed of $$ 5.0 $$ revolutions per second. The speed of sound in air is $$ 330\, m/s $$.
If the observer moves towards the source with a constant speed of $$ 20\,m/s $$ , along the radial line to the centre , the fractional change in the apparent frequency over the frequency that the source will have if considered at rest at the centre will be 


A
6 %
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B
3 %
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C
2 %
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D
9 %
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Solution

The correct option is A $$ 6 $$ %
The frequency of the source considered stationary at the centre is $$ n_0 = 500\,Hz $$  when the observer moves towards the centre with constant speed  $$ u $$ , the apparent frequency is 
              $$ n_a = \left ( \dfrac{V + u}{V} \right ) 500 = \dfrac{330 + 20}{330} \times 500 = \dfrac{35}{33} \times 500 $$ 
 Change in frequency is given by 
           $$ n_a - n_0 = \left ( \dfrac{35}{33} \times 500 - 500  \right ) = \dfrac{2 \times 500}{33} \,hz $$ 
 Fractional change of frequency is 
         $$\left (\dfrac{n_a - n_0}{n_0}  \right ) = \left (\dfrac{2 \times 500}{33}  \right ) \times  \dfrac{1}{500} $$ 
                                    $$ = \dfrac{2}{33} = 0.06 $$ 
Hence , percentage change is $$ 6 $$ % 

Physics

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