Question

A boy sitting on a swing which is moving to an angle of ${30}^0$ from the vertical is blowing a whistle which is of frequency 1000 Hz. The whistle is 2m from the point of support of the swing. If a girl stands infront of the swing, the maximum and minimum frequencies she will hear are

(velocity of sound = 330 m/s, g = 9.8 m/s${}^2$)

• A. $1000,990Hz$
• B. $1007,1000Hz$
• C. $1007,993Hz$
• D. $1100,900Hz$

Answer 1

The correct option is C $1007,993Hz$

Total potential energy will convert into kinetic energy, at the bottom most point, and thus swing attain its maximum speed at the lowest point. At this point, we took velocity of source $v_s$ for exremum, for maximum frequency towards the girl, and minimum freq away from the girl, See figure.
given datas are, $l=2m$, angle $\theta ={30}^o$, $\nu =1000Hz$, and velocity of sound in air $V=330m{s}^{-1}$,
$v_s=\sqrt{2gl(1-\cos \theta )}=\sqrt{2\times 9.8\times 2\times (1-\cos 30)}=\sqrt{19.6(2-\sqrt{3})}=\sqrt{19.6\times 0.268}=2.30m{s}^{-1}$
for maximum freq ${\nu }_{max}$ that appears to girl, apply Doppler's effect,
${\nu }_{max}=\frac{V}{V-v_s}\nu =\frac{330}{330-2.30}\times 1000Hz=1007Hz$
for minimum freq ${\nu }_{min}$ that appears to girl, apply Dopler's effect
${\nu }_{min}=\frac{V}{V+v_s}\nu =\frac{330}{330+2.30}\times 1000Hz=993Hz$
Option C is nearly correct.