Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at along the perpendicular bisec

Question

Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at $60 km/h$ along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let $ν (t)$ represent the beat frequency measured by a person sitting in the car at time $t$ . Let $ν_{P}$ , $ν_{Q}$ and $ν_{R}$ be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is $330 {m s}^{- 1}$ . Which of the following statement(s) is(are) true regarding the sound heard by the person? [2016]

Smart Achievers · Accepted Answer

(1, 2, 3)

$(1) ν_{P} = (ν_{N} - ν_{M}) [\frac{v + v_{c} \cos θ}{v}] = 121 - 118 [\frac{v + v_{c} \cos θ}{v}]$

$ν_{Q} = (ν_{N} - ν_{M}) = 121 - 118 = 3$

$ν_{R} = (ν_{N} - ν_{M}) [\frac{v - v_{c} \cos θ}{v}] = (121 - 118) [\frac{v - v_{c} \cos θ}{v}]$

$∴$ $ν_{P} + ν_{R} = 2 ν_{Q}$

In general, when the car is passing through A,

$ν = 3 [\frac{v + v_{c} \cos α}{v}]$ ...(i)

$∴$ $\frac{d ν}{d α} = - 3 [\frac{v_{c} \sin α}{v}]$ $| \frac{d ν}{d α} |$ $is maximum when \sin α = 1$

i.e., $α = 90 °$ $(at Q)$

From Eq. (i), $\frac{d ν}{d t} = \frac{3 v_{c}}{v} (- \sin α) \frac{d α}{d t}$ ....(ii)

Also, $\tan α = \frac{10}{x}$ $∴$ $\sec^{2} α \frac{d α}{d t} = - \frac{10}{x^{2}} \frac{d x}{d t}$

$∴$ $\frac{d α}{d t} = \frac{- 10 v}{x^{2} \sec^{2} α}$ ...(iii)

From Eqs. (ii) and (iii),

$\frac{d ν}{d t} = - \frac{3 v_{c}}{v} \sin α (\frac{- 10 v}{x^{2} \sec^{2} α}) = \frac{30 v_{c} \sin α}{x^{2} \sec^{2} α}$

$∴$ $\frac{d ν}{d t} = \frac{30 v_{c} \sin α}{{(10 \cot α)}^{2} \sec^{2} α} = 0.3 v_{c} \sin^{3} α$

At $α = 90 °$

$\frac{d ν}{d t} = 0.3 v_{c}$

$∴$ ${(\frac{d ν}{d t})}_{\max} = 0.3 v_{c}$

$\frac{d ν}{d t} = \max$

Solution

1 $ν_{P} + ν_{R} = 2 ν_{Q}$

2 The rate of change in beat frequency is maximum when the car passes through Q.

3 The plot below represents schematically the variation of beat frequency with time.

4 The plot below represents schematically the variation of beat frequency with time.

Want Us to Email you About Special Offers & Updates?