Water is flowing continuously from a tap having an internal diameter $8 \times 10^{-3} \mathrm{~m}$. The water velocity as it leaves the tap is $0.4 \mathrm{ms}^{-1}$. The diameter of the water stream at a distance $2 \times 10^{-1} \mathrm{~m}$ below the lap is close to :

(C) $3.6 \times 10^{-3} \mathrm{~m}$

(A) $7.5 \times 10^{-3} \mathrm{~m}$

(B) $9.6 \times 10^{-3} \mathrm{~m}$

(D) $5.0 \times 10^{-3} \mathrm{~m}$

JEE Main · 2011 · 10 Apr 2011 · Offline · Fluid Mechanics

Water is flowing continuously from a tap having an internal diameter

8 \times 1 0^{- 3} m

. The water velocity as it leaves the tap is

0.4 ms^{- 1}

. The diameter of the water stream at a distance

2 \times 1 0^{- 1} m

below the lap is close to :

Answer: (C)

As the water falls it speeds up, and continuity then narrows the stream. First find the speed

2 \times 1 0^{- 1} m

below the tap from

v_{b}^{2} = v_{t}^{2} + 2 g h = (0.4)^{2} + 2 (9.8) (0.2) = 0.16 + 3.92 \approx 4.08

, so

v_{b} \approx 2 m s^{- 1}

. Continuity

A_{t} v_{t} = A_{b} v_{b}

means the squared diameter scales inversely with speed,

d_{b} = d_{t} v_{t} / v_{b} = 8 \times 1 0^{- 3} 0.4/2 = 8 \times 1 0^{- 3} \times 0.447

, giving

d_{b} \approx 3.6 \times 1 0^{- 3} m

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option