JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Fluid Mechanics

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A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are

20 mm

and

1 mm

, respectively. The upper end of the container is open to the atmosphere.

If the density of air is

ρ_{a}

and that of the liquid

ρ_{ℓ}

, then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

(A) $\frac{ρ _{a}}{ρ _{ℓ}}$ ✓ Correct
(B) $ρ_{a} ρ_{ℓ}$
(C) $\frac{ρ _{ℓ}}{ρ _{a}}$
(D) $ρ_{ℓ}$

Answer: (A)

Solution

Using Bernoulli's equation, the pressure difference created by the fast-moving air causes the liquid to rise. The velocity of liquid ejection is proportional to

\frac{Δ P}{ρ _{ℓ}}

, where

Δ P \propto ρ_{a} v^{2}

.

Therefore, the liquid velocity

v_{ℓ} \propto \frac{ρ _{a}}{ρ _{ℓ}}

.

The volume flow rate is

Q = A v_{ℓ} \propto \frac{ρ _{a}}{ρ _{ℓ}}

.

In the limiting case where

ρ_{ℓ} \to \infty

, the flow rate approaches zero, confirming the relationship.

JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Single correct option

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JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Fluid Mechanics

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