If the density of air is , and that of the liquid , then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to [2014]

Question

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.

Q. 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 [2014]

Smart Achievers · Accepted Answer

(1)

$P_{X} - P_{Y} = \frac{1}{2} ρ_{a} v_{a}^{2}$

$P_{Z} - P_{Y} = \frac{1}{2} ρ_{l} v_{l}^{2}$

$But P_{Z} = P_{X}$

$∴$ $\frac{1}{2} ρ_{l} v_{l}^{2} = \frac{1}{2} ρ_{a} v_{a}^{2}$ $\Rightarrow v_{l} = \sqrt{\frac{ρ_{a}}{ρ_{l}}} \times v_{a}$

$∴$ $Volume flow rate \propto \sqrt{\frac{ρ_{a}}{ρ_{l}}}$

Solution

1 $\sqrt{\frac{ρ_{a}}{ρ_{l}}}$

2 $\sqrt{ρ_{a} ρ_{l}}$

3 $\sqrt{\frac{ρ_{l}}{ρ_{a}}}$

4 $ρ_{l}$

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Solution

1 ρaρl

2 ρaρl

3 ρlρa

4 ρl

Ans. (1) PX-PY=12ρava2 PZ-PY=12ρlvl2 But PZ=PX ∴ 12ρlvl2=12ρava2⇒vl=ρaρl×va ∴ Volume flow rate∝ρaρl

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1 $\sqrt{\frac{ρ_{a}}{ρ_{l}}}$

2 $\sqrt{ρ_{a} ρ_{l}}$

3 $\sqrt{\frac{ρ_{l}}{ρ_{a}}}$

4 $ρ_{l}$