SATHEE: Properties of Matter 4 Question 17

Properties of Matter 4 Question 17

19. A liquid of density $900 kg / m^{3}$ is filled in a cylindrical tank of upper radius $0.9 m$ and lower radius $0.3 m$ . A capillary tube of length $l$ is attached at the bottom of the tank as shown in the figure. The capillary has outer radius $0.002 m$ and inner radius $a$ . When pressure $p$ is applied at the top of the tank volume flow rate of the liquid is $8 \times 10^{- 6} m^{3} / s$ and if capillary tube is detached, the liquid comes out from the tank with a velocity $10 m / s$ .

Determine the coefficient of viscosity of the liquid .

[Given, $π a^{2} = 10^{- 6} m^{2}$ and $a^{2} / l = 2 \times 10^{- 6} m$ ]

(2003, 4M)

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Answer:

Correct Answer: 19. $\frac{1}{720} N - s / m^{2}$

Solution:

When the tube is not there,

$\begin{aligned} p + p_{0} + \frac{1}{2} ρ v_{1}^{2} + ρ g H = \frac{1}{2} ρ v_{2}^{2} + p_{0} \\ ∴ p + ρ g H = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2}) \\ A_{1} v_{1} = A_{2} v_{2} \\ v_{1} = \frac{A_{2} v_{2}}{A_{1}} \\ ∴ p + ρ g H = \frac{1}{2} \times ρ [v_{2}^{2} - {(\frac{A_{2}}{A_{1}} v_{2})}^{2}] \\ = \frac{1}{2} \times ρ \times v_{2}^{2} [1 - {(\frac{π (0.3)^{2}}{π (0.9)^{2}})}^{2}] \\ = \frac{1}{2} \times ρ \times (10)^{2} [1 - \frac{1}{81}] \\ = \frac{4 \times 10^{3} ρ}{81} \\ = \frac{4 \times 10^{3} \times 900}{81} \\ = \frac{4}{9} \times 10^{5} N / m^{2} \end{aligned}$

This is also the excess pressure $Δ p$ .

By Poiseuille’s equation, the rate of flow of liquid in the capillary tube

$Q = \frac{π (Δ p) a^{4}}{8 η l}$

$\begin{aligned} ∴ & 8 \times 10^{- 6} & = \frac{(π a^{2}) (Δ p)}{8 η} (\frac{a^{2}}{l}) \\ ∴ η & = \frac{(π a^{2}) (Δ p) (\frac{a^{2}}{l})}{8 \times 8 \times 10^{- 6}} \end{aligned}$

Substituting the values, we have

$\begin{aligned} η & = \frac{(10^{- 6}) (\frac{4}{9} \times 10^{5}) (2 \times 10^{- 6})}{8 \times 8 \times 10^{- 6}} \\ = \frac{1}{720} N - s / m^{2} \end{aligned}$

Properties of Matter 4 Question 17

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Properties of Matter 4 Question 17

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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