
The reading of a meter which reads pressure is fitted in a closed pipe is $\text{5}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{N}{{\text{m}}^{-2}}$ on the opening the value of the pipe, the reading of that meter reduces to $\text{5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{N}{{\text{m}}^{-2}}$. The speed of water flowing in the pipe is?
Answer
139.8k+ views
Hint: In order to solve this question we will apply Bernoulli's principle when the pipe was closed and after it was opened. Only the value of the initial and final pressure is given in the question. It is to be assumed that the atmospheric pressure remains constant. Bernoulli’s equation can be summarized as the total pressure is the sum of static pressure and dynamic pressure.
Formula used:
${{\text{P}}_{\text{i}}}\ \text{=}\ {{\text{P}}_{\text{f}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}^{\text{2}}}$
Here ${{\text{P}}_{\text{i}}}$ is a Initial static pressure
${{\text{P}}_{\text{f}}}$ is a final pressure
$\text{P}$ is the density of water
$\text{V}$ is the velocity of water.
Complete step by step solution:
Using Bernoulli’s principle we get
${{\text{P}}_{\text{1}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{P}}_{\text{1}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{2}\ }}\ \ \text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\left( \text{1} \right)$
Initially the value is closed, so velocity of water I.e. ${{\text{V}}_{\text{1}}}\text{=0}$
$\text{ }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}\ }}\text{=}\ {{\text{h}}_{\text{2}\ }}\ \text{=0}$
Both are flowing at same reference point

Now, putting the value ${{\text{V}}_{\text{1}}}\text{=}\ \text{0}\ \text{ }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}}}\text{=}{{\text{h}}_{\text{2}\ }}\text{=0}$
We get,
${{\text{P}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}$
Here ${{\text{P}}_{\text{1}}}$ is the initial pressure
${{\text{P}}_{\text{2}}}$ is the final pressure when the valve is open
So, further
$\Rightarrow {{\text{P}}_{\text{1}}}\text{=}{{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\ \text{=}\ \dfrac{\text{2}\left( {{\text{P}}_{\text{1}}}\text{-}{{\text{P}}_{\text{2}}} \right)}{\text{P}}$
Here $\text{ }\!\!\rho\!\!\text{ }\ \text{=}\ \text{1000kg/}{{\text{m}}^{\text{3}}}$
${{\text{P}}_{\text{1}}}\ \text{=}\ \text{5}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{N/}{{\text{M}}^{\text{2}}}$
${{\text{P}}_{\text{2}}}\ \text{=}\ \text{5}\text{.0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{N/}{{\text{M}}^{\text{2}}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\ \text{=}\ \dfrac{\text{2}\left( \text{5}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{-5}\text{.0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}} \right)}{\text{1}{{\text{0}}^{\text{3}}}}$
$=\ \dfrac{2\left( 0.5\times {{10}^{15}} \right)}{{{10}^{3}}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\text{=}\dfrac{\text{1}{{\text{0}}^{\text{15}}}}{\text{1}{{\text{0}}^{\text{3}}}}\ \text{=}\ \text{1}{{\text{0}}^{\text{15-3}}}\ \text{=}\ \text{1}{{\text{0}}^{\text{12}}}$
So,
${{\text{V}}_{\text{2}}}\text{=}\ \text{1}{{\text{0}}^{\text{6}}}\text{m/s}$
So, the speed of the water is $\text{1}{{\text{0}}^{\text{6}}}\text{m/s}$.
Note: Bernoulli’s principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid’s potential energy.
Principle: Within a horizontal flow of fluid, points of higher fluid speed will have less pressure than points of slower fluid speed.
Here is a way to express kinetic energy is to do work on it.
This is expressed by the work energy principle
$\text{ }\!\!\Delta\!\!\text{ }{{\text{W}}_{\text{external}}}\ \text{= }\!\!\Delta\!\!\text{ K}\ \text{=}\ \dfrac{\text{1}}{\text{2}}\text{m}{{\text{V}}_{\text{f}}}^{\text{2}}\text{-}\dfrac{\text{1}}{\text{2}}\text{m}{{\text{V}}_{\text{i}}}^{\text{2}}$
Bernoulli’s equation is usually used in isentropic fluids.
In order to solve we have to use Bernoulli’s equation which is given by
${{\text{P}}_{\text{1}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{P}}_{\text{1}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{2}\ }}$
Here the value is closed initially so ${{\text{V}}_{\text{1}\ }}\text{=0 }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}\ }}\text{ }\!\!\And\!\!\text{ }{{\text{h}}_{\text{2}\ }}\text{=0,}$ So, by putting these values we can do this question.
Formula used:
${{\text{P}}_{\text{i}}}\ \text{=}\ {{\text{P}}_{\text{f}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}^{\text{2}}}$
Here ${{\text{P}}_{\text{i}}}$ is a Initial static pressure
${{\text{P}}_{\text{f}}}$ is a final pressure
$\text{P}$ is the density of water
$\text{V}$ is the velocity of water.
Complete step by step solution:
Using Bernoulli’s principle we get
${{\text{P}}_{\text{1}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{P}}_{\text{1}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{2}\ }}\ \ \text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\left( \text{1} \right)$
Initially the value is closed, so velocity of water I.e. ${{\text{V}}_{\text{1}}}\text{=0}$
$\text{ }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}\ }}\text{=}\ {{\text{h}}_{\text{2}\ }}\ \text{=0}$
Both are flowing at same reference point

Now, putting the value ${{\text{V}}_{\text{1}}}\text{=}\ \text{0}\ \text{ }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}}}\text{=}{{\text{h}}_{\text{2}\ }}\text{=0}$
We get,
${{\text{P}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}$
Here ${{\text{P}}_{\text{1}}}$ is the initial pressure
${{\text{P}}_{\text{2}}}$ is the final pressure when the valve is open
So, further
$\Rightarrow {{\text{P}}_{\text{1}}}\text{=}{{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\ \text{=}\ \dfrac{\text{2}\left( {{\text{P}}_{\text{1}}}\text{-}{{\text{P}}_{\text{2}}} \right)}{\text{P}}$
Here $\text{ }\!\!\rho\!\!\text{ }\ \text{=}\ \text{1000kg/}{{\text{m}}^{\text{3}}}$
${{\text{P}}_{\text{1}}}\ \text{=}\ \text{5}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{N/}{{\text{M}}^{\text{2}}}$
${{\text{P}}_{\text{2}}}\ \text{=}\ \text{5}\text{.0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{N/}{{\text{M}}^{\text{2}}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\ \text{=}\ \dfrac{\text{2}\left( \text{5}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}}\text{-5}\text{.0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{15}}} \right)}{\text{1}{{\text{0}}^{\text{3}}}}$
$=\ \dfrac{2\left( 0.5\times {{10}^{15}} \right)}{{{10}^{3}}}$
${{\text{V}}_{\text{2}}}^{\text{2}}\text{=}\dfrac{\text{1}{{\text{0}}^{\text{15}}}}{\text{1}{{\text{0}}^{\text{3}}}}\ \text{=}\ \text{1}{{\text{0}}^{\text{15-3}}}\ \text{=}\ \text{1}{{\text{0}}^{\text{12}}}$
So,
${{\text{V}}_{\text{2}}}\text{=}\ \text{1}{{\text{0}}^{\text{6}}}\text{m/s}$
So, the speed of the water is $\text{1}{{\text{0}}^{\text{6}}}\text{m/s}$.
Note: Bernoulli’s principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid’s potential energy.
Principle: Within a horizontal flow of fluid, points of higher fluid speed will have less pressure than points of slower fluid speed.
Here is a way to express kinetic energy is to do work on it.
This is expressed by the work energy principle
$\text{ }\!\!\Delta\!\!\text{ }{{\text{W}}_{\text{external}}}\ \text{= }\!\!\Delta\!\!\text{ K}\ \text{=}\ \dfrac{\text{1}}{\text{2}}\text{m}{{\text{V}}_{\text{f}}}^{\text{2}}\text{-}\dfrac{\text{1}}{\text{2}}\text{m}{{\text{V}}_{\text{i}}}^{\text{2}}$
Bernoulli’s equation is usually used in isentropic fluids.
In order to solve we have to use Bernoulli’s equation which is given by
${{\text{P}}_{\text{1}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{P}}_{\text{1}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{1}}}\ \text{=}\ {{\text{P}}_{\text{2}}}\text{+}\dfrac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{V}}_{\text{2}}}^{\text{2}}\text{+ }\!\!\rho\!\!\text{ g}{{\text{h}}_{\text{2}\ }}$
Here the value is closed initially so ${{\text{V}}_{\text{1}\ }}\text{=0 }\!\!\And\!\!\text{ }\ {{\text{h}}_{\text{1}\ }}\text{ }\!\!\And\!\!\text{ }{{\text{h}}_{\text{2}\ }}\text{=0,}$ So, by putting these values we can do this question.
Recently Updated Pages
Average fee range for JEE coaching in India- Complete Details

Difference Between Rows and Columns: JEE Main 2024

Difference Between Length and Height: JEE Main 2024

Difference Between Natural and Whole Numbers: JEE Main 2024

Algebraic Formula

Difference Between Constants and Variables: JEE Main 2024

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

JEE Main Exam Marking Scheme: Detailed Breakdown of Marks and Negative Marking

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

Other Pages
Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line
