I E Irodov Physical Fundamentals Of Mechanics (Hydrodynamics) Q. 1.336

I E Irodov Physical Fundamentals of Mechanics, is a popular advanced Physics book for students preparing for JEE Advanced, Physics Olympiads and other competitive examinations.

Hydrodynamics chapter covers fluid flow, continuity equation, Reynolds number, viscosity, pressure and behaviour of liquids inside pipes. These problems enable students to better understand Fluid Mechanics and to apply mathematical relations to changing pipe dimensions.

On this page, we give a simple step-by-step solution to the problem I E Irodov Physical Fundamentals of Mechanics (Hydrodynamics) Question 1.336, with the concepts and formulas used.

I E Irodov Hydrodynamics Question 1.336

Question:

The cross-sectional radius of a pipeline decreases gradually according to:

[
r=r_0e^{-ax}
]

where:

[
a=0.50\text{ m}^{-1}
]

and (x) is the distance from the pipeline inlet.

Find the ratio of Reynolds numbers for two cross-sections separated by:

[
\Delta x=3.2\text{ m}
]

Concepts Used in I E Irodov Q. 1.336

This problem is based on fluid flow through a pipe whose radius changes with distance.

Important concepts include:

• Reynolds number
• Continuity equation
• Steady fluid flow
• Cross-sectional area
• Flow velocity
• Exponential variation of radius
• Incompressible fluid flow

For steady flow of an incompressible liquid, the volume flow rate is constant. Hence, the liquid velocity increases as the pipe radius decreases.

Easy Solution of I E Irodov Hydrodynamics Q. 1.336

Given:

Radius of the pipeline:

[
r=r_0e^{-ax}
]

Value of (a):

[
a=0.50\text{ m}^{-1}
]

Distance between the two cross-sections:

[
\Delta x=3.2\text{ m}
]

We need to find the ratio of the Reynolds numbers at the two cross-sections.

Let:

• (r_1) = radius at the first cross-section
• (r_2) = radius at the second cross-section
• (v_1) = fluid velocity at the first cross-section
• (v_2) = fluid velocity at the second cross-section
• (\mathrm{Re}_1) = Reynolds number at the first cross-section
• (\mathrm{Re}_2) = Reynolds number at the second cross-section

Assume that the second cross-section is located downstream from the first one by a distance (\Delta x).

Step 1: Write the Reynolds Number Formula

For fluid flow through a circular pipe, the Reynolds number is:

[
\mathrm{Re}=\frac{\rho vD}{\eta}
]

Since the pipe diameter is:

[
D=2r
]

we can write:

[
\mathrm{Re}=\frac{2\rho vr}{\eta}
]

For the same fluid, density (\rho) and viscosity (\eta) remain constant.

Therefore:

[
\mathrm{Re}\propto vr
]

Step 2: Apply the Continuity Equation

For steady flow of an incompressible liquid:

[
Av=\text{constant}
]

The cross-sectional area of the pipe is:

[
A=\pi r^2
]

Therefore:

[
\pi r^2v=\text{constant}
]

This means:

[
v\propto\frac{1}{r^2}
]

As the radius decreases, the fluid velocity increases.

Step 3: Determine How Reynolds Number Depends on Radius

We know:

[
\mathrm{Re}\propto vr
]

and:

[
v\propto\frac{1}{r^2}
]

Therefore:

[
\mathrm{Re}\propto\frac{1}{r^2}\times r
]

Hence:

[
\mathrm{Re}\propto\frac{1}{r}
]

The Reynolds number is inversely proportional to the radius of the pipeline.

Step 4: Find the Ratio of the Two Radii

At the first cross-section located at (x_1):

[
r_1=r_0e^{-ax_1}
]

At the second cross-section located at:

[
x_2=x_1+\Delta x
]

the radius is:

[
r_2=r_0e^{-a(x_1+\Delta x)}
]

Therefore:

[
r_2=r_0e^{-ax_1}e^{-a\Delta x}
]

Since:

[
r_1=r_0e^{-ax_1}
]

we get:

[
r_2=r_1e^{-a\Delta x}
]

Thus:

[
\frac{r_1}{r_2}=e^{a\Delta x}
]

Step 5: Find the Reynolds Number Ratio

Since:

[
\mathrm{Re}\propto\frac1r
]

the ratio of the Reynolds numbers is:

\frac{r_1}{r_2}
]

Therefore:

e^{a\Delta x}
]

Substitute:

[
a=0.50\text{ m}^{-1}
]

and:

[
\Delta x=3.2\text{ m}
]

Hence:

e^{(0.50)(3.2)}
]

e^{1.6}
]

Using:

[
e^{1.6}\approx4.95
]

we obtain:

[
\frac{\mathrm{Re}_2}{\mathrm{Re}_1}\approx4.95
]

Final Answer

The ratio of the Reynolds number at the downstream cross-section to that at the upstream cross-section is:

[
\boxed{
\frac{\mathrm{Re}_2}{\mathrm{Re}_1}=e^{a\Delta x}
}
]

For the given values:

[
\boxed{
\frac{\mathrm{Re}_2}{\mathrm{Re}_1}\approx4.95
}
]

Therefore, the Reynolds number at the second cross-section is approximately:

4.95 times the Reynolds number at the first cross-section.

Simple Explanation of the Answer

The pipe gets smaller as the fluid moves forward.

Because the liquid is assumed incompressible , the same amount of liquid must flow through any cross section in the same amount of time . Hence the fluid moves faster in the constricted region of the pipe.

The Reynolds number depends on the fluid velocity and the radius of the pipe:

[
\mathrm{Re}\propto vr
]

However, the velocity increases as:

[
v\propto\frac1{r^2}
]

Therefore:

[
\mathrm{Re}\propto\frac1r
]

The radius decreases exponentially. The Reynolds number therefore increases exponentially. Over a length of (3.2) meters it increases by a factor of about 4.95.

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Key Takeaways

Common Mistakes Students Should Avoid

Students commonly make these mistakes:

• Assuming that Reynolds number is directly proportional to the pipe radius
• Forgetting to apply the continuity equation
• Using (v\propto1/r) instead of (v\propto1/r^2)
• Ignoring that the diameter is proportional to the radius
• Reversing the Reynolds number ratio
• Using (e^{-a\Delta x}) for the downstream-to-upstream Reynolds number ratio
• Forgetting that the same liquid has constant density and viscosity

Always identify which cross-section is upstream and which one is downstream before writing the final ratio.

Why Practice I E Irodov Hydrodynamics Problems?

I E Irodov Hydrodynamics problems help students understand how fluid properties and pipeline dimensions affect fluid motion.

Benefits include:

• Better understanding of Reynolds number
• Stronger application of the continuity equation
• Improved knowledge of laminar and turbulent flow
• Better mathematical reasoning
• Improved advanced numerical-solving ability
• Stronger preparation for Fluid Mechanics problems

Students should solve selected I E Irodov questions after completing standard JEE-level Fluid Mechanics concepts and numerical problems.

Is I E Irodov Q. 1.336 a JEE Main Question?

I E Irodov Hydrodynamics Q. 1.336 is more advanced than a typical JEE Main question.

The problem combines:

• Reynolds number
• Continuity equation
• A pipeline with changing radius
• Exponential variation
• Ratio analysis

It is more suitable for:

• JEE Advanced aspirants
• Physics Olympiad students
• Advanced Fluid Mechanics practice
• Students comfortable with exponential relations

For JEE Main, students should first focus on NCERT concepts, the official syllabus, previous year questions and standard numerical problems.

Is Hydrodynamics Important for JEE?

Fluid Mechanics is an important part of Physics preparation, but the number of questions can vary between examinations.

Important Fluid Mechanics topics include:

• Fluid pressure
• Buoyant force
• Archimedes’ principle
• Equation of continuity
• Bernoulli’s theorem
• Viscosity
• Reynolds number
• Laminar and turbulent flow
• Stokes’ law
• Terminal velocity
• Surface tension

Students should understand the concepts and solve previous year questions instead of relying on a fixed question-count prediction.

Important Topics in I E Irodov Physical Fundamentals of Mechanics

How Should Students Use I E Irodov for JEE Advanced?

Recommended preparation sequence:

I E Irodov should be used as an advanced problem-solving resource after students have developed strong fundamentals.

I E Irodov for NRI JEE Students

Selected I E Irodov Hydrodynamics problems can be used by NRI students preparing for JEE from USA, UAE, Saudi Arabia, Qatar, Oman, Kuwait and Bahrain for advanced Physics practice.

Students who are following AP, IB, British, American or other international syllabus first compare their school syllabus with JEE syllabus.

A useful preparation sequence is:

• Complete JEE Fluid Mechanics concepts
• Solve chapter-wise JEE questions
• Practice JEE previous year papers
• Attempt JEE Advanced numerical problems
• Solve selected I E Irodov questions
• Review difficult concepts with a teacher or mentor

This method helps NRI students use advanced reference books efficiently.

Related I E Irodov Hydrodynamics Questions

After completing Question 1.336, students can also practise:

• I E Irodov Hydrodynamics Q. 1.324
• I E Irodov Hydrodynamics Q. 1.328
• I E Irodov Hydrodynamics Q. 1.335
• I E Irodov Hydrodynamics Q. 1.339

Each question should link to its corresponding TestPrepKart solution page.

Practice Next

After solving this question, students can continue with:

• Reynolds number numerical problems
• Continuity equation questions
• Laminar and turbulent flow problems
• Bernoulli’s theorem questions
• Viscosity problems
• JEE Advanced Fluid Mechanics previous year papers
• Physics revision notes
• JEE Advanced mock tests

Useful JEE, DASA & CIWG Resources for I E Irodov Physics

FAQs on I E Irodov Hydrodynamics Q. 1.336

1. What is the answer to I E Irodov Q. 1.336?

The ratio of the downstream Reynolds number to the upstream Reynolds number is:

[
\boxed{
\frac{\mathrm{Re}_2}{\mathrm{Re}_1}\approx4.95
}
]

2. Which concepts are used in I E Irodov Q. 1.336?

The problem uses Reynolds number, the continuity equation, changing cross-sectional area and exponential variation of the pipe radius.

3. Why does fluid velocity increase when the pipe radius decreases?

For steady incompressible flow:

[
Av=\text{constant}
]

A smaller cross-sectional area requires a higher fluid velocity to maintain the same volume flow rate.

4. How does Reynolds number depend on the pipe radius?

For constant fluid density, viscosity and flow rate:

[
\mathrm{Re}\propto\frac1r
]

Therefore, Reynolds number increases when the pipeline radius decreases.

5. Why is the Reynolds number ratio exponential?

The pipe radius varies as:

[
r=r_0e^{-ax}
]

Since Reynolds number is inversely proportional to radius:

[
\mathrm{Re}\propto e^{ax}
]

Therefore, the ratio between two locations separated by (\Delta x) is:

[
e^{a\Delta x}
]

6. Is I E Irodov Q. 1.336 useful for JEE Main?

It is more advanced than a typical JEE Main question and is better suited to JEE Advanced and advanced Fluid Mechanics practice.

7. What does a higher Reynolds number indicate?

A higher Reynolds number indicates that inertial effects are becoming stronger relative to viscous effects. It may also indicate a greater tendency for the flow to move toward turbulence.