Which unit is equivalent to 1 pascal (Pa)?

Correct Option: B. Solution: 1 pascal (Pa) is equivalent to 1 N m⁻².

Mechanical Properties of Fluids

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Summary

Chapter 9: Mechanical Properties of Fluids

Summary

Fluids can flow and do not have a definite shape.
Liquids are incompressible and have a free surface; gases are compressible and expand to fill their container.
Pressure is defined as force per unit area:
- Average pressure, $P_{av} = \frac{F}{A}$
Pascal's law states that pressure in a fluid at rest is the same at all points at the same height.
The pressure in a fluid varies with depth:
- $P = P_a + pgh$
Continuity equation for incompressible fluid flow:
- $V A = \text{constant}$
Bernoulli's principle:
- $P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}$
Coefficient of viscosity, $n$ $n$ , relates shear stress to shear strain rate:
- $n = \frac{\text{shear stress}}{\text{shear strain rate}}$
Stokes' law for viscous drag force:
- $F = 6\pi n a v$
Surface tension is a force per unit length acting at the interface of a liquid.

Key Formulas/Definitions

Physical Quantity	Symbol	Dimensions	Unit	Remarks
Pressure	P	[M T⁻²]	pascal (Pa)	1 atm = 1.013 X 10⁵ Pa, Scalar
Density	p	[M L⁻³]	kg m⁻³	Scalar
Specific Gravity	No	-	-	$\frac{P_{substance}}{P_{water}}$
Coefficient of viscosity	n	[M L⁻¹ T⁻¹]	Pa s or poise	Scalar
Surface Tension	S	[M T⁻²]	N m⁻¹	Scalar

Learning Objectives

Understand the basic properties of fluids, including:
- Definition of fluids as substances that can flow.
- Differences between fluids and solids.
Explore the concept of pressure in fluids:
- Definition of pressure as force per unit area.
- Pascal's law and its implications for fluid mechanics.
Study the behavior of fluids in motion:
- Streamline flow and its characteristics.
- Bernoulli's principle and its applications.
Investigate the properties of fluids:
- Viscosity and its effect on fluid flow.
- Surface tension and its significance in fluid behavior.
Apply concepts to real-world scenarios:
- Analyze fluid behavior in various contexts, such as capillaries and hydraulic systems.
- Solve problems related to fluid dynamics and pressure.

Detailed Notes

Chapter Nine: Mechanical Properties of Fluids

9.1 Introduction

Fluids are defined as substances that can flow, distinguishing them from solids.
Key properties of fluids include:
- No definite shape (unlike solids)
- Fixed volume for solids and liquids; gases fill their containers.
- Compressibility: Solids and liquids have lower compressibility compared to gases.

9.2 Pressure

Definition: Pressure is defined as force per unit area.
Units:
- Pascal (Pa) = N m⁻²
- 1 atm = 1.01 x 10⁵ Pa
- 1 bar = 10⁵ Pa
- 1 torr = 133 Pa = 0.133 kPa
- 1 mm of Hg = 1 torr = 133 Pa
Pascal's Law: Pressure in a fluid at rest is the same at all points at the same height.
Pressure Variation:
- Formula: P = Pa + pgh (where p is the fluid density)
Continuity Equation: V A = constant (mass conservation in incompressible fluid flow).

9.3 Streamline Flow

A streamline is a path traced by a fluid particle in steady flow.
Streamlines do not intersect in steady flow.

9.4 Bernoulli's Principle

Statement: Along a streamline, the sum of pressure (P), kinetic energy per unit volume (pv²/2), and potential energy per unit volume (pgy) is constant.
Equation: P + pv²/2 + pgy = constant
This principle applies to non-viscous fluid motion in steady state.

9.5 Viscosity

Definition: The coefficient of viscosity (n) is the ratio of shear stress to the rate of shear strain.
Stokes' Law: F = 6πnav (viscous drag force on a sphere in a fluid).

9.6 Surface Tension

Definition: Surface tension is the force per unit length acting at the interface of a liquid.
It represents the extra energy of molecules at the surface compared to those in the interior.

Points to Ponder

Pressure is a scalar quantity, not a vector.
Pressure exists at all points in a fluid, not just on solid surfaces.

Exercises

Explain why blood pressure is greater at the feet than at the brain.
Discuss the behavior of fluids under pressure and the implications for various applications.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Pressure: Students often confuse pressure as a vector quantity due to its definition involving force. Remember, pressure is a scalar quantity defined as force per unit area, specifically the normal component of force.
Ignoring Fluid Properties: Failing to recognize that fluids can flow and do not have a fixed shape can lead to incorrect applications of principles like Bernoulli's.
Assuming Incompressibility: Many students apply equations assuming fluids are incompressible without considering the context. While liquids are largely incompressible, gases are not, and this affects calculations.
Confusing Shear Stress and Shear Strain: It's important to differentiate between shear stress (force per unit area) and shear strain (deformation due to shear stress). This confusion can lead to incorrect applications of viscosity concepts.

Exam Tips

Understand Key Principles: Familiarize yourself with Pascal's law and Bernoulli's principle, as these are frequently tested. Know how to apply them in various scenarios.
Practice Pressure Calculations: Work on problems involving pressure differences and hydrostatic pressure to solidify your understanding of the concepts.
Visualize Fluid Flow: Draw diagrams to represent fluid flow and forces acting on fluids. This can help clarify concepts like streamlines and pressure distribution.
Review Viscosity and Surface Tension: Make sure to understand the definitions and units of viscosity and surface tension, as well as their implications in real-world scenarios.
Check Units: Always ensure that your units are consistent, especially when dealing with pressure, density, and viscosity calculations.

Practice & Assessment

Multiple Choice Questions

1400 Pa

1800 Pa

1600 Pa

2200 Pa

Correct Answer: A

Solution:

Bernoulli's principle states that for an incompressible, non-viscous fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline. Thus,

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Assuming the fluid density

\rho

is constant and neglecting height differences,

2000 + \frac{1}{2} \rho (3)^2 = P_2 + \frac{1}{2} \rho (5)^2

. Solving for

P_2

, we find

P_2 = 2000 - \frac{1}{2} \rho (5^2 - 3^2) = 1400

Pa, assuming

\rho = 1

for simplification. Thus, the correct answer is option a.

1 N m

1 N m⁻²

1 N m²

1 N m⁻³

Correct Answer: B

Solution:

1 pascal (Pa) is equivalent to 1 N m⁻².

98,000 Pa

101,300 Pa

1,000,000 Pa

198,000 Pa

Correct Answer: A

Solution:

The pressure at depth is given by

P = P_a + \rho gh

. Here,

P_a

is atmospheric pressure,

\rho = 1000 \text{ kg/m}^3

g = 9.8 \text{ m/s}^2

, and

h = 10 \text{ m}

. So,

P = 0 + 1000 \times 9.8 \times 10 = 98,000 \text{ Pa}

2500 Pa

2000 Pa

1500 Pa

1000 Pa

Correct Answer: A

Solution:

Using Bernoulli's equation:

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Solving for

P_2

P_2 = 1500 + \frac{1}{2} \times 1000 \times (2^2 - 1^2) = 2500

Pa.

500 Pa

1000 Pa

1500 Pa

2500 Pa

Correct Answer: A

Solution:

Using Bernoulli's principle:

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Let

\rho = 1000 \ kg/m^3

. Given

P_1 = 2000 \ Pa

v_1 = 3 \ m/s

, and

v_2 = 6 \ m/s

, we solve for

P_2

P_2 = 2000 + \frac{1}{2} \times 1000 \times (3^2 - 6^2) = 500 \ Pa

500 Pa

1000 Pa

1500 Pa

2000 Pa

Correct Answer: A

Solution:

Using Bernoulli's equation:

P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2

. Given

P_1 = 1500 \text{ Pa}

v_1 = 2 \text{ m/s}

v_2 = 4 \text{ m/s}

, and assuming

\rho = 1000 \text{ kg/m}^3

, we solve for

P_2

1500 + \frac{1}{2} \times 1000 \times 2^2 = P_2 + \frac{1}{2} \times 1000 \times 4^2

, leading to

P_2 = 500 \text{ Pa}

The pressure in a fluid decreases as its velocity increases.

The pressure in a fluid increases as its velocity increases.

The pressure in a fluid is independent of its velocity.

The pressure in a fluid is always constant.

Correct Answer: A

Solution:

Bernoulli's principle states that as we move along a streamline, the sum of the pressure, the kinetic energy per unit volume, and the potential energy per unit volume remains constant. Thus, an increase in velocity results in a decrease in pressure.

Pressure only

Velocity only

The sum of pressure, kinetic energy per unit volume, and potential energy per unit volume

Density only

Correct Answer: C

Solution:

According to Bernoulli's principle, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

Viscosity increases with temperature.

Viscosity decreases with temperature.

Viscosity remains constant with temperature.

Viscosity becomes zero at high temperatures.

Correct Answer: B

Solution:

As temperature rises, the atoms of the liquid become more mobile and the coefficient of viscosity decreases.

Its volume changes significantly with pressure.

It has a fixed shape and cannot flow.

Its volume remains constant regardless of pressure changes.

It expands to fill the entire volume of its container.

Correct Answer: C

Solution:

An incompressible fluid has a constant volume that does not change significantly with pressure.

It causes the liquid to form a flat surface.

It causes the liquid to rise or fall in the tube.

It has no effect on the liquid in the tube.

It causes the liquid to evaporate quickly.

Correct Answer: B

Solution:

Surface tension causes the liquid to rise or fall in a capillary tube, depending on the interaction between the liquid and the tube material.

Gravity

Surface tension

Viscosity

Air pressure

Correct Answer: B

Solution:

The capillary rise is due to surface tension, which is the force per unit length acting at the liquid-air interface.

Fluids have no definite shape and offer little resistance to shear stress.

Fluids have high compressibility compared to solids.

Fluids have a fixed volume under all conditions.

Fluids have a higher density than solids.

Correct Answer: A

Solution:

Fluids can flow because they have no definite shape and offer very little resistance to shear stress, unlike solids.

45 Pa

90 Pa

180 Pa

360 Pa

Correct Answer: B

Solution:

The excess pressure inside a spherical droplet is given by

\Delta P = \frac{2S}{r}

. Substituting

S = 0.045 \text{ N/m}

and

r = 0.002 \text{ m}

, we find

\Delta P = \frac{2 \times 0.045}{0.002} = 90 \text{ Pa}

14.6 cm

2.98 cm

5.96 cm

7.30 cm

Correct Answer: B

Solution:

The height

h

of the capillary rise is given by the formula:

h = \frac{2S \cos \theta}{\rho g r}

. Substituting the values:

h = \frac{2 \times 0.073 \times 1}{1000 \times 9.8 \times 0.0005} = 2.98 \text{ cm}

The pressure inside the bubble is equal to the atmospheric pressure.

The pressure inside the bubble is less than the atmospheric pressure.

The pressure inside the bubble is greater than the atmospheric pressure.

The pressure inside the bubble is independent of the atmospheric pressure.

Correct Answer: C

Solution:

The pressure inside a bubble of gas in a liquid is greater than the atmospheric pressure due to the excess pressure given by

2S/r

, where

S

is the surface tension.

9000 Pa

900 Pa

90 Pa

9 Pa

Correct Answer: A

Solution:

Pressure is calculated as force per unit area:

P = \frac{F}{A} = \frac{90 \text{ N}}{0.01 \text{ m}^2} = 9000 \text{ Pa}

Pressure only

Kinetic energy only

The sum of pressure, kinetic energy per unit volume, and potential energy per unit volume

Potential energy only

Correct Answer: C

Solution:

Bernoulli's principle states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

The pressure inside is greater than atmospheric pressure.

The pressure inside is equal to atmospheric pressure.

The pressure inside is less than atmospheric pressure.

The pressure inside is unaffected by the meniscus shape.

Correct Answer: C

Solution:

When the liquid meniscus is concave, the pressure inside the tube is less than the atmospheric pressure.

Streamlines can intersect.

Streamlines represent paths of fluid particles.

Streamlines indicate areas of high pressure.

Streamlines are only applicable to gases.

Correct Answer: B

Solution:

In a steady flow, streamlines represent the paths of fluid particles and do not intersect.

13,500 N

40,500 N

4,500 N

1,350 N

Correct Answer: B

Solution:

The force exerted by the hydraulic lift is given by Pascal's law, which states that pressure is transmitted undiminished in an enclosed fluid. The pressure applied on the small piston is transmitted to the large piston. Therefore, the force on the large piston is calculated as follows:

F_2 = P imes A_2 = 1.5 \times 10^3 imes \pi \times (0.15)^2 = 40,500 \text{ N}

The sum decreases along a streamline.

The sum remains constant along a streamline.

The sum increases along a streamline.

The sum fluctuates randomly along a streamline.

Correct Answer: B

Solution:

Bernoulli's principle states that the sum of the pressure, the kinetic energy per unit volume, and the potential energy per unit volume remains constant along a streamline.

72 Pa

144 Pa

36 Pa

288 Pa

Correct Answer: B

Solution:

The excess pressure inside a spherical droplet is given by the formula

\Delta P = \frac{2S}{r}

, where

S

is the surface tension and

r

is the radius of the droplet. Substituting the given values,

\Delta P = \frac{2 \times 0.072}{0.001} = 144 \text{ Pa}

Pascal

Newton

Joule

Watt

Correct Answer: A

Solution:

The unit of pressure is the pascal (Pa), which is equivalent to N/m².

The pressure is transmitted undiminished to every point of the fluid.

The pressure decreases as it moves through the fluid.

The pressure only affects the surface of the fluid.

The pressure is absorbed by the walls of the container.

Correct Answer: A

Solution:

Pascal's law states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.

Pressure is the same at all points at the same height.

Pressure increases with height.

Pressure decreases with depth.

Pressure is a vector quantity.

Correct Answer: A

Solution:

According to Pascal's law, pressure in a fluid at rest is the same at all points which are at the same height.

490 N

1470 N

4900 N

4410 N

Correct Answer: A

Solution:

Using Pascal's law, the force applied on the small piston is transmitted to the larger piston. The force on the larger piston is

F_2 = mg = 1500 \times 9.8

. The force on the small piston is

F_1 = \frac{A_1}{A_2}F_2

, where

A_1 = \pi (0.1)^2

and

A_2 = \pi (0.3)^2

. Therefore,

F_1 = \frac{(0.1)^2}{(0.3)^2} \times 1500 \times 9.8 = 490 \text{ N}

Pressure decreases with increasing depth.

Pressure remains constant at all depths.

Pressure increases with increasing depth.

Pressure is independent of depth.

Correct Answer: C

Solution:

The pressure in a fluid increases with depth according to the expression

P = P_a + \rho gh

, where

\rho

is the density of the fluid.

Fluids offer high resistance to shear stress.

Fluids change shape easily under shear stress.

Fluids do not change shape under shear stress.

Fluids become solid under shear stress.

Correct Answer: B

Solution:

Fluids offer very little resistance to shear stress, allowing them to change shape easily.

6 m/s

12 m/s

9 m/s

15 m/s

Correct Answer: B

Solution:

According to the continuity equation for incompressible flow,

A_1v_1 = A_2v_2

. Therefore,

\pi (0.1/2)^2 \times 3 = \pi (0.05/2)^2 \times v_2

. Solving for

v_2

v_2 = \frac{(0.1)^2}{(0.05)^2} \times 3 = 12 \text{ m/s}

1500 Pa

1000 Pa

2000 Pa

2500 Pa

Correct Answer: A

Solution:

Using Bernoulli's equation:

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Substituting the given values:

3000 + \frac{1}{2} \times 1000 \times (2)^2 = P_2 + \frac{1}{2} \times 1000 \times (4)^2

. Solving for

P_2

P_2 = 3000 + 2000 - 8000 = 1500 \text{ Pa}

A path that fluid particles follow over time.

A line where fluid velocity is zero.

A line where fluid density is maximum.

A line where fluid pressure is minimum.

Correct Answer: A

Solution:

A streamline is a path traced by a fluid particle under steady flow conditions, where the tangent to the streamline at any point gives the direction of the fluid velocity at that point.

500 Pa

1000 Pa

1500 Pa

2000 Pa

Correct Answer: B

Solution:

Using Bernoulli's principle, P₁ + 0.5ρv₁² = P₂ + 0.5ρv₂². Assuming ρ = 1 kg/m³ for simplicity, 1500 + 0.5(1)(2)² = P₂ + 0.5(1)(4)². Solving for P₂ gives P₂ = 1000 Pa.

The pressure decreases with depth.

The pressure remains constant with depth.

The pressure increases with depth.

The pressure fluctuates randomly with depth.

Correct Answer: C

Solution:

The pressure in a fluid increases with depth according to the expression

P = P_a + \rho gh

, where

\rho

is the density of the fluid.

Fluids offer very little resistance to shear stress.

Fluids have a definite shape and resist shear stress strongly.

Fluids compress significantly under shear stress.

Fluids expand indefinitely under shear stress.

Correct Answer: A

Solution:

Fluids, unlike solids, offer very little resistance to shear stress, allowing them to flow.

Fluids have a definite shape and resist shear stress.

Fluids have no definite shape and offer little resistance to shear stress.

Fluids are highly compressible and have a fixed volume.

Fluids cannot flow and have a fixed shape.

Correct Answer: B

Solution:

Fluids, unlike solids, do not have a definite shape and offer very little resistance to shear stress, allowing them to flow.

300 N

900 N

1200 N

1500 N

Correct Answer: B

Solution:

According to Pascal's law, the pressure applied to the small piston is transmitted undiminished to the larger piston. The force on the larger piston is given by

F_2 = F_1 \times \frac{A_2}{A_1}

, where

A_1 = \pi (0.05)^2

and

A_2 = \pi (0.15)^2

. Thus,

F_2 = 100 \times \frac{0.15^2}{0.05^2} = 900 \text{ N}

18,000 N

6,000 N

54,000 N

12,000 N

Correct Answer: C

Solution:

According to Pascal's law, the pressure applied on the small piston is transmitted undiminished to the larger piston. The force exerted on the larger piston can be calculated using the formula:

F_2 = P \times A_2

where

A_2 = \pi (0.3)^2

and

P = 2.0 \times 10^3

N/m². Therefore,

F_2 = 2.0 \times 10^3 \times \pi \times (0.3)^2 = 54,000

Pressure is higher at the bottom of the fluid.

Pressure is the same at all points at the same height.

Pressure decreases with depth.

Pressure is only exerted on solid surfaces.

Correct Answer: B

Solution:

Pascal's law states that pressure in a fluid at rest is the same at all points which are at the same height.

1.5 m/s

3 m/s

6 m/s

9 m/s

Correct Answer: C

Solution:

According to the equation of continuity for incompressible fluids,

A_1v_1 = A_2v_2

. Given

A_1 = 0.02 \text{ m}^2

v_1 = 3 \text{ m/s}

, and

A_2 = 0.01 \text{ m}^2

, we find

v_2 = \frac{A_1v_1}{A_2} = \frac{0.02 \times 3}{0.01} = 6 \text{ m/s}

It increases.

It decreases.

It remains constant.

It fluctuates randomly.

Correct Answer: B

Solution:

As temperature rises, the atoms of the liquid become more mobile, and the coefficient of viscosity decreases.

Bar

Pascal

Torr

Atmosphere

Correct Answer: B

Solution:

The unit of pressure in the International System of Units (SI) is the pascal (Pa), which is equivalent to N m⁻².

4 m/s

8 m/s

16 m/s

32 m/s

Correct Answer: B

Solution:

Using the principle of conservation of mass for incompressible fluids, the flow rate must be constant:

A_1 v_1 = A_2 v_2

. Given

A_1 = \pi (0.1/2)^2

and

A_2 = \pi (0.05/2)^2

, we find

v_2 = v_1 \frac{A_1}{A_2} = 2 \times \frac{(0.1/2)^2}{(0.05/2)^2} = 8 \text{ m/s}

1225 N

2450 N

490 N

980 N

Correct Answer: C

Solution:

Using Pascal's law, the pressure applied on the small piston is equal to the pressure on the large piston. Therefore,

F_1/A_1 = F_2/A_2

, where

F_2 = 2000 \times 9.8

N. Solving for

F_1

, we get

F_1 = F_2 \times (A_1/A_2) = 2000 \times 9.8 \times (\pi \times 0.05^2)/(\pi \times 0.2^2) = 490 \text{ N}

The velocity of the fluid at the hole is independent of the height of the fluid column.

The pressure at the hole is equal to atmospheric pressure.

The velocity of the fluid at the hole increases with the height of the fluid column.

The pressure at the hole is greater than atmospheric pressure.

Correct Answer: B

Solution:

According to Bernoulli's principle, the pressure at the hole is equal to atmospheric pressure because the fluid is exposed to the atmosphere as it exits the container. The velocity of the fluid at the hole is related to the height of the fluid column, but the pressure is atmospheric.

Fluids can flow and have no definite shape.

Fluids are always compressible.

Fluids have a fixed shape and volume.

Fluids cannot exert pressure.

Correct Answer: A

Solution:

Fluids, which include liquids and gases, can flow and do not have a definite shape of their own.

The velocity of fluid particles varies randomly at a given point.

The velocity of fluid particles remains constant over time at a given point.

The velocity of fluid particles is always zero at a given point.

The velocity of fluid particles increases exponentially at a given point.

Correct Answer: B

Solution:

In a streamline flow, the velocity of each passing fluid particle remains constant in time at any given point.

8000 Pa

4000 Pa

2000 Pa

500 Pa

Correct Answer: A

Solution:

Using the continuity equation and Bernoulli's principle, the velocity at the second section is doubled due to the halved diameter. Thus,

v_2 = 6

m/s. Applying Bernoulli's equation:

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Solving for

P_2

P_2 = 2000 + \frac{1}{2} \times 1000 \times (3^2 - 6^2) = 8000

Pa.

500 Pa

1000 Pa

1500 Pa

2000 Pa

Correct Answer: A

Solution:

Using Bernoulli's principle,

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Solving for

P_2

, we get

P_2 = 2000 + \frac{1}{2} \rho (3^2 - 6^2)

. Assuming

\rho = 1000 \text{ kg/m}^3

P_2 = 2000 + 0.5 \times 1000 \times (9 - 36) = 500 \text{ Pa}

1500 N

4500 N

5000 N

3000 N

Correct Answer: B

Solution:

According to Pascal's law, the pressure applied to a confined fluid is transmitted undiminished throughout the fluid. Therefore, F₁/A₁ = F₂/A₂. The area of the small piston is A₁ = π(0.1)² and the area of the large piston is A₂ = π(0.3)². Solving for F₂ gives F₂ = F₁(A₂/A₁) = 500 N (π(0.3)²/π(0.1)²) = 4500 N.

0.012 N

0.025 N

0.015 N

0.018 N

Correct Answer: C

Solution:

Stokes' law for viscous drag force is given by:

F = 6 \pi \eta a v

. Substituting the given values:

F = 6 \times \pi \times 0.1 \times 0.02 \times 0.5 = 0.015 \text{ N}

10 m/s

8 m/s

12 m/s

6 m/s

Correct Answer: A

Solution:

According to the principle of continuity for incompressible flow, the product of cross-sectional area and velocity at any two points in the flow must be constant. Therefore, A₁v₁ = A₂v₂. Substituting the given values: (0.05 m²)(4 m/s) = (0.02 m²)(v₂). Solving for v₂ gives v₂ = 10 m/s.

It remains constant at all points in space.

It varies randomly at different points.

It remains constant at a given point over time.

It is always zero.

Correct Answer: C

Solution:

In a streamline flow, the velocity of each passing fluid particle remains constant in time at any given point.

-0.58 cm

0.58 cm

-1.16 cm

1.16 cm

Correct Answer: A

Solution:

The height of capillary rise or fall is given by the formula:

h = \frac{2S \cos \theta}{\rho g r}

. Substituting the given values:

h = \frac{2 \times 0.485 \times \cos(140°)}{13,600 \times 9.8 \times 0.00025}

. Calculating the cosine of 140° gives a negative value, indicating a fall:

h = -0.0058

m or -0.58 cm.

1 m/s

2 m/s

4 m/s

8 m/s

Correct Answer: D

Solution:

According to the equation of continuity for incompressible flow,

A_1v_1 = A_2v_2

. Given that the diameters are 0.04 m and 0.02 m, the areas are proportional to the square of the diameters. Therefore,

v_2 = v_1 \times \left(\frac{d_1}{d_2}\right)^2 = 2 \times \left(\frac{0.04}{0.02}\right)^2 = 8 \text{ m/s}

2.94 cm

3.92 cm

4.80 cm

5.88 cm

Correct Answer: A

Solution:

The height of capillary rise is given by

h = \frac{2S \cos \theta}{\rho g r}

. Substituting the given values,

h = \frac{2 \times 0.072 \times \cos 0}{1000 \times 9.8 \times 0.0005} = 2.94 \text{ cm}

5000 N

1000 N

500 N

2000 N

Correct Answer: A

Solution:

According to Pascal's law, the pressure is the same on both pistons:

\frac{F_1}{A_1} = \frac{F_2}{A_2}

. Given

F_1 = 200 \ N

A_1 = 0.02 \ m^2

, and

A_2 = 0.5 \ m^2

, we solve for

F_2

F_2 = \frac{F_1 \times A_2}{A_1} = \frac{200 \times 0.5}{0.02} = 5000 \ N

A force per unit volume acting within the fluid.

A force per unit length acting in the plane of the interface between the liquid and the bounding surface.

A force per unit area acting perpendicular to the surface.

A force that acts only in gases.

Correct Answer: B

Solution:

Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of the interface between the liquid and the bounding surface.

1:3

1:9

3:1

9:1

Correct Answer: D

Solution:

The force exerted by the large piston is proportional to the square of the radius ratio, so the ratio is

(15/5)^2 = 9:1

Liquids are generally incompressible.

Gases are less compressible than liquids.

Solids are more compressible than gases.

All fluids have the same compressibility.

Correct Answer: A

Solution:

Liquids are considered to be incompressible because their volume does not change significantly with pressure.

The pressure inside the bubble increases.

The pressure inside the bubble decreases.

The pressure inside the bubble remains constant.

The pressure inside the bubble becomes zero.

Correct Answer: B

Solution:

As a bubble rises, the pressure due to the liquid column above it decreases, leading to a decrease in pressure inside the bubble.

750 Pa

1500 Pa

3000 Pa

500 Pa

Correct Answer: A

Solution:

Using Bernoulli's principle,

P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2

. Assuming

\rho = 1000 \text{ kg/m}^3

P_2 = P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2) = 1500 + \frac{1}{2} \times 1000 \times (2^2 - 4^2) = 750 \text{ Pa}

1 N m²

1 N m⁻²

1 N

1 N m

Correct Answer: B

Solution:

The unit of pressure, pascal (Pa), is defined as 1 N m⁻².

Pascal's Law

Archimedes' Principle

Bernoulli's Principle

Newton's Third Law

Correct Answer: C

Solution:

Bernoulli's principle states that as we move along a streamline, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant. This principle is used to explain the lift force on an airplane wing.

1.5 m/s

3 m/s

6 m/s

9 m/s

Correct Answer: C

Solution:

According to the equation of continuity for incompressible fluids,

A_1 v_1 = A_2 v_2

. Given

A_1 = 0.02 \text{ m}^2

v_1 = 3 \text{ m/s}

A_2 = 0.01 \text{ m}^2

, solve for

v_2

v_2 = \frac{A_1 v_1}{A_2} = \frac{0.02 \times 3}{0.01} = 6 \text{ m/s}

True or False

Correct Answer: False

Solution:

In a steady flow, the velocity of a fluid particle at any given point remains constant over time.

Correct Answer: True

Solution:

Surface tension is caused by the higher potential energy of molecules at the surface, which is not balanced by neighboring molecules as it is in the interior.

Correct Answer: False

Solution:

Bernoulli's principle does not hold perfectly in the presence of viscous drag. The principle is an approximation that assumes no viscosity, which is not true for real fluids.

Correct Answer: True

Solution:

As the temperature rises, the atoms of the liquid become more mobile, and the coefficient of viscosity decreases.

Correct Answer: False

Solution:

Surface tension is a scalar quantity, defined as force per unit length or energy per unit area acting at the interface between a liquid and another phase.

Correct Answer: False

Solution:

In a steady flow, streamlines cannot intersect because it would imply that a fluid particle has two different velocities at the same point.

Correct Answer: False

Solution:

The excess pressure inside a bubble of gas in a liquid is given by 2S/r. The formula 4S/r applies to a bubble of liquid in a gas.

Correct Answer: True

Solution:

This expression shows how pressure increases with depth in a fluid due to the weight of the fluid above.

Correct Answer: True

Solution:

As the temperature rises, the atoms of the liquid become more mobile, reducing the coefficient of viscosity.

Correct Answer: False

Solution:

Surface tension arises due to the excess potential energy of molecules at the interface between a fluid and another substance, not just between two fluids.

Correct Answer: True

Solution:

According to Pascal's law, the pressure in a fluid at rest is the same at all points which are at the same height.

Correct Answer: False

Solution:

Fluids, which include liquids and gases, do not have a definite shape. Liquids have a fixed volume, while gases fill the entire volume of their container.

Correct Answer: True

Solution:

Surface tension is indeed a force per unit length (or surface energy per unit area) acting at the interface between the liquid and the bounding surface.

Correct Answer: False

Solution:

According to Pascal's law, the pressure in a fluid at rest is the same at all points that are at the same height.

Correct Answer: False

Solution:

Bernoulli's principle does not hold in the presence of viscous drag on the fluid. The work done by this dissipative viscous force must be taken into account.

Correct Answer: True

Solution:

With an increase in temperature, the atoms in a liquid become more mobile, leading to a decrease in the coefficient of viscosity.

Correct Answer: False

Solution:

As temperature rises, the viscosity of a liquid typically decreases because the molecules become more mobile.

Correct Answer: False

Solution:

Bernoulli's principle applies to ideal fluid motion without viscous drag. In the presence of viscosity, the dissipative forces must be considered, which affects the pressure and energy conservation described by Bernoulli's equation.

Correct Answer: False

Solution:

Fluids do not have a definite shape of their own. They take the shape of their container.

Correct Answer: True

Solution:

This expression represents how pressure in a fluid increases with depth due to the weight of the fluid above.

Correct Answer: True

Solution:

A streamline is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point in a steady flow.

Correct Answer: True

Solution:

The coefficient of viscosity is defined as the ratio of shear stress to the time rate of shearing strain.

Correct Answer: False

Solution:

Unlike solids, liquids do not have a definite shape. They take the shape of their container.

Correct Answer: True

Solution:

This is a consequence of the continuity equation, which states that the product of cross-sectional area and velocity is constant for incompressible fluid flow.

Correct Answer: True

Solution:

Fluids, unlike solids, change shape easily under shear stress because their resistance to shear stress is about a million times smaller than that of solids.

Correct Answer: True

Solution:

The capillary rise is indeed due to surface tension and is inversely proportional to the radius of the tube, meaning it is larger for smaller radii.

Correct Answer: True

Solution:

As temperature rises, the atoms in a liquid become more mobile, reducing the coefficient of viscosity.

Correct Answer: True

Solution:

The pascal (Pa) is the SI unit of pressure and is equivalent to one newton per square meter (N m⁻²).

Correct Answer: False

Solution:

In a steady flow, streamlines do not intersect because each point in the fluid has a unique velocity.

Correct Answer: True

Solution:

In steady flow, the velocity of fluid particles at any given point does not change with time, although it may vary from point to point in space.

Correct Answer: True

Solution:

The pressure in a fluid varies with depth according to the expression

P = P_a + \rho gh

, where

\rho

is the density of the fluid.

Correct Answer: False

Solution:

Surface tension arises due to the excess potential energy of molecules at the interface between two substances, at least one of which is a fluid. It is not a property of a single fluid alone.

Correct Answer: False

Solution:

In a steady flow, the velocity of a fluid particle remains constant at a given point over time, but it can vary from one point in space to another.

Correct Answer: False

Solution:

In a steady flow, streamlines do not intersect. If they did, it would imply that a fluid particle at the intersection would have two different velocities, which is impossible.

Correct Answer: True

Solution:

According to Pascal's law, the pressure in a fluid at rest is the same at all points which are at the same height.

Correct Answer: True

Solution:

Surface tension arises because molecules at the surface have higher potential energy than those in the interior, due to the imbalance of intermolecular forces.

Correct Answer: False

Solution:

Fluids, unlike solids, do not have a definite shape of their own. They take the shape of their container.

Correct Answer: False

Solution:

In streamline flow, the velocity of fluid particles at a given point remains constant over time, but the velocity can vary at different points in space. Streamlines indicate the path of fluid particles, and while the flow is steady, it doesn't mean the velocity is uniform across different spatial locations.

Correct Answer: True

Solution:

In a steady flow, the velocity of a fluid particle may change as it moves from one point to another, but at any given point, the velocity remains constant over time.

Correct Answer: True

Solution:

The pressure difference is given by the formula

\Delta P = \frac{2S}{r}

, where

S

is the surface tension and

r

is the radius of the tube.

Correct Answer: True

Solution:

The unit of pressure, pascal (Pa), is indeed equivalent to N m⁻².

Correct Answer: True

Solution:

The pressure inside a bubble is higher than the atmospheric pressure due to the surface tension of the liquid, which creates an excess pressure inside the bubble.

Correct Answer: True

Solution:

Fluids can flow and do not have a fixed shape, unlike solids. This property allows them to conform to the shape of their container.

Correct Answer: True

Solution:

Viscosity is a measure of a fluid's resistance to deformation at a given rate, such as when it is sheared or stretched.

Correct Answer: True

Solution:

The continuity equation for incompressible fluids states that the product of the cross-sectional area and velocity is constant along a streamline, ensuring the same volume flow rate at any point in the pipe.

Correct Answer: False

Solution:

The unit of pressure is the pascal (Pa), which is equivalent to N/m², whereas the unit of force is the newton (N).

Correct Answer: True

Solution:

Bernoulli's principle is an application of the conservation of energy to fluid motion, stating that

P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}

along a streamline.

Correct Answer: True

Solution:

Fluids, unlike solids, can easily change shape when subjected to shear stress. The shearing stress of fluids is about a million times smaller than that of solids, indicating their low resistance to shear stress.

Correct Answer: True

Solution:

Bernoulli's principle is about the conservation of energy in a fluid flow, stating that the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.

Correct Answer: False

Solution:

Surface tension arises due to the excess potential energy of molecules at the interface between two substances, at least one of which is a fluid.

Correct Answer: True

Solution:

The pressure difference across a liquid surface in a capillary tube is indeed due to surface tension, which causes the capillary rise or depression depending on the contact angle.

Correct Answer: True

Solution:

Water is often considered incompressible because its volume does not change significantly under pressure.

Correct Answer: False

Solution:

The pressure inside a capillary tube at the liquid surface is lower than the atmospheric pressure due to the capillary rise caused by surface tension.

Correct Answer: True

Solution:

Fluids, unlike solids, offer very little resistance to shear stress, which allows them to flow. This is a fundamental property that distinguishes fluids from solids.

Mechanical Properties of Fluids

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Summary

Chapter 9: Mechanical Properties of Fluids

Summary

Key Formulas/Definitions

Learning Objectives

Detailed Notes

Chapter Nine: Mechanical Properties of Fluids

9.1 Introduction

9.2 Pressure

9.3 Streamline Flow

9.4 Bernoulli's Principle

9.5 Viscosity

9.6 Surface Tension

Points to Ponder

Exercises

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Exam Tips

Practice & Assessment

Multiple Choice Questions

Q1.In a steady flow of an incompressible fluid, the velocity at point A is 3 m/s and the pressure is 2000 Pa. At point B, the velocity is 5 m/s. Using Bernoulli's equation, what is the pressure at point B?

Correct Answer: A

Q2.Which unit is equivalent to 1 pascal (Pa)?

Correct Answer: B

Q3.What is the pressure at a depth of 10 meters in a lake, assuming the density of water is 1000 kg/m³ and gravitational acceleration is 9.8 m/s²?

Correct Answer: A

Correct Answer: A

Q5.A fluid flows steadily through a pipe with varying diameters. If the pressure at a point where the velocity is 3 m/s is 2000 Pa, what is the pressure at another point where the velocity is 6 m/s, assuming the flow is inviscid and horizontal?

Correct Answer: A

Q6.In a fluid dynamics experiment, a fluid flows steadily through a pipe of varying diameter. If the pressure at a point where the velocity is 2 m/s is 1500 Pa, what is the pressure at a point where the velocity is 4 m/s? Assume the fluid is incompressible and inviscid.

Correct Answer: A

Q7.Which of the following is a consequence of Bernoulli's principle?

Correct Answer: A

Q8.In a streamline flow, what remains constant along a streamline?

Correct Answer: C

Q9.What is the effect of temperature on the viscosity of a liquid?

Correct Answer: B

Q10.Which of the following is a characteristic of an incompressible fluid?

Correct Answer: C

Q11.What is the effect of surface tension on a liquid in a capillary tube?

Correct Answer: B

Q12.A capillary tube is dipped in water, and the water rises in the tube. What causes this capillary rise?

Correct Answer: B

Q13.What is the primary reason that fluids can flow, unlike solids?

Correct Answer: A

Q14.A spherical droplet of oil is suspended in water. If the surface tension of oil is 0.045 N/m and the radius of the droplet is 0.002 m, what is the excess pressure inside the droplet compared to the outside?

Correct Answer: B

Q15.A capillary tube of radius 0.5 mm is dipped vertically into a container of water. If the surface tension of water is 0.073 N/m, and the contact angle is zero, what is the height to which water will rise in the tube?

Correct Answer: B

Q16.What happens to the pressure inside a bubble of gas in a liquid compared to the atmospheric pressure?

Correct Answer: C

Q17.In a hydraulic system, the pressure applied to a small piston is transmitted to a larger piston. If the small piston has a force of 90 N applied to it and the area of the small piston is 0.01 m², what is the pressure applied?

Correct Answer: A

Q18.According to Bernoulli's principle, what remains constant along a streamline?

Correct Answer: C

Q19.What happens to the pressure inside a capillary tube when the liquid meniscus is concave?

Correct Answer: C

Q20.In a steady flow, what is true about streamlines?

Correct Answer: B

Q21.A hydraulic lift is used to lift a car weighing 1350 kg. The small piston has a radius of 5 cm, and the large piston has a radius of 15 cm. If the pressure applied on the small piston is 1.5 x 10³ N, what is the force exerted on the large piston?

Correct Answer: B

Q22.According to Bernoulli's principle, what happens to the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume along a streamline?

Correct Answer: B

Q23.A spherical droplet of water is suspended in air. If the surface tension of water is 0.072 N/m and the radius of the droplet is 0.001 m, what is the excess pressure inside the droplet compared to the outside?

Correct Answer: B

Q24.Which of the following is a unit of pressure?

Correct Answer: A

Q25.According to Pascal's law, what happens when pressure is applied to an enclosed fluid?

Correct Answer: A

Q26.Which of the following statements is true about the pressure in a fluid at rest?

Correct Answer: A

Q27.A hydraulic lift uses a small piston of radius 0.1 m to lift a car weighing 1500 kg using a larger piston of radius 0.3 m. What force must be applied to the small piston to lift the car? (Assume g=9.8 m/s2g = 9.8 \text{ m/s}^2g=9.8 m/s2 and ignore friction.)

Correct Answer: A

Q28.What is the relationship between pressure and depth in a fluid?

Correct Answer: C

Q29.What is the effect of shear stress on fluids?

Correct Answer: B