## Chapter 3 Topic C Viscosity And Bernoulli’s Theorem Synopsis

- Viscosity is the property of a fluid by virtue of which it tries to reduce the relative motion between its two adjacent layers.
- Viscosity is a general property of the fluid. An ideal fluid has no viscosity. It is also called the internal friction of a fluid.
- Terminal velocity is defined as the maximum uniform velocity with which a falling body falls through a viscous medium.
- If a liquid is flowing through a pipe, then the volume of liquid flowing per second through any cross-section of the pipe is called the rate of flow of that liquid.

**Bernoulli’s Theorem:**

In case of the streamline flow of an ideal fluid, the summation of its kinetic energy, potential energy, and pressure energy (energy due to pressure) per unit volume at every point of a streamline is constant.

The equation of Bernoulli’s theorem is given by

**Read and Learn More WBBSE Solutions for Class 9 Physical Science and Environment**

\(\frac{1}{2} \rho v^2+\rho g h+P\) = constant

or, \(\frac{v^2}{2 g}+h+\frac{p}{\rho g}\) = constant

or, velocity head + elevation head + pressure head = constant

## Chapter 3 Topic C Viscosity And Bernoulli’s Theorem Short And Long Answer Type Questions

**Question 1. What is viscosity?**

**Answer:**

Viscosity is the property of a fluid by which it tries to reduce the relative velocity between two adjacent layers.

**Question 2. What is the relationship between the viscosity and mobility of a liquid?**

**Answer:**

Viscosity is a general property of matter. It is different for different liquids. When viscosity of a fluid increases, mobility decreases.

**Example:** Honey has greater viscosity than water.

**Question 3. Why is viscosity also called internal friction?**

**Answer:**

When a body moves or is about to move on another body or on a surface, then the opposing force that works against this motion or tendency of motion is called friction.

On the other hand, the property by virtue of which a liquid tries to reduce the relative velocity between two adjacent layers is called viscosity of the liquid. Because of this type of similarity between viscosity and friction, viscosity is also called internal friction of a liquid in many cases.

**Question 4. What are the differences between viscosity and friction?**

**Answer:**

The differences between viscosity and friction are:

**Question 5. The viscous force depends on which factors?**

**Answer:**

Viscous force depends on

- Nature of the liquid,
- Area of the layers in contact,
- Velocity gradient between the layers.

**Question 6. What is velocity gradient? What are the unit and dimensional formula of velocity gradient?**

**Answer:**

- Velocity gradient is the change in velocity between adjacent layers with distance perpendicular to the flow of fluids.
- Dimensional formula of velocity gradient is T
^{-1}and its unit in SI is s^{-1}.

**Question 7. What is meant by an ideal fluid?**

**Answer:**

While flowing, a fluid tries to reduce the relative motion between its two adjacent layers due to viscous force. But for an ideal fluid, no such force acts between layers. The ideal fluid is hence non-viscous and streamlined.

**Question 8. What do you mean by streamline flow or laminar flow?**

**Answer:**

If the magnitude and direction of the flow always remain unchanged at any point of the flow line during flow of a fluid, then that flow is called streamline flow or laminar flow. In this condition, there is not any collision among the particles of the liquid.

**Question 9. What do you mean by a streamline? What is the type of streamline in case of streamline motion of a liquid through a right circular cylindrical tube?**

**Answer:**

In case of a streamline motion, the path in which a particle of the fluid flows is called a streamline. In case of a streamline motion, any particle of the fluid always has the velocity of the preceding particle. A tangent drawn at any point of a streamline expresses the direction of the velocity of the fluid at that point.

In case of streamline motion of a liquid through a right circular cylindrical tube, streamlines are parallel to the axis of the tube.

**Question 10. What are the characteristics of a streamline?**

**Answer:**

Characteristics of a streamline are:

- A streamline may be a straight line or a curved line.
- A tangent drawn at any point of a streamline indicates the direction of the velocity of the fluid at that point.
- Two streamlines never intersect with each other.
- Velocity of the fluid increases at that place in the tube where the streamlines are very close to each other and velocity decreases at that place where the streamlines maintain greater distances between them.

**Question 11. What do you mean by turbulent flow?**

**Answer:**

At any point of a flow line during the flow of a fluid, if the magnitude and direction of the flow change in a haphazard way, then the flow is called turbulent flow.

Suppose, a liquid flows through a tube. During this condition, if the liquid particles collide with each other continuously and also move simultaneously, then this type of flow is called turbulent flow.

During this time, whirls are created at some places inside the liquid.

**Question 12. Why do two streamlines never intersect each other?**

**Answer: **

If a tangent is drawn at any point of a streamline, the tangent indicates the direction of the velocity of the fluid at that point. Now, suppose two streamlines intersect each other at the point A.

Two tangents can be drawn at the point of intersection. As a result, two directions of the velocity of the fluid are obtained at the point A, which is not possible. Therefore, two streamlines can never intersect each other.

**Question 13. What do you mean by terminal velocity?**

**Answer:**

The velocity of a small body that falls through a viscous medium due to gravity increases steadily at the beginning. But as its velocity increases, value of the viscous resistance inside the perpendicular layers of the adjacent fluid of the body also increases.

Along with this, the buoyancy of the medium acts on the body in an upward direction. As a result, downward acceleration gradually decreases. At a particular point of time, when value of resistance force due to viscosity and buoyancy are equal to the gravitational force, the resultant force on the body becomes zero.

Then the body falls through the medium with a steady or uniform velocity. This uniform velocity is called terminal velocity.

Thus, terminal velocity is defined as the highest uniform velocity with which a body finally falls through an infinitely spread viscous medium.

**Question 14. Draw the velocity-time graph of a body falling through an infinitely spread viscous medium.**

**Answer: **

The velocity-time graph of a body falling through an infinitely spread viscous medium is shown here. Here the acceleration of the body decreases gradually. After a certain time the body falls with a constant velocity. This is the terminal velocity, shown in the figure with v_{0}.

**Question 15. write down Bernoulli’s theorem.**

**Answer:**

In case of the streamline flow of an ideal fluid, summation of its kinetic energy, potential energy, and pressure energy (energy due to pressure) per unit volume at every point of a streamline is constant.

**Question 16. Write down Bernoulli’s theorem on the basis of conservation of energy.**

**Answer:**

In case of streamline motion of an ideal fluid, the net mechanical energy (i.e., summation of its kinetic energy, potential energy and energy due to pressure per unit volume) at any point of a streamline is always constant.

**Question 17. State the mathematical form of Bernoulli’s theorem.**

**Answer:**

Suppose, an ideal liquid is flowing in streamline flow through a tube. Let us assume that streamline flow of a liquid takes place through a tube of non-uniform cross section.

If v is the velocity of the liquid at any point on the streamline, kinetic energy of unit volume = 1/2ρv^{2} (where p is the density of liquid).

If h is the height of that point from a particular reference level, potential energy in unit volume = ρgh and if P is the pressure at that point, then according to Bernoulli’s theorem,

1/2ρv^{2} + ρgh + P = constant……..(1)

If equation (1) is divided by ρg, we get

\(\frac{v^2}{2 g}+h+\frac{P}{\rho g}=\text { constant }\)……(2)

In equation (2), \(\frac{v^2}{2 g}\) is called the velocity head, h is called the elevation head and \(\frac{P}{\rho g}\) is called the pressure head.

In the velocities of flow of the liquid at points A and B are v_{1} and v_{2}, respectively. h_{1} and h_{2} are the heights of the points A and B from a particular reference level (CD) and pressures at those two points are P_{1} and P_{2}, respectively.

Then, according to Bernoulli’s theorem,

\(\frac{1}{2} \rho v_1^2+\rho g h_1+P_1=\frac{1}{2} \rho v_2^2+\rho g h_2+P_2\)**Question 18. Write down Bernoulli’s theorem with respect to the horizontal flow of a fluid.**

**Answer:**

Bernoulli’s equation is given by

\(\frac{1}{2} \rho v^2+\rho g h+P\) = constant;

where ρ is the density of the liquid, v is its velocity and P is its pressure at depth h.

Here, the kinetic energy, potential energy, and pressure energy of unit volume of the fluid are 1/2ρv^{2}, ρhg, and P, respectively.

For horizontal flow of the fluid, Bernoulli’s theorem can be written as \(\frac{1}{2} \rho v^2+P\) = constant

So at the place where kinetic energy of the fluid is more, pressure is less, and vice versa.

**Question 19. On the basis of velocity and pressure of the fluid, how do you describe Bernoulli’s theorem?**

**Answer:**

In respect of a fluid flowing in a horizontal way, pressure of the fluid is less where its velocity is more and vice versa—this is the essence of Bernoulli’s theorem.

**Question 20. With the help of Bernoulli’s theorem, calculate the pressure of water at a depth of h for a stationary liquid.**

**Answer:**

Let us assume that a liquid of density ρ is in a vessel at a steady condition. Now at a depth h from the free surface, a point A is taken.

Pressure of liquid at the point A has to be calculated. If the bottom surface of the liquid is taken as the reference surface, height of point A is h_{1}. Now as the liquid is still, so the velocities of the liquid at points A and B are zero i.e., v_{A} = v_{B} = 0.

If the atmospheric pressure is P_{a}, then pressure at point B, P_{B} = P_{a}

If P is the pressure due to the liquid at point A, then total pressure at point A, P_{A} = P_{a} + P

By applying Bernoulli’s theorem, we get

\(\frac{1}{2} \rho v_A ^2+\rho g h_1+P_a=\frac{1}{2} \rho v_B^2+\rho g\left(h+h_1\right)+P_B\)or, \(\rho g h_1+P_a+P=\rho g h+\rho g h_1+P_a\)

or, ρ = hρg

**Question 21. Why is it dangerous to stand near a fast-moving train?**

**Answer:**

One should not stand near a fast-moving train. The air near the train starts flowing at a very high speed due to the high speed of the train. Consequently, pressure in that region decreases compared to the air pressure of the surrounding region.

This excess surrounding pressure behind the person tends to push the person towards the train and may cause a serious accident.

**Question 22. Why is the tin roof shade of a house blown away during stormy wind?**

**Answer:**

When there is a stormy wind, velocity of air and hence, its kinetic energy increases and thus pressure of air decreases. As the air inside the room remains more or less still, atmospheric pressure becomes greater than the outside.

As a result, there is an upward thrust on the roof shade. When this force is more than a specific value, the roof shade is blown away.

**Question 23. Why does the velocity of water through a pipe increase if the nozzle of the pipe is slightly closed by a finger?**

**Answer:**

If the nozzle of the pipe is slightly closed by a finger, a stream of waterfalls at a greater distance. This means velocity of water stream has increased. According to the equation of continuity, av = constant, where a is the area of cross-section and v is the velocity of the fluid.

When the nozzle of the pipe is closed by a finger, area of the cross-section of the pipe decreases, causing the velocity of the water stream to increase.

**Question 24. If you blow between two pages of a book, the pages stick together instead of spreading apart – explain with reason.**

**Answer:**

If air is blown between two pages of a book, speed of air increases between them. Hence, according to Bernoulli’s theorem air pressure between the pages becomes less than outside. Due to this difference in pressure, the pages come closer and stick together.

**Question 25. If two boats in a river move side by side, they tend to come closer – explain with reason.**

**Answer:**

If two boats in a river move side by side, the speed of water between the boats becomes more than that of the other sides. According to Bernoulli’s theorem, the pressure at the other sides of the boats becomes more than that between them. Therefore the boats tend to come closer.

## Chapter 3 Topic C Viscosity And Bernoulli’s Theorem Very Short Answer Type Questions Choose The Correct Answer

Question 1. On a plane horizontal surface, some amount of water, shampoo, and tar are poured slowly. They move with different speeds and the tar stops at first. This is because, among these three, tar has the minimum

- Viscosity
- Surface tension
- Fluidity
- Elasticity

Answer: 1. Viscosity

Question 2. Which of the following quantities is similar to friction?

- Viscosity
- Surface tension
- Buoyancy
- All of these

Answer: 1. Viscosity

Question 3. Which of the following properties is applicable only for a flowing liquid material?

- Elasticity
- Surface tension
- Malleability
- Viscosity

Answer: 4. Viscosity

Question 4. Which of the following statements is incorrect?

- Like friction, viscosity is a force against the motion
- More the viscosity of a liquid, lesser is its mobility
- After reaching its terminal velocity, a body starts falling with uniform velocity
- If the area of a liquid surface decreases, value of viscous force also increases

Answer: 4. If the area of a liquid surface decreases, value of viscous force also increases

Question 5. If there is a laminar flow of a liquid through a pipe similar to the shape of a right circular cylinder, the streamlines look like

**Answer: **1.

Question 6. Laminar flow of a liquid through a pipe of a non-uniform shape is shown. If velocities of the flowing liquid at points A and B are given by V_{A} and V_{B} respectively, then

- V
_{A}> V_{B} - V
_{A}< V_{B} - V
_{A}= V_{B} - Cannot be determined

Answer: 1. V_{A} > V_{B}

Question 7. There is laminar flow of a liquid through a horizontal tube of non-uniform shape. Velocity of liquid is V at the point where area of the cross section is A. What is the velocity at the point where the cross-section is A/2?

- V/2
- V
- 2V
- 4V

Answer: 3. 2V

Question 8. lf the velocity of water through a pipe is 1 m/s, velocity head is [g = 10 m/s^{2}]

- 1 cm
- 2 cm
- 4 cm
- 5 cm

Answer: 4. 5 cm

Question 9. Stir a liquid kept in a vessel and then leave it to itself. After some time the motion subsides because of

- Viscosity
- Surface tension
- Elasticity
- Buoyancy

Answer: 1. Viscosity

Question 10. Bernoulli’s theorem is based on the law of

- Conservation of momentum
- Conservation of mass
- Conservation of angular momentum
- Conservation of energy

Answer: 4. Conservation of energy

Question 11. Action of a sprayer depends on

- Bernoulli’s theorem
- Jurin’s law
- Avogadro’s theorem
- Stoke’s law

Answer: 1. Bernoulli’s theorem

Question 12. A body falling through a viscous liquid attains the terminal velocity. Afterwards it falls with an acceleration equal to

- g
- 0
- -g
- g/2

Answer: 2. 0

Question 13. Bernoulli’s theorem is applicable for

- Viscous fluid
- Nonviscous fluid
- Incompressible and nonviscous fluid
- Compressible fluid

Answer: 3. Incompressible and nonviscous fluid

## Chapter 3 Topic C Viscosity And Bernoullis Theorem Answer In Brief

**Question 1. Bernoulli’s theorem is established on which conservation law?**

**Answer:** Bernoulli’s theorem is established on the law of conservation of energy.

**Question 2. If temperature is increased, does the viscosity of a liquid increase or decrease?**

**Answer:** Viscosity of a liquid decreases, if temperature is increased.

**Question 3. If temperature is increased, does the viscosity of a gas increase or decrease?**

**Answer:** Viscosity of a gas increases, if temperature is increased.

**Question 4. Do two streamlines ever intersect with each other?**

**Answer:** No, two streamlines never intersect with each other.

**Question 5. What type of a fluid gives rise to a whirlwind?**

**Answer:** Turbulent flow of a fluid gives rise to a whirlwind.

**Question 6. What is the velocity of a liquid layer in contact with the bottom surface when laminar flow of liquid takes place over a firmly fixed horizontal plane?**

**Answer:** Velocity of the liquid layer in contact with the bottom surface is zero.

**Question 7. What is that velocity called when a body falls through a viscous medium with maximum uniform velocity?**

**Answer:** That velocity is called terminal velocity.

**Question 8. If a body is moving with terminal velocity through a viscous medium and density of the body is greater than the density of the medium, then what is the direction of terminal velocity?**

**Answer:** As the density of the body is greater than the density of the medium, direction of terminal velocity is perpendicularly downward.

**Question 9. lf a body is moving with terminal velocity through a long viscous medium and density of the body is less than the density of the medium, then what the direction of terminal velocity?**

**Answer:** As the density of the body is less than the density of the medium, direction of terminal velocity is perpendicularly upward.

**Question 10. What is the relationship between viscosity of a liquid and its mobility?**

**Answer:** When viscosity of a liquid increases, its mobility decreases.

**Question 11. Bernoulli’s theorem is fully applicable for what type of fluid?**

**Answer:** Bernoulli’s theorem is fully applicable for an ideal fluid.

**Question 12. What are the characteristics of an ideal fluid?**

**Answer:** An ideal fluid is incompressible and non-viscous.

**Question 13. When a car is running very fast, it is found that light polythene packets, etc. keep flying behind the running car. This phenomenon takes place due to which principle?**

**Answer:** This happens due to Bernoulli’s theorem.

## Chapter 3 Topic C Viscosity And Bernoullis Theorem Fill In the Blanks

Question 1. Water is ______ viscous than kerosene.

Answer: More

Question 2. ________ of a fluid is called its internal friction.

Answer: Viscosity

Question 3. When viscosity of a liquid increases, its _______ decreases.

Answer: Mobility

Question 4. When a small ball of iron falls through water, three forces act on it, namely gravitational force, buoyant force and ________ force.

Answer: Viscous

Question 5. Water is _______ viscous than kerosene.

Answer: More

Question 6. Ideal fluid has no _______

Answer: Viscosity

Question 7. The maximum velocity of a fluid, up to which the flow of the fluid is ________ and beyond which the flow becomes _________ is regarded as the critical velocity for that fluid.

Answer: Streamline, turbulent

Question 8. Two _________ never intersect each other.

Answer: Streamlines

Question 9. Raindrops fall to the ground with _______ velocity.

Answer: Terminal

## Chapter 3 Topic C Viscosity And Bernoullis Theorem State Whether True Or False

Question 1. Bernoulli’s theorem follows the law of conservation of energy.

Answer: True

Question 2. A smooth, uninterrupted flow in ordered layers, without any energy transfer between the layers is called laminar flow.

Answer: True

Question 3. Mobility of water is greater than that of honey.

Answer: True

Question 4. Two streamlines can intersect each other.

Answer: False

Question 5. Viscosity is called internal friction of a liquid.

Answer: True

Question 6. Viscosity of a fluid decreases with the rise in temperature.

Answer: True

Question 7. For a streamline flow of an fluid the sum of the velocity head, elevation head, and pressure head always remain constant at any point in the fluid.

Answer: True

Question 8. If the relative motion between the layers in contact in a flowing liquid decreases, viscosity decreases.

Answer: True

## Chapter 3 Topic C Viscosity And Bernoullis Theorem Numerical Examples

**Useful information**

If the cross-sectional area at any place of a tube is a and the velocity of the fluid at that place is v, then rate of flow of liquid at that place is av.

According to the equation of continuity av = constant.

If v = velocity of a liquid, h = height from any standard level, P = pressure, ρ = density of the liquid, g = acceleration due to gravity,

- according to Bernoulli’s theorem, \(\frac{v^2}{2 g}+h+\frac{p}{\rho g}=\) constant
- for horizontal flow of liquid, \(\frac{1}{2} \rho v^2+P\) = constant
- velocity head = \(\frac{v^2}{2 g}\), elevation head = h and pressure head = \(\frac{P}{\rho g}\)

**Question 1. Velocities of air below and above the surface of wings of a model plane are v and 3v, respectively. If the area of a wing is A and density of air is ρ, what is the dynamic lift?**

**Answer:**

Velocity of air below the surface of the wing of a model plane, v_{1} = v, and velocity of air above the surface of wing of a model plane, v_{2} = 3v.

Suppose, pressure of air below and above the surface of wing are P_{1} and P_{2}, respectively.

If we take the wing as horizontal and thickness of wing as negligible, then by Bernoulli’s theorem,

\(\frac{1}{2} \rho v_1^2+P_1=\frac{1}{2} \rho v_2^2+P_2\)or, \(\frac{1}{2} \rho v^2+P_1=\frac{1}{2} \rho \cdot 9 v^2+P_2\)

Hence the dynamic lift,

F = (P_{1} – P_{2}) . A = 4ρv^{2}A

**Question 2. Water flows through a horizontal pipe. At one point of the pipe, velocity of water is v and its pressure is P. If velocity of water at another point having same height as the first is 2v, then what is the pressure at that point?**

**Answer:**

At the first point, velocity of water, v_{1} = v and pressure of water, P_{1} = P.

At the second point, velocity of water, v_{2} = 2v.

Suppose, pressure of water at the second point be P_{2}.

Since both the points are situated at the same height, so according to Bernoulli’s theorem,

\(\frac{1}{2} \rho v_1^2+P_1=\frac{1}{2} \rho v_2^2+P_2\)or, \(P_2=P_1+\frac{1}{2} \rho\left(v_1^2-v_2^2\right)=P+\frac{1}{2} \rho\left(v^2-4 v^2\right)\)

= \(P-\frac{3}{2} \rho v^2\)

**Question 3. Water is flowing through a horizontal pipe with non-uniform cross section. At two points A and B inside the pipe which are at the same height, velocity of water are 20 cm/s and 50 cm/s, respectively. What is the difference of pressure between the points A and B?**

**Answer:**

Velocity of water at point A, v_{1} = 20 cm/s

The velocity of water at point B, v_{2} = 50 cm/s

Density of water, ρ = 1 g/cm^{3}

Suppose, pressure at points A and B are P_{1} and P_{2}, respectively.

Since points A and B are at the same height, then according to Bernoulli’s theorem,

\(\frac{1}{2} \rho v_1^2+P_1=\frac{1}{2} \rho v_2^2+P_2\)or, \(P_1-P_2=\frac{1}{2} \rho\left(v_2^2-v_1^2\right)=\frac{1}{2} \times 1 \times\left(50^2-20^2\right)\)

or, \(P_1-P_2=1050 \mathrm{dyn} / \mathrm{cm}^2\)

**Question 4. Find the velocity of flow of water at a point where the velocity head is 0.4 m.**

**Answer:**

Velocity head =\(\frac{v^2}{2 g}\), where, v = velocity of water and g = acceleration due to gravity.

Here, \(0.4=\frac{v^2}{2 \times 9.8}\)

or, v = \(\sqrt{0.4 \times 2 \times 9.8}=2.8 \mathrm{~m} / \mathrm{s}\)

∴ The velocity of flow of water is 2.8 m/s.