Notes : Pressure Exerted by a Liquid Column (Hydrostatic Pressure)

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Learn Pressure Exerted by a Liquid Column (Hydrostatic Pressure) with complete NCERT Class 11 Physics notes. Includes derivation of P = ρgh, formula, applications, blood pressure, FAQs, MCQs, true/false, fill in the blanks, and short  long answer questions.

Pressure Exerted by a Liquid Column (Hydrostatic Pressure)

Liquids at rest exert pressure on every surface in contact with them. This pressure acts normally (perpendicular) to the surface and increases with depth because the weight of the liquid above the point increases. The pressure exerted by a liquid due to its own weight is called hydrostatic pressure.

What is Pressure?

Pressure is defined as the normal force acting per unit area of a surface.

Mathematical Expression

$$P=\frac{F}{A}$$

Where

  • P = Pressure (Pa)
  • F = Force (N)
  • A = Area (m²)

SI Unit: Pascal (Pa)

$$1\;Pa=1\;N/m^2$$

Pressure Exerted by a Liquid

A liquid at rest exerts pressure on the bottom and walls of its container. The pressure is produced due to the weight of the liquid column above the point under consideration. Therefore, pressure increases as the depth increases.

Derivation of Pressure Exerted by a Liquid Column

Consider a vessel filled with a liquid of density ρ. Let an imaginary liquid column of height h and cross-sectional area A be drawn inside the liquid.

Given

  • Cross-sectional area = A
  • Height (Depth) = h
  • Density of liquid = ρ
  • Acceleration due to gravity = g

Step 1: Volume of the Liquid Column

The volume of a cylindrical liquid column is

$$V=Ah$$

Step 2: Mass of the Liquid Column

Mass is equal to density multiplied by volume.

$$m=\rho V$$

Substituting the value of volume,

$$m=\rho Ah$$

Step 3: Weight of the Liquid Column

Weight is given by

$$F=mg$$

Substituting the value of mass,

$$F=\rho Ahg$$

This force acts on the area A due to the weight of the liquid column.

Step 4: Pressure on the Bottom Surface

Pressure is defined as force per unit area.

$$P=\frac{F}{A}$$

Substituting the value of force,

$$P=\frac{\rho Ahg}{A}$$

Cancelling A,

$$\boxed{P=\rho gh}$$

Final Expression

The pressure exerted by a liquid column of height h is

$$\boxed{P=\rho gh}$$

Meaning of Symbols

Symbol Meaning SI Unit
P Pressure Pascal (Pa)
ρ Density of Liquid kg m-3
g Acceleration due to Gravity m s-2
h Depth of Liquid m

Factors Affecting Liquid Pressure

1. Depth of Liquid

Pressure is directly proportional to depth.

$$P\propto h$$

Greater the depth, greater will be the pressure.

2. Density of Liquid

Pressure is directly proportional to density.

$$P\propto \rho$$

A denser liquid exerts greater pressure.

3. Acceleration Due to Gravity

Pressure is directly proportional to gravitational acceleration.

$$P\propto g$$

Factors on Which Liquid Pressure Does Not Depend

  • Shape of the container
  • Size of the container
  • Cross-sectional area of the container
  • Amount of liquid (if depth remains the same)

Important Conclusions

  • Pressure increases with depth.
  • Pressure increases with density.
  • Pressure increases with gravitational acceleration.
  • Pressure is the same at all points lying on the same horizontal level.
  • Pressure is independent of the shape of the container.

Applications of Hydrostatic Pressure

  • Dams are made thicker at the bottom.
  • Divers experience greater pressure at greater depths.
  • Water tanks are installed at a height to provide sufficient water pressure.
  • Hydraulic machines work based on pressure transmission.
  • Submarines are designed to withstand high water pressure.

Relevant Question

Why is blood pressure greater at the feet than at the brain?

The pressure due to a liquid column is given by

$$P=\rho gh$$

The height of the blood column is greater at the feet than at the brain. Since pressure is directly proportional to height, blood pressure is greater at the feet than at the brain.

Blood Pressure

Blood pressure is commonly measured in millimetres of mercury (mm Hg).

  • Systolic Pressure: Approximately 120 mm Hg (during contraction of the heart).
  • Diastolic Pressure: Approximately 80 mm Hg (during relaxation of the heart).

Normal blood pressure:

$$\boxed{120/80\ \text{mm Hg}}$$

Frequently Asked Questions (FAQs)

1. What is liquid pressure?

Liquid pressure is the normal force exerted by a liquid per unit area on any surface in contact with it.

2. What is the formula for pressure exerted by a liquid column?

$$P=\rho gh$$

3. On which factors does liquid pressure depend?

  • Depth of the liquid ($h$)
  • Density of the liquid ($\rho$)
  • Acceleration due to gravity ($g$)

4. Does liquid pressure depend on the shape of the container?

No. Liquid pressure depends only on depth, density, and gravity. It is independent of the shape and size of the container.

5. Why does pressure increase with depth?

As depth increases, the height and weight of the liquid column above the point increase, resulting in greater pressure.

6. What is the SI unit of pressure?

The SI unit of pressure is Pascal (Pa).

7. Why is blood pressure greater at the feet than at the brain?

The height of the blood column is greater at the feet. Therefore, according to $$P=\rho gh$$ blood pressure is greater at the feet.

8. What is the normal blood pressure of a healthy adult?

The normal blood pressure is 120/80 mm Hg.

Multiple Choice Questions (MCQs)

  1. The pressure exerted by a liquid column is

    A. $P=\rho h$
    B. $P=\rho gh$
    C. $P=gh$
    D. $P=\rho g$

    Answer: B

  2. The SI unit of pressure is

    A. Newton
    B. Joule
    C. Pascal
    D. Watt

    Answer: C

  3. Liquid pressure increases with

    A. Decrease in depth
    B. Increase in depth
    C. Shape of the container
    D. Area of the container

    Answer: B

  4. Pressure at the same depth in a liquid is

    A. Different
    B. Zero
    C. Same in all directions
    D. Depends on the container shape

    Answer: C

  5. Which factor does not affect liquid pressure?

    A. Density
    B. Gravity
    C. Depth
    D. Shape of the container

    Answer: D

  6. Blood pressure is commonly measured in

    A. Pascal
    B. Newton
    C. mm of Hg
    D. Bar

    Answer: C

  7. The pressure at the bottom of a vessel is

    A. Minimum
    B. Zero
    C. Maximum
    D. Equal to atmospheric pressure

    Answer: C

  8. According to $P=\rho gh$, pressure is directly proportional to

    A. Area only
    B. Height and density
    C. Shape of the vessel
    D. Volume only

    Answer: B

  9. If the density of a liquid doubles, the pressure at the same depth becomes

    A. Half
    B. Double
    C. One-fourth
    D. Unchanged

    Answer: B

  10. The pressure due to a liquid column is called

    A. Atmospheric pressure
    B. Hydrostatic pressure
    C. Gas pressure
    D. Osmotic pressure

    Answer: B

True or False

1. Liquid pressure increases with depth. True

2. Liquid pressure depends on the shape of the container. False

3. The SI unit of pressure is Pascal. True

4. Pressure at the same depth is equal in all directions. True

5. Denser liquids exert greater pressure. True

6. Blood pressure is measured in mm Hg. True

7. Pressure is inversely proportional to depth. False

8. Water pressure is maximum at the bottom of a container. True

9. Liquid pressure depends on gravitational acceleration. True

10. The expression for liquid pressure is $P=\rho gh$. True

Fill in the Blanks

  1. The pressure exerted by a liquid column is $\rho gh$.
  2. The SI unit of pressure is Pascal.
  3. Liquid pressure increases with depth.
  4. Pressure is directly proportional to the density of the liquid.
  5. Blood pressure is measured in mm Hg.
  6. The pressure during heart contraction is called systolic pressure.
  7. The pressure during heart relaxation is called diastolic pressure.
  8. Pressure is equal to force divided by area.
  9. The acceleration due to gravity is represented by g.
  10. Liquid pressure is independent of the shape of the container.

Very Short Answer Questions

1. Define pressure.

Answer: Pressure is the normal force acting per unit area.

2. Write the SI unit of pressure.

Answer: Pascal (Pa).

3. Write the formula for pressure exerted by a liquid column.

Answer:

$$P=\rho gh$$

4. Name the factors affecting liquid pressure.

Answer: Depth, density of the liquid, and acceleration due to gravity.

5. Does liquid pressure depend on the shape of the container?

Answer: No.

6. What is the SI unit of density?

Answer: kg m-3.

7. What is the normal blood pressure of a healthy adult?

Answer: 120/80 mm Hg.

8. Which instrument is used to measure blood pressure?

Answer: Sphygmomanometer.

Short Answer Questions

1. Derive the expression for pressure exerted by a liquid column.

Answer:

Consider a liquid column of cross-sectional area A, height h, density ρ, and acceleration due to gravity g.

Volume:

$$V=Ah$$

Mass:

$$m=\rho V=\rho Ah$$

Weight:

$$F=mg=\rho Ahg$$

Pressure:

$$P=\frac{F}{A}$$

Substituting the value of force,

$$P=\frac{\rho Ahg}{A}=\rho gh$$

Hence,

$$\boxed{P=\rho gh}$$

2. State the factors affecting liquid pressure.

Answer:

  • Depth of the liquid.
  • Density of the liquid.
  • Acceleration due to gravity.

3. Why does liquid pressure increase with depth?

Answer:

As depth increases, the weight of the liquid column above the point increases. Hence, the pressure also increases.

4. Why does liquid pressure not depend on the shape of the container?

Answer:

According to the equation

$$P=\rho gh$$

pressure depends only on the density of the liquid, depth, and gravity. It does not contain the shape or area of the container.

5. Explain why blood pressure is greater at the feet than at the brain.

Answer:

The height of the blood column is greater at the feet than at the brain. According to

$$P=\rho gh$$

pressure is directly proportional to height. Therefore, blood pressure is greater at the feet.

Long Answer Questions

1. Derive the expression for pressure exerted by a liquid column and explain the factors affecting it.

Answer:

Consider a liquid column of cross-sectional area A and height h.

Volume of liquid:

$$V=Ah$$

Mass of liquid:

$$m=\rho Ah$$

Weight of liquid:

$$F=\rho Ahg$$

Pressure:

$$P=\frac{F}{A}$$

Substituting,

$$P=\frac{\rho Ahg}{A}=\rho gh$$

Therefore,

$$\boxed{P=\rho gh}$$

Factors affecting liquid pressure:

  • Pressure is directly proportional to depth.
  • Pressure is directly proportional to density.
  • Pressure is directly proportional to gravitational acceleration.
  • Pressure is independent of the shape and size of the container.

2. Explain the applications of hydrostatic pressure.

Answer:

  • Dams are constructed thicker at the bottom because pressure is greatest there.
  • Divers experience higher pressure as they go deeper into water.
  • Water tanks are installed at greater heights to increase water pressure.
  • Hydraulic machines work using the transmission of pressure through liquids.
  • Submarines are designed to withstand large hydrostatic pressure at great depths.

Key Points for Revision

  • Pressure is force acting per unit area.
  • SI unit of pressure is Pascal (Pa).
  • The pressure exerted by a liquid column is called hydrostatic pressure.
  • The formula for hydrostatic pressure is:

$$\boxed{P=\rho gh}$$

  • Pressure increases with depth.
  • Pressure increases with density.
  • Pressure increases with gravitational acceleration.
  • Pressure is the same at all points on the same horizontal level.
  • Pressure does not depend on the shape or size of the container.
  • Blood pressure is measured in mm Hg.
  • Normal blood pressure of a healthy adult is 120/80 mm Hg.

NCERT Exam Points

  • Hydrostatic pressure depends only on depth, density, and gravity.
  • Pressure at the same depth is equal in all directions.
  • The pressure at the bottom of a liquid-filled container is maximum.
  • Liquids exert pressure normally on the walls and bottom of the container.
  • The expression for hydrostatic pressure is $P=\rho gh$.
  • The shape of the container does not affect liquid pressure.
  • Dams are made thicker at the base due to higher pressure at greater depths.
  • Blood pressure is higher at the feet because the height of the blood column is greater.

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