Notes : Open Tube Manometer - Derivation , Working , Formula

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Master Open Tube Manometer with these complete Class 11 Physics NCERT notes. Learn its definition, construction, working principle, derivation, formulas, gauge pressure, solved explanations, FAQs, MCQs, true/false, fill in the blanks, and very short, short, and long questions with answers. Ideal for CBSE, State Boards, JEE Main, NEET, and CUET preparation - physicskund 

Open Tube Manometer

An Open Tube Manometer is a pressure measuring device used to determine the pressure of a gas or liquid by comparing it with the atmospheric pressure. It consists of a U-shaped glass tube partially filled with a manometric liquid such as mercury or water. One end of the tube is connected to the vessel containing the fluid, while the other end remains open to the atmosphere.

Definition

An open tube manometer is a U-shaped pressure measuring instrument in which one limb is connected to the fluid whose pressure is to be measured and the other limb is open to the atmosphere. It measures the pressure difference between the fluid and the atmospheric pressure.

Construction

  • A transparent U-shaped glass tube.
  • The tube is partially filled with a manometric liquid (usually mercury).
  • One limb is connected to the vessel containing the gas or liquid.
  • The other limb is open to the atmosphere.
  • The atmospheric pressure acting on the open limb is represented by \(P_a\).
  • The pressure inside the vessel is represented by \(P\).
  • The difference in liquid levels is represented by \(h\).

Important Parts of the Figure

Symbol Meaning
\(P\) Pressure of the gas inside the vessel
\(P_a\) Atmospheric pressure acting on the open end
\(A\) Point inside the left limb
\(B\) Point inside the right limb
\(h\) Difference between liquid levels
\(\rho\) Density of the manometer liquid
\(g\) Acceleration due to gravity

Working Principle

The open tube manometer works on the principle of hydrostatic pressure.

Principle: Pressure at the same horizontal level in the same stationary liquid is always equal.

Therefore,

\[ P_A=P_B \]

This equation is the basis of the derivation of the pressure formula.

Case 1 : Pressure Inside the Vessel is Greater than Atmospheric Pressure

Open Tube Manometer

Suppose,

\[ P>P_a \]

Step 1

The gas inside the vessel exerts pressure on the liquid in the left limb.

Step 2

Since the gas pressure is greater than atmospheric pressure, it pushes the liquid downward in the connected limb.

Step 3

The liquid rises in the right limb because liquids transmit pressure equally.

  • Liquid level falls in the left limb.
  • Liquid level rises in the right limb.
  • The difference between the two levels becomes \(h\).

Pressure at Point A

Point A is directly under the gas pressure.

Therefore,

\[ P_A=P \]

Pressure at Point B

Atmospheric pressure acts on the open surface.

When we move downward through a liquid by a height \(h\), pressure increases by $ \rho gh$

Hence,

\[ P_B=P_a+\rho gh \]

Derivation of Pressure Equation

According to hydrostatic equilibrium,

\[ P_A=P_B \]

Substituting the values,

\[ P=P_a+\rho gh \]

Subtracting atmospheric pressure from both sides,

\[ P-P_a=\rho gh \]

The quantity

\[ P-P_a \]

is called the Gauge Pressure.

Therefore,

\[ \boxed{P_g=\rho gh} \]

Why Does Pressure Increase by \(\rho gh\)?

Whenever we move downward inside a liquid:

  • The amount of liquid above us increases.
  • The weight of the liquid column increases.
  • Greater weight produces greater pressure.
  • The increase in pressure is given by $\rho gh$

Case 2 : Pressure Inside the Vessel is Less than Atmospheric Pressure

Open Tube Manometer

Suppose,

$P$ < $P_a$

In this case atmospheric pressure is greater than the gas pressure.

  • Atmospheric pressure pushes the liquid downward in the open limb.
  • The liquid rises in the connected limb.
  • The level difference remains \(h\).

Again,

\[ P_A=P_B \]

Pressure at point A,

\[ P_A=P \]

Pressure at point B,

\[ P_B=P_a-\rho gh \]

Therefore,

\[ \boxed{P=P_a-\rho gh} \]

Formula Summary

Condition Formula
Gas pressure greater than atmospheric pressure \[ P=P_a+\rho gh \]
Gas pressure less than atmospheric pressure \[ P=P_a-\rho gh \]
Gauge pressure \[ P_g=P-P_a=\rho gh \]

Applications

  • Measurement of gas pressure.
  • Measurement of liquid pressure.
  • Calibration of pressure gauges.
  • Laboratory experiments.
  • Hydraulic system testing.
  • Fluid mechanics experiments.

Advantages

  • Simple construction.
  • Easy to operate.
  • Highly accurate.
  • No external power supply required.
  • Low maintenance.

Limitations

  • Not suitable for measuring very high pressures.
  • Requires a long tube for large pressure differences.
  • Mercury is toxic and must be handled carefully.
  • Not convenient for portable applications.

Frequently Asked Questions (FAQs)

Q1. What is an Open Tube Manometer?

An open tube manometer is a U-shaped pressure measuring device used to measure the pressure of a gas or liquid by comparing it with atmospheric pressure.

Q2. On which principle does an Open Tube Manometer work?

It works on the principle of hydrostatic pressure, which states that pressure at the same horizontal level in the same stationary liquid is equal.

\[ P_A=P_B \]

Q3. Why is one end of the manometer kept open?

One end is kept open so that atmospheric pressure acts on the liquid surface, allowing comparison between the gas pressure and atmospheric pressure.

Q4. What does the height difference \(h\) represent?

The quantity \(h\) represents the difference in the liquid levels of the two limbs due to the pressure difference.

Q5. What is Gauge Pressure?

Gauge pressure is the pressure above atmospheric pressure.

\[ P_g=P-P_a=\rho gh \]

Q6. Which liquid is commonly used in an Open Tube Manometer?

Mercury is commonly used because of its high density, low vapour pressure, and because it does not wet glass.

Q7. What happens if the gas pressure is greater than atmospheric pressure?

The liquid level falls in the connected limb and rises in the open limb.

Q8. What happens if the gas pressure is less than atmospheric pressure?

The liquid level rises in the connected limb and falls in the open limb.

Multiple Choice Questions (MCQs)

  1. An Open Tube Manometer is used to measure

    A. Absolute pressure
    B. Gauge pressure
    C. Vacuum only
    D. Atmospheric pressure only

    Answer: B. Gauge pressure

  2. The open end of the manometer is exposed to

    A. Vacuum
    B. Gas pressure
    C. Atmospheric pressure
    D. Liquid pressure

    Answer: C. Atmospheric pressure

  3. The pressure difference measured by an Open Tube Manometer is

    A. \(\rho h\)
    B. \(\rho gh\)
    C. \(gh\)
    D. \(\rho g\)

    Answer: B. \(\rho gh\)

  4. If the gas pressure is greater than atmospheric pressure, the liquid level in the connected limb

    A. Rises
    B. Falls
    C. Remains unchanged
    D. Overflows

    Answer: B. Falls

  5. The SI unit of pressure is

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

    Answer: C. Pascal

  6. Pressure at the same horizontal level in a stationary liquid is

    A. Zero
    B. Equal
    C. Maximum
    D. Minimum

    Answer: B. Equal

  7. Gauge pressure is equal to

    A. \(P+P_a\)
    B. \(P-P_a\)
    C. \(P_a-P\)
    D. \(\rho g\)

    Answer: B. \(P-P_a\)

  8. The commonly used manometer liquid is

    A. Alcohol
    B. Mercury
    C. Petrol
    D. Kerosene

    Answer: B. Mercury

Fill in the Blanks

  1. An Open Tube Manometer compares pressure with atmospheric pressure.
  2. The pressure acting on the open limb is represented by \(P_a\).
  3. The pressure difference is measured using the height \(h\).
  4. Pressure at the same horizontal level in a stationary liquid is equal.
  5. Gauge pressure is given by \(P_g=\rho gh\).
  6. The SI unit of pressure is Pascal (Pa).
  7. Mercury is preferred because of its high density.
  8. Pressure increases when moving downward inside a liquid.
  9. The density of the manometer liquid is represented by \(\rho\).
  10. The acceleration due to gravity is represented by \(g\).

True or False

1. An Open Tube Manometer measures gauge pressure.

Answer: True

2. Pressure at the same horizontal level in a stationary liquid is equal.

Answer: True

3. Atmospheric pressure acts on the open limb of the manometer.

Answer: True

4. Gauge pressure is equal to \(P+P_a\).

Answer: False

5. Mercury is commonly used as the manometer liquid.

Answer: True

6. Pressure decreases while moving downward in a liquid.

Answer: False

7. If \(P>P_a\), the liquid level rises in the connected limb.

Answer: False

8. The pressure difference depends on \(\rho\), \(g\), and \(h\).

Answer: True

9. One limb of the Open Tube Manometer remains open to the atmosphere.

Answer: True

10. The SI unit of pressure is Joule.

Answer: False

11. Gauge pressure can be negative when the gas pressure is less than atmospheric pressure.

Answer: True

12. An Open Tube Manometer is mainly used to compare pressure with atmospheric pressure.

Answer: True

Very Short Answer Questions (1 Mark)

Q1. What is an Open Tube Manometer?

An Open Tube Manometer is a U-shaped device used to measure the pressure of a gas or liquid by comparing it with atmospheric pressure.

Q2. What is the SI unit of pressure?

The SI unit of pressure is Pascal (Pa).

Q3. What is atmospheric pressure?

Atmospheric pressure is the pressure exerted by the Earth's atmosphere on all objects. It is represented by \(P_a\).

Q4. What is Gauge Pressure?

Gauge pressure is the excess pressure over atmospheric pressure.

\[ P_g=P-P_a=\rho gh \]

Q5. Which liquid is commonly used in an Open Tube Manometer?

Mercury is commonly used as the manometer liquid.

Q6. What does the symbol \(h\) represent?

It represents the difference in the liquid levels of the two limbs.

Q7. Which principle is used in an Open Tube Manometer?

It works on the principle that pressure at the same horizontal level in the same stationary liquid is equal.

Q8. Write the pressure equation when gas pressure is greater than atmospheric pressure.

\[ P=P_a+\rho gh \]

Short Answer Questions (2 –3 Marks)

Q1. Explain the construction of an Open Tube Manometer.

Answer:

  • It consists of a transparent U-shaped glass tube.
  • The tube is partially filled with mercury or another manometric liquid.
  • One limb is connected to the vessel containing the gas.
  • The other limb remains open to the atmosphere.
  • The pressure difference is measured from the difference in liquid levels.

Q2. Explain the working principle of an Open Tube Manometer.

Answer:

The manometer works on the hydrostatic principle that pressure at the same horizontal level in the same stationary liquid is equal.

\[ P_A=P_B \]

The difference in liquid levels produces the pressure difference, allowing the gas pressure to be calculated.

Q3. Why is mercury preferred as the manometer liquid?

Answer:

  • High density.
  • Low vapour pressure.
  • Does not wet glass.
  • Produces a shorter liquid column.
  • Provides accurate measurements.

Q4. What happens when the gas pressure is greater than atmospheric pressure?

Answer:

  • The gas pushes the liquid downward in the connected limb.
  • The liquid rises in the open limb.
  • A height difference \(h\) is produced.
  • The pressure equation becomes

\[ P=P_a+\rho gh \]

Q5. What happens when the gas pressure is less than atmospheric pressure?

Answer:

  • Atmospheric pressure becomes greater than the gas pressure.
  • The liquid rises in the connected limb.
  • The liquid falls in the open limb.
  • The pressure equation becomes

\[ P=P_a-\rho gh \]

Long Answer Questions (5 Marks)

Q1. Explain the construction, working and derivation of an Open Tube Manometer.

Answer:

An Open Tube Manometer is a U-shaped glass tube partially filled with mercury or another manometric liquid. One limb is connected to the vessel whose pressure is to be measured, while the other limb is open to the atmosphere.

Suppose the pressure inside the vessel is \(P\) and the atmospheric pressure is \(P_a\).

If

\[ P>P_a \]

the gas pushes the liquid downward in the connected limb, causing the liquid to rise in the open limb. The difference in the liquid levels is \(h\).

Choose two points A and B at the same horizontal level.

According to hydrostatic equilibrium,

\[ P_A=P_B \]

Pressure at point A,

\[ P_A=P \]

Pressure at point B,

\[ P_B=P_a+\rho gh \]

Therefore,

\[ P=P_a+\rho gh \]

Subtracting \(P_a\),

\[ P-P_a=\rho gh \]

Hence,

\[ P_g=\rho gh \]

Thus, the pressure of the gas can be determined by measuring the height difference \(h\).

Q2. Derive the equation of an Open Tube Manometer.

Answer:

At the same horizontal level,

\[ P_A=P_B \]

Pressure at A,

\[ P_A=P \]

Pressure at B,

\[ P_B=P_a+\rho gh \]

Equating the pressures,

\[ P=P_a+\rho gh \]

Therefore,

\[ P-P_a=\rho gh \]

The gauge pressure is

\[ P_g=\rho gh \]

Q3. Write the advantages and limitations of an Open Tube Manometer.

Advantages:

  • Simple construction.
  • Easy to use.
  • Highly accurate.
  • No external power supply required.
  • Low maintenance cost.

Limitations:

  • Not suitable for very high pressures.
  • Requires a long tube for large pressure differences.
  • Mercury is toxic.
  • Not convenient for portable applications.

Important Formula Revision

Pressure greater than atmospheric pressure

\[ P=P_a+\rho gh \]

Pressure less than atmospheric pressure

\[ P=P_a-\rho gh \]

Gauge Pressure

\[ P_g=P-P_a=\rho gh \]

Hydrostatic Principle

\[ P_A=P_B \]

Exam Tips (NCERT)

  • Always choose two points at the same horizontal level.
  • Write the pressure at each point separately.
  • Use the relation \(P_A=P_B\).
  • Substitute the pressure expressions correctly.
  • Remember that pressure increases by \(\rho gh\) when moving downward in a liquid.
  • Write the final equation with proper units and symbols.
  • For derivation questions, clearly mention the assumptions and define every symbol used.

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