Notes : Variation of Pressure with Depth (Gauge Pressure) - PhysicsKund

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Variation of Pressure with Depth (Gauge Pressure) - Physicskund 

Introduction

When we dive deeper into a swimming pool or a river, we feel more pressure on our ears. This happens because the weight of the liquid above us increases with depth. Therefore, the pressure inside a liquid increases as we move deeper.

Variation of Pressure with Depth (Gauge Pressure) - Physicskund

Consider a Small Liquid Element

Consider a small cylindrical element of liquid of cross-sectional area A and height \(\Delta h\).

Let,

  • Pressure on the top face = \(p_1\)
  • Pressure on the bottom face = \(p_2\)
  • Density of the liquid = \(\rho\)

Since the liquid is at rest, the cylindrical element is also in equilibrium.

Forces Acting on the Liquid Element

1. Pressure Force on the Top Face

Pressure on the top surface is \(p_1\). Therefore, the downward force acting on the top face is

\[ F_1=p_1A \]

2. Pressure Force on the Bottom Face

Pressure on the bottom surface is \(p_2\). Therefore, the upward force acting on the bottom face is

\[ F_2=p_2A \]

Since the bottom face is at a greater depth,

\[ p_2>p_1 \]

3. Weight of the Liquid Element

The liquid element has mass, so its weight acts vertically downward.

\[ W=mg \]

Condition of Equilibrium

As the liquid is at rest,

Upward Force = Downward Forces

\[ F_2=F_1+W \]

Substituting the values,

\[ p_2A=p_1A+mg \]

Rearranging,

\[ (p_2-p_1)A=mg \]

Mass of the Liquid Element

Mass of the liquid element is

\[ m=\rho\times\text{Volume} \]

The volume of the cylindrical element is

\[ A\Delta h \]

Therefore,

\[ m=\rho A\Delta h \]

Substituting in the above equation,

\[ (p_2-p_1)A=\rho A\Delta hg \]

Cancelling \(A\) from both sides,

\[ p_2-p_1=\rho g\Delta h \]

or

\[ \boxed{\Delta p=\rho g\Delta h} \]

Interpretation of the Result

The equation

\[ \Delta p=\rho g\Delta h \]

shows that the increase in pressure depends on:

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

Thus, pressure increases as the depth increases.

Differential Form

For a very small change in depth,

\[ \Delta h\rightarrow0 \]

Therefore,

\[ \frac{dp}{dh}=\rho g \]

or

\[ dp=\rho g\,dh \]

Pressure at a Depth \(h\)

At the liquid surface,

\[ h=0,\qquad p=P_a \]

where \(P_a\) is the atmospheric pressure.

At a depth \(h\),

\[ p=p \]

Integrating,

\[ \int_{P_a}^{p}dp=\rho g\int_{0}^{h}dh \]

After integration,

\[ p-P_a=\rho gh \]

Hence,

\[ \boxed{p=P_a+\rho gh} \]

Gauge Pressure

The pressure produced only due to the liquid column is called Gauge Pressure.

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

Absolute Pressure

The total pressure inside the liquid is called Absolute Pressure.

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

Key Points

  • Pressure increases with depth.
  • Pressure is directly proportional to depth.
  • Pressure depends on the density of the liquid.
  • Pressure depends on acceleration due to gravity.
  • Pressure at the same depth is the same everywhere in a liquid.
  • Gauge Pressure is equal to \(\rho gh\).
  • Absolute Pressure is equal to \(P_a+\rho gh\).

Formula Box

\[ \boxed{\Delta p=\rho g\Delta h} \]

\[ \boxed{\frac{dp}{dh}=\rho g} \]

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

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

Frequently Asked Questions (FAQs)

Q1. Why does pressure increase with depth?

Pressure increases with depth because the weight of the liquid above increases.

Q2. What is gauge pressure?

Gauge pressure is the pressure exerted by the liquid column above atmospheric pressure.

Q3. What is absolute pressure?

Absolute pressure is the sum of atmospheric pressure and gauge pressure.

Q4. Does pressure depend on the shape of the container?

No. Pressure depends only on the depth, density of the liquid, and acceleration due to gravity.

Q5. Write the expression for pressure at a depth h.

\[ p=P_a+\rho gh \]

Multiple Choice Questions (MCQs)

  1. The pressure in a liquid increases with:

    (A) Area    (B) Depth    (C) Volume    (D) Shape

    Answer: (B) Depth

  2. The SI unit of pressure is:

    (A) Joule    (B) Newton    (C) Pascal    (D) Watt

    Answer: (C) Pascal

  3. Gauge pressure is equal to:

    (A) \(P_a+\rho gh\)

    (B) \(\rho gh\)

    (C) \(P_a\)

    (D) \(\rho g\)

    Answer: (B) \(\rho gh\)

  4. Pressure at the same depth in a liquid is:

    (A) Different    (B) Zero    (C) Same    (D) Infinite

    Answer: (C) Same

  5. Absolute pressure is equal to:

    (A) \(\rho gh\)

    (B) \(P_a\)

    (C) \(P_a+\rho gh\)

    (D) Zero

    Answer: (C) \(P_a+\rho gh\)

True or False

1. Pressure in a liquid increases with depth. Answer: True

2. Gauge pressure includes atmospheric pressure. Answer: False

3. Pressure at the same depth is the same everywhere in a liquid. Answer: True

4. Pressure depends on the shape of the container. Answer: False

5. Absolute pressure is always greater than gauge pressure. Answer: True

Fill in the Blanks

  1. Pressure in a liquid increases with depth.
  2. The SI unit of pressure is Pascal (Pa).
  3. Gauge pressure is equal to \(\rho gh\).
  4. Absolute pressure is equal to \(P_a+\rho gh\).
  5. Pressure at the same depth is equal at all points.

Very Short Answer Questions

Q1. What is pressure?

Pressure is the normal force acting per unit area.

Q2. What is the SI unit of pressure?

Pascal (Pa).

Q3. What is gauge pressure?

The pressure due to the liquid column above atmospheric pressure.

Q4. What is atmospheric pressure?

The pressure exerted by the atmosphere on the Earth's surface.

Q5. Write the formula for gauge pressure.

\[ P_g=\rho gh \]

Short Answer Questions

Q1. Why does pressure increase with depth?

Pressure increases with depth because the weight of the liquid column above increases, exerting a greater force per unit area.

Q2. Differentiate between gauge pressure and absolute pressure.

Gauge Pressure Absolute Pressure
\(\rho gh\) \(P_a+\rho gh\)
Excludes atmospheric pressure. Includes atmospheric pressure.

Q3. On what factors does pressure in a liquid depend?

Pressure depends on the density of the liquid, acceleration due to gravity, and depth.

Long Answer Questions

Q1. Derive the expression for variation of pressure with depth in a liquid.

Consider a small cylindrical liquid element in equilibrium. Applying the condition of equilibrium and substituting the mass of the liquid element, we obtain:

\[ p_2-p_1=\rho g\Delta h \]

For a liquid surface where pressure is atmospheric pressure, integration gives:

\[ p=P_a+\rho gh \]

Hence, the gauge pressure is:

\[ P_g=\rho gh \]

Q2. Explain the terms gauge pressure and absolute pressure with suitable formulae.

Gauge Pressure:

\[ P_g=\rho gh \]

Absolute Pressure:

\[ P=P_a+\rho gh \]

Absolute pressure is the total pressure inside the liquid, whereas gauge pressure is the pressure due only to the liquid column.

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