Lesson 5.2: Fluid Statics (Manometry, Buoyancy, Stability of Floating Bodies)

Fluid Statics deals with fluids at rest and is essential for designing dams, ships, tanks, and pressure measurement systems. GATE and PSU exams often test manometry, buoyancy, and stability concepts.

🔹 1. Introduction

• Definition: Fluid statics studies pressure distribution in fluids at rest.

• Applications: Hydraulic structures, storage tanks, ships, submarine design

• Key Concepts:

  1. Pressure variation with depth: p=p0+ρghp = p_0 + \rho g hp=p0​+ρgh

  2. Hydrostatic force on surfaces

  3. Centre of pressure

🔹 2. Manometry

• Definition: Measurement of fluid pressure using a manometer.

• Types of Manometers:

  1. U-Tube Manometer → simple pressure difference

  2. Inclined Manometer → precise measurement for low pressures

  3. Differential Manometer → measures pressure difference between two points

• Formula (U-Tube):

$$\Delta p=\rho gh$$

Where h = height difference in fluid column

• Example: Find pressure difference using mercury U-tube manometer with 0.2 m difference (ρ=13600kg/m3\rho = 13600 kg/m^3ρ=13600kg/m3).

$$\Delta p=\rho gh=13600\ast 9.81\ast 0.2\approx 26.7kPa$$

🔹 3. Buoyancy

• Definition: Upthrust on a body submerged or floating in fluid

FB=ρgVF_B = \rho g VFB​=ρgV

Where V = volume of displaced fluid

• Floating Body: Weight = Buoyant force

• Submerged Body: Buoyant force acts upward, fully supports body if weight ≤ FBF_BFB​

• Example: Volume 0.05 m³ of wood floats in water. Weight = 400 N. Buoyant force = $\rho gV=1000\ast 9.81\ast 0.05=490.5N$ → body floats

🔹 4. Stability of Floating Bodies

• Metacentre (M): Point about which floating body oscillates when tilted

• Stability Criteria:

  1. Stable: Metacentre above center of gravity (G) → body returns to original position

  2. Unstable: M below G → body overturns

  3. Neutral: M coincides with G → body remains tilted

• Metacentric Height (GM): Measure of stability

$$GM=BM-BG$$

Where BM = metacentric radius, BG = distance from center of buoyancy to center of gravity

• Example: A ship with GM = 1.2 m → stable. If GM < 0 → unstable.

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2016):
A U-tube manometer shows mercury level difference 0.25 m. Calculate pressure difference.

$$\Delta p=13600\ast 9.81\ast 0.25\approx 33.4kPa$$

Example 2 (PSU Exam):
Floating rectangular block, weight 500 N, volume 0.06 m³, submerged in water. Determine stability using GM = 0.3 m → stable.

🔹 6. Practice Exercises

1. Calculate pressure difference using U-tube and inclined manometers.

2. Determine buoyant force on submerged and floating bodies.

3. Find metacentric height and assess stability of floating ship.

4. Compute centre of pressure on vertical submerged surface.

5. Explain differences between stable, unstable, and neutral equilibrium.

🔹 7. Summary

• Manometry: Measures fluid pressure, U-tube and differential types

• Buoyancy: Upthrust = weight of displaced fluid, floating vs submerged

• Stability: Metacentre, metacentric height, stable/unstable/neutral equilibrium

• Applications: Ships, tanks, submarines, hydraulic structures