Unit 10 - Grade 11-12 Physics

Fluid Mechanics

Study how liquids and gases behave through density, pressure, buoyancy, hydraulic systems, fluid flow, continuity, Bernoulli's principle, viscosity, and real-world applications in engineering and natural systems.

Lesson roadmap

What Students Should Master in This Unit

Fluid mechanics explains how liquids and gases exert pressure, create buoyant forces, move through pipes, and exchange pressure, speed, and height. This unit connects forces, energy, density, and real-world engineering applications.

Describe fluid properties

Use density, specific gravity, pressure, compressibility, viscosity, and flow rate correctly.

Analyze pressure and buoyancy

Apply hydrostatic pressure, Pascal's principle, Archimedes' principle, and floating rules.

Model moving fluids

Use continuity and Bernoulli's equation to connect area, speed, pressure, and height.

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Jump to a Topic

Fluid Properties Density and Specific Gravity Pressure Hydrostatic Pressure Pascal's Principle Buoyancy Floating and Sinking Fluid Flow Continuity Equation Bernoulli's Principle Viscosity and Drag Simulation Labs Lab Skills Worked Examples Practice Problems
Liquids and gases

1. What Is a Fluid?

A fluid is a substance that can flow. Liquids and gases are fluids because they do not keep a fixed shape. They deform continuously when forces act on them.

What makes liquids and gases fluids Fluids flow and take the shape of their containers liquid: fixed volume gas: fills container
Liquids keep nearly the same volume but change shape, while gases change shape and volume to fill available space.
Liquid Fixed volume, changing shape Usually treated as nearly incompressible in Grade 11-12 physics.
Gas Changing volume and shape Compressible and expands to fill its container.
Ideal fluid model incompressible, nonviscous, steady flow A simplified model used for continuity and Bernoulli problems.

Important Fluid Vocabulary

  • Density: mass per unit volume.
  • Pressure: force per unit area.
  • Buoyant force: upward force a fluid exerts on an immersed object.
  • Viscosity: internal resistance to flow.
  • Steady flow: fluid speed at each point stays constant over time.
  • Laminar flow: smooth layered flow.
  • Turbulent flow: chaotic flow with swirls and eddies.
Mass packed into volume

2. Density and Specific Gravity

Density tells how much mass is contained in a certain volume. It is one of the main properties used to predict whether objects float or sink.

Density compares mass packed into volume Same volume, different mass means different density lower density higher density ρ = m / V density tells how tightly mass is packed
For the same volume, the sample with more mass has greater density and is more likely to sink in a less dense fluid.
Density ρ = m / V ρ is density, m is mass, V is volume.
Mass from density m = ρV Useful when volume and density are known.
Specific gravity SG = ρobject / ρwater No units. Water has SG = 1.

Common Densities

Substance Approximate Density Meaning
Air1.2 kg/m3Much less dense than liquids.
Ice917 kg/m3Less dense than liquid water, so it floats.
Oil900 kg/m3Usually floats on water.
Fresh water1000 kg/m3Common reference density.
Seawater1025 kg/m3Slightly denser than fresh water.
Aluminum2700 kg/m3Sinks as a solid block in water.
Mercury13600 kg/m3Very dense liquid metal.
Force spread over area

3. Pressure

Pressure is force divided by area. The same force creates greater pressure when applied over a smaller area.

Pressure depends on contact area The same force creates more pressure on a smaller area F large area, lower pressure same F small area, higher pressure P = F / A
Pressure increases when the same force is concentrated over a smaller surface area.
Pressure P = F / A Unit: pascal, Pa = N/m2.
Force from pressure F = PA Pressure acts perpendicular to a surface.
Atmospheric pressure 1 atm = 1.013 × 105 Pa Often rounded to 101 kPa.

Pressure Types

  • Absolute pressure: total pressure measured relative to a vacuum.
  • Gauge pressure: pressure above atmospheric pressure.
  • Vacuum pressure: pressure below atmospheric pressure.
Real-world idea: Snowshoes reduce pressure on snow by increasing contact area. Sharp knives increase pressure by decreasing contact area.
Pressure from depth

4. Hydrostatic Pressure

In a fluid at rest, pressure increases with depth because deeper points support more fluid above them.

Pressure increases with depth Deeper points have greater fluid pressure h lower pressure higher pressure ΔP = ρgh
Hydrostatic pressure depends on density, gravity, and depth below the surface, not on the shape of the container.
Gauge pressure at depth ΔP = ρgh h is depth below the surface.
Absolute pressure at depth P = Patm + ρgh Includes atmospheric pressure above the fluid.
Pressure difference P2 - P1 = ρg(h2 - h1) Use for two depths in the same fluid.

Important Hydrostatic Facts

  • Pressure at the same depth in the same connected fluid is the same.
  • Pressure does not depend on container shape.
  • Pressure acts equally in all directions at a point.
  • Pressure increases if density, gravity, or depth increases.
Common mistake: Depth h is measured downward from the fluid surface, not from the bottom of the container.
Hydraulic systems

5. Pascal's Principle

Pascal's principle says that a pressure change applied to an enclosed fluid is transmitted equally throughout the fluid. This is the basis of hydraulic lifts and hydraulic brakes.

Pascal's principle in a hydraulic lift A pressure change is transmitted through an enclosed fluid small input force larger output force F1/A1 = F2/A2
A small piston can create a large output force when the pressure acts on a larger area.
Equal pressure P1 = P2 For connected fluid at the same height.
Hydraulic force ratio F1/A1 = F2/A2 A larger area can produce a larger force.
Work relationship F1d1 = F2d2 Hydraulics multiply force, not energy.
Professional connection: Hydraulic systems are used in vehicle brakes, lifts, construction equipment, aircraft controls, and industrial machines.
Upward force from fluids

6. Buoyancy and Archimedes' Principle

Any object in a fluid experiences an upward buoyant force because pressure is greater at the bottom of the object than at the top.

Buoyant force comes from pressure difference Buoyancy is the upward force from displaced fluid FB W = mg FB = ρfluidgVdisp bottom pressure is greater than top pressure
The buoyant force equals the weight of the fluid displaced by the submerged part of the object.
Archimedes' principle FB = ρfluidgVdisp Buoyant force equals weight of displaced fluid.
Apparent weight Wapp = W - FB Why objects feel lighter underwater.
Weight of object W = mg = ρobjectgVobject Useful when density and volume are known.

What Controls Buoyant Force?

  • The density of the fluid.
  • The volume of fluid displaced.
  • The gravitational field strength.
  • Not the object's mass directly, except through how much fluid it displaces.
Equilibrium in fluids

7. Floating, Sinking, and Submerged Objects

An object floats, sinks, or stays suspended depending on the relationship between weight and buoyant force.

Floating, sinking, and neutral buoyancy Compare buoyant force with weight to predict motion floats: FB = W sinks: W > FB neutral: ρobject = ρfluid
An object floats, sinks, or remains suspended based on how buoyant force compares with its weight.
Floating equilibrium FB = W For a floating object at rest.
Fraction submerged Vsub/Vobject = ρobject/ρfluid Only for floating objects.
Neutral buoyancy ρobject = ρfluid Object neither rises nor sinks.
Condition Result Force or Density Comparison
Object floatsRises until partly submergedρobject < ρfluid
Object sinksMoves downwardρobject > ρfluid
Object is neutrally buoyantStays suspendedρobject = ρfluid
Floating object at restNo vertical accelerationFB = W
Fully submerged objectDisplaces its full volumeVdisp = Vobject
Moving fluids

8. Fluid Flow

Fluid flow describes how much fluid moves through a region each second. Flow can be steady or unsteady, laminar or turbulent, compressible or incompressible.

Laminar and turbulent flow Flow can be smooth or chaotic laminar: smooth layers turbulent: swirls and mixing
Laminar flow is smooth and predictable, while turbulent flow contains mixing, eddies, and rapidly changing speeds.
Volume flow rate Q = V/t Volume per second, unit m3/s.
Flow rate through area Q = Av A is cross-sectional area, v is fluid speed.
Mass flow rate mass flow rate = ρAv Mass per second passing a point.

Laminar vs. Turbulent Flow

  • Laminar flow is smooth and predictable.
  • Turbulent flow is irregular and mixes layers of fluid.
  • Higher speed, rough surfaces, and low viscosity make turbulence more likely.
Conservation of mass

9. Continuity Equation

For steady incompressible flow, the same volume of fluid must pass through every part of a pipe each second. If the pipe narrows, the fluid speeds up.

Continuity in a narrowing pipe When area decreases, fluid speed increases A1 A2 wide pipe: slower v1 narrow pipe: faster v2 A1v1 = A2v2
For steady incompressible flow, volume flow rate stays the same through every cross-section of the pipe.
Continuity equation A1v1 = A2v2 For incompressible steady flow.
Flow rate Q = A1v1 = A2v2 Same everywhere in a closed pipe.
Pipe narrows A decreases, v increases Speed rises in narrower sections.
Common mistake: A smaller pipe area does not mean less fluid per second in steady incompressible flow. It means the fluid moves faster.
Energy conservation in fluids

10. Bernoulli's Principle

Bernoulli's equation is an energy conservation statement for ideal fluid flow. Pressure energy, kinetic energy per volume, and gravitational potential energy per volume can trade off.

Bernoulli pressure-speed tradeoff In faster flow, pressure is often lower higher P lower P slower flow faster flow P + 1/2ρv2 + ρgy = constant
Bernoulli's principle connects pressure, speed, and height as forms of mechanical energy per volume.
Bernoulli equation P + 1/2ρv2 + ρgy = constant For steady, incompressible, nonviscous flow.
Two-point form P1 + 1/2ρv12 + ρgy1 = P2 + 1/2ρv22 + ρgy2 Compare two points in the same streamline.
Torricelli's speed v = √(2gh) Speed of fluid leaving a hole below the surface.

Bernoulli Meaning

  • If a horizontal fluid speeds up, pressure usually decreases.
  • If fluid moves to a higher height, pressure and/or speed may decrease.
  • Bernoulli is most accurate when viscosity and turbulence are small.
Real fluid effects

11. Viscosity, Drag, and Terminal Speed

Real fluids are not perfectly ideal. Viscosity causes internal friction, and drag forces oppose motion through a fluid.

Drag and terminal speed Drag grows until the net force can become zero FD + FB mg terminal speed: FD + FB = mg net force is zero, so speed stops increasing
Drag opposes motion through a fluid and increases with speed until forces can balance at terminal speed.
Viscosity idea higher viscosity = harder to flow Honey is more viscous than water.
Quadratic drag model FD = 1/2CρAv2 Often used at higher speeds.
Terminal speed condition FD + FB = mg Net force becomes zero, so speed stops increasing.

Real-World Examples

  • A parachute increases area, which increases drag and reduces terminal speed.
  • Oil flows more slowly than water because it has greater viscosity.
  • Streamlined shapes reduce drag in air and water.
Simulation labs

12. Simulation Labs for This Unit

These official PhET simulations help students visualize density, buoyancy, pressure at depth, fluid pressure, and flow behavior.

Fluid simulation lab workflow Use fluid simulations to connect variables, graphs, and explanations Change depth, area, density Measure pressure, speed, force Explain use formulas and trends A strong lab answer names the controlled variable, changed variable, and measured result.
Simulations become professional lab evidence when students change one variable at a time and explain the trend with physics.
Density

Explore how mass and volume determine density and predict whether objects sink or float in different fluids.

Lab idea: keep volume constant, change mass, and observe how density affects floating.
Open Simulation
Buoyancy

Investigate buoyant force, displaced volume, object density, fluid density, apparent weight, and floating conditions.

Lab idea: compare the buoyant force on the same object in different fluids.
Open Simulation
Under Pressure

Visualize how pressure changes with depth, fluid density, gravity, and container shape.

Lab idea: measure pressure at equal depths in differently shaped containers.
Open Simulation
Fluid Pressure and Flow

Explore pressure, flow speed, pipe area, and the relationship between fluid motion and pressure.

Lab idea: narrow a pipe section and compare flow speed and pressure before and after the constriction.
Open Simulation
Investigation skills

13. Fluid Mechanics Lab Skills

Fluid labs often require careful measurement because small changes in volume, depth, or cross-sectional area can strongly affect results.

Fluid mechanics lab measurements Good fluid labs depend on careful measurements depth h area A flow speed v measure mass, volume, depth, area, time, and force with units
Before calculating, label every measured quantity with units and convert volume and area carefully.

Common Labs

  • Density lab using mass and water displacement.
  • Pressure vs. depth investigation.
  • Hydraulic lift or syringe Pascal's principle lab.
  • Buoyancy and apparent weight lab with a spring scale.
  • Floating-object fraction-submerged lab.
  • Flow rate and continuity lab using different tube diameters.
  • Bernoulli-style pressure and speed investigation.

Useful Measurements

  • Mass in kilograms.
  • Volume in cubic meters or milliliters converted carefully.
  • Depth below fluid surface in meters.
  • Area in square meters.
  • Force in newtons.
  • Flow rate in m3/s or L/s.
  • Fluid speed in m/s.
Unit warning: 1 mL = 1 cm3 = 1 × 10-6 m3. Many fluid mistakes come from volume conversion errors.
Worked examples

14. Worked Examples

Example 1: Density

A metal block has mass 0.540 kg and volume 2.00 × 10-4 m3. Find density.

ρ = m/V = 0.540/(2.00 × 10-4) = 2700 kg/m3.

Example 2: Pressure

A 600 N force acts on an area of 0.030 m2. Find pressure.

P = F/A = 600/0.030 = 2.0 × 104 Pa.

Example 3: Pressure at depth

Find gauge pressure 3.0 m below the surface of fresh water.

ΔP = ρgh = (1000)(9.8)(3.0) = 2.94 × 104 Pa.

Example 4: Hydraulic lift

A hydraulic system has A1 = 0.010 m2, A2 = 0.50 m2, and F1 = 200 N. Find F2.

F1/A1 = F2/A2.

F2 = (200)(0.50/0.010) = 1.0 × 104 N.

Example 5: Buoyant force

An object displaces 0.0020 m3 of water. Find buoyant force.

FB = ρgV = (1000)(9.8)(0.0020) = 19.6 N upward.

Example 6: Apparent weight

An object weighs 30 N in air and displaces 0.0010 m3 of water when submerged. Find apparent weight.

FB = (1000)(9.8)(0.0010) = 9.8 N.

Wapp = W - FB = 30 - 9.8 = 20.2 N.

Example 7: Fraction submerged

A wood block has density 600 kg/m3 and floats in water. What fraction is submerged?

Vsub/V = ρobject/ρwater = 600/1000 = 0.60.

60% of the block is submerged.

Example 8: Continuity

Water flows through a pipe with A1 = 0.040 m2 at v1 = 2.0 m/s. The pipe narrows to A2 = 0.010 m2. Find v2.

A1v1 = A2v2.

v2 = (0.040)(2.0)/(0.010) = 8.0 m/s.

Example 9: Bernoulli pressure change

A horizontal pipe has water pressure 200 kPa where speed is 2.0 m/s. The pipe narrows and speed becomes 8.0 m/s. Find the new pressure.

P1 + 1/2ρv12 = P2 + 1/2ρv22.

P2 = 200000 + 1/2(1000)(2.02 - 8.02) = 1.70 × 105 Pa.

Example 10: Torricelli speed

Water exits a small hole 1.5 m below the surface. Estimate exit speed.

v = √(2gh) = √[2(9.8)(1.5)] = 5.4 m/s.

Independent practice

15. Practice Problems

Try each problem first. Then open the answer check and compare formulas, substitutions, units, and reasoning.

1. A liquid has mass 2.4 kg and volume 0.0030 m3. Find density.

Answer

ρ = m/V = 2.4/0.0030 = 800 kg/m3.

2. An object has density 500 kg/m3 and mass 10 kg. Find volume.

Answer

V = m/ρ = 10/500 = 0.020 m3.

3. A 1000 N force acts on 0.25 m2. Find pressure.

Answer

P = F/A = 1000/0.25 = 4000 Pa.

4. Find gauge pressure 5.0 m below the surface of fresh water.

Answer

ΔP = ρgh = (1000)(9.8)(5.0) = 4.9 × 104 Pa.

5. Estimate absolute pressure 10 m below the surface of fresh water. Use Patm = 1.01 × 105 Pa.

Answer

P = Patm + ρgh = 1.01 × 105 + (1000)(9.8)(10) = 1.99 × 105 Pa.

6. Water pressure on a hatch is 30 kPa. The hatch area is 0.80 m2. Find force.

Answer

F = PA = (30000)(0.80) = 24000 N.

7. In a hydraulic lift, F1 = 150 N, A1 = 0.020 m2, and A2 = 0.50 m2. Find F2.

Answer

F2 = F1(A2/A1) = 150(0.50/0.020) = 3750 N.

8. A hydraulic input piston moves 0.20 m. A1 = 0.020 m2 and A2 = 0.50 m2. Estimate output piston movement.

Answer

Volume moved is equal: A1d1 = A2d2.

d2 = (0.020)(0.20)/(0.50) = 0.0080 m.

9. An object displaces 0.0040 m3 of water. Find buoyant force.

Answer

FB = ρgV = (1000)(9.8)(0.0040) = 39.2 N.

10. An object weighs 80 N and experiences a 25 N buoyant force. Find apparent weight.

Answer

Wapp = W - FB = 80 - 25 = 55 N.

11. An object has density 1200 kg/m3. Will it float or sink in fresh water?

Answer

It sinks because 1200 kg/m3 is greater than water's density of 1000 kg/m3.

12. A floating object has density 750 kg/m3. What fraction is submerged in water?

Answer

Vsub/V = ρobject/ρwater = 750/1000 = 0.75. It is 75% submerged.

13. A 3.0 kg floating object is in water. What volume of water must it displace?

Answer

For floating, FB = mg, so ρwatergV = mg. V = m/ρ = 3.0/1000 = 0.0030 m3.

14. Water flows through A1 = 0.020 m2 at v1 = 3.0 m/s, then into A2 = 0.0060 m2. Find v2.

Answer

v2 = A1v1/A2 = (0.020)(3.0)/(0.0060) = 10 m/s.

15. A pipe has area 0.0050 m2 and fluid speed 4.0 m/s. Find flow rate.

Answer

Q = Av = (0.0050)(4.0) = 0.020 m3/s.

16. In a horizontal constriction, fluid speed increases. What usually happens to pressure?

Answer

Pressure decreases, according to Bernoulli's principle.

17. Water in a horizontal pipe has P1 = 150 kPa, v1 = 3.0 m/s, and v2 = 9.0 m/s. Find P2.

Answer

P2 = P1 + 1/2ρ(v12 - v22).

P2 = 150000 + 1/2(1000)(9 - 81) = 114000 Pa = 114 kPa.

18. Water exits a hole 2.0 m below the surface. Estimate exit speed.

Answer

v = √(2gh) = √[2(9.8)(2.0)] = 6.3 m/s.

19. How do gauge pressure and absolute pressure differ?

Answer

Gauge pressure is pressure above atmospheric pressure. Absolute pressure includes atmospheric pressure.

20. Why does a parachute reduce terminal speed?

Answer

It increases area, which increases drag force at a given speed, so terminal speed becomes lower.

Final review

16. What to Know Before Moving On

  • Density is mass per volume: ρ = m/V.
  • Pressure is force per area: P = F/A.
  • Fluid pressure increases with depth: ΔP = ρgh.
  • Pascal's principle explains hydraulic force multiplication.
  • Buoyant force equals the weight of displaced fluid: FB = ρfluidgVdisp.
  • Floating objects satisfy FB = W.
  • A floating object's submerged fraction is ρobject/ρfluid.
  • Continuity says A1v1 = A2v2 for steady incompressible flow.
  • Bernoulli's equation connects pressure, speed, and height.
  • Real fluids include viscosity, turbulence, drag, and energy losses.
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