🎯⭐ INTERACTIVE LESSON

Buoyancy and Archimedes' Principle

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Buoyancy and Archimedes' Principle - Complete Interactive Lesson

Part 1: What is Buoyancy?

🛟 What Is Buoyancy?

Part 1 of 7 — Fluids: Buoyancy

Why does a steel ship float but a steel coin sink? The answer is buoyancy — the upward force a fluid exerts on any object submerged or floating in it. This part introduces the qualitative idea before we hit Archimedes' equation.

In this lesson you will learn:

The physical origin of buoyancy (pressure difference)
Why buoyancy always points UP
How buoyancy depends on fluid (not object) density
How buoyancy compares to gravity in a static situation

Where Buoyancy Comes From

The pressure on the bottom of a submerged object is greater than the pressure on the top, because pressure increases with depth ( $P = \rho g h$ ).

The downward force on the top is $F_{top} = P_{top} A$ . The upward force on the bottom is $F_{b o t} = P_{}$ , with .

The net result is an UPWARD force:

$F_b = F_{bot} - F_{top} = (\rho_{fluid}\, g\, h_{bot} - \rho_{fluid}\, g\, h_{top})A = \rho_{fluid}\, g\, V_{disp}$

This is the buoyant force.

Two Possible Outcomes (static)

Comparison	Result
$F_b > W$	Net force up — object accelerates upward / floats higher
$F_b = W$

Important Reality Checks

Buoyancy depends on the fluid's density, not the object's.
Volume submerged matters — only the displaced volume contributes.
Air is also a fluid — every object in air experiences a small buoyant force (usually negligible).

Buoyancy Concepts 🎯

Quick Buoyancy Calculations 🧮 (g = 10, $\rho_{water} = 1000$ )

A fully submerged object displaces 0.020 m³ of water. Buoyant force (N)?
A block, half-submerged, displaces 0.0050 m³. Buoyant force (N)?
A balloon of volume 1.0 m³ in air ( $\rho_{air} = 1.2$ ). Buoyant force from air (N)?

Buoyancy Reasoning 🔍

Exit Quiz — Intro to Buoyancy ✅

Part 2: Archimedes' Principle

🏛 Archimedes' Principle

Part 2 of 7 — Fluids: Buoyancy

Archimedes (~250 BC) realized that the buoyant force equals the weight of the fluid displaced. This single equation is the engine of every floating/sinking problem on the AP.

In this lesson you will learn:

The exact statement of Archimedes' Principle
The equation $F_b = \rho_{fluid}\, g\, V_{disp}$

Part 3: Floating vs Sinking

🚢 Floating vs Sinking

Part 3 of 7 — Fluids: Buoyancy

The simplest rule of buoyancy: compare the object's average density to the fluid's density. If $\rho_{obj} < \rho_{fluid}$ , it floats. AP loves multi-step questions where you must apply this insight before doing math.

Part 4: Submerged Object Calculations

🪨 Submerged Object Calculations

Part 4 of 7 — Fluids: Buoyancy

Now we'll work problems where the object is FULLY submerged — held under, tied to a string, or sitting on the bottom. The key is balancing weight, buoyancy, and any other vertical forces (tension, normal, applied).

In this lesson you will learn:

The free-body diagram for a submerged object
Apparent weight: $W_{app} = W - F_b$

Part 5: Floating Object Calculations

🛶 Floating Object Calculations

Part 5 of 7 — Fluids: Buoyancy

When an object floats in equilibrium, the buoyant force equals the object's weight. From this single equation we can solve for unknowns: density, volume submerged, draft depth, or the load a boat can carry.

In this lesson you will learn:

The floating-equilibrium equation $F_b = W$
"Draft" depth and waterline calculations
How extra cargo changes submerged volume
The maximum load a vessel can carry before sinking

Equilibrium Equation for Floating

$ρ_{f l u i d} g V_{}$

Part 6: Problem-Solving Workshop

🛠 Buoyancy Problem-Solving Workshop

Part 6 of 7 — Fluids: Buoyancy

This workshop combines submerged- and floating-object problems with multiple forces (tension, normal, applied). AP loves to mix these so you must construct careful free-body diagrams.

Workshop Strategy:

Sketch the FBD: weight (down), buoyancy (up), other forces.
Identify $V_{disp}$ : full $V_{obj}$ if fully submerged, or only the part below the waterline.

Part 7: Synthesis & AP Review

🎯 Synthesis & AP Review — Buoyancy

Part 7 of 7 — Fluids: Buoyancy

You now have the full toolkit: density, displaced volume, equilibrium, free-body diagrams. AP buoyancy questions reward students who can quickly identify $V_{disp}$ and write the right force equation.

Big Ideas Recap:

$F_b = \rho_{fluid}\, g\, V_{disp}$ (Archimedes)

F_{b o t} = P_{b o t} A

How to identify

V_{disp}

in three cases (fully submerged, floating, partial)

Common pitfalls (substituting object density)

Archimedes' Principle

The buoyant force on an object is equal in magnitude to the weight of the fluid the object displaces.

$\boxed{F_b = \rho_{fluid}\, g\, V_{disp}}$

$\rho_{fluid}$ : density of the SURROUNDING fluid (kg/m³)
$V_{disp}$ : volume of fluid displaced (m³)
m/s² (or 10 in many AP problems)

The Three Scenarios

Scenario	$V_{disp}$
Fully submerged	= total object volume $V_{obj}$

Why "Displaced Volume" — Not Object Volume

If only half a block is underwater, only that half displaces water. The portion above the surface displaces air, which contributes nearly nothing to buoyancy (very small $\rho_{air}$ ).

Quick Numerical Anchor

1 m³ of water weighs ~10,000 N (using $g \approx 10$ , $\rho = 1000$ ).
So 1 m³ submerged in water → buoyant force ~10,000 N.

Archimedes' Principle Concepts 🎯

Archimedes Calculations 🧮 (g = 10, $\rho_w = 1000$ , $\rho_{sw} = 1030$ )

A 0.30 m³ object is fully submerged in fresh water. Buoyant force (N)?
Same object, this time fully submerged in seawater. Buoyant force (N)?
A boat displaces 8.0 m³ of water when floating. Its weight (N)?

Archimedes Reasoning 🔍

Exit Quiz — Archimedes' Principle ✅

In this lesson you will learn:

The density rule for floating
Why a steel ship floats (effective vs material density)
The "fraction submerged" formula
Edge cases (neutral buoyancy)

The Density Rule

Compare object's AVERAGE density to fluid:

Comparison	Behavior
$\rho_{obj} < \rho_{fluid}$	Floats (partially submerged)
$\rho_{obj} = \rho_{fluid}$	Neutral buoyancy (hovers anywhere)
$\rho_{obj} > \rho_{fluid}$	Sinks

Fraction Submerged Formula (floating object)

For a floating object in equilibrium ( $F_b = W$ ):

$\rho_{fluid} g V_{sub} = \rho_{obj} g V_{obj}$

$\boxed{\frac{V_{sub}}{V_{obj}} = \frac{\rho_{obj}}{\rho_{fluid}}}$

The fraction submerged equals the density ratio.

Examples

Object	$\rho_{obj}$ (kg/m³)	Fraction in fresh water
Cork	240	0.24 (24% submerged)
Wood (oak)	700	0.70 (70%)
Ice	917	0.92 (92% — "tip of the iceberg")
Water	1000	1.00 (just at surface)

Why a Steel Ship Floats

Steel itself ( $\rho \approx 7800$ ) sinks. But a hollow ship has lots of trapped air inside its hull, lowering its average density below water's. As long as $\rho_{avg} < 1000$ , it floats.

A coin (solid steel) has no air → density = 7800 → sinks.

Floating vs Sinking Concepts 🎯

Floating Calculations 🧮 (g = 10, $\rho_w = 1000$ , $\rho_{sw} = 1030$ )

An object of density 600 kg/m³ floats in water. Fraction submerged (decimal)?
A 0.20 m³ block of density 800 kg/m³ floats in water. Volume submerged (m³)?
Same block in seawater. Fraction submerged (decimal, 4 sig figs)?

Density vs Buoyancy 🔍

Exit Quiz — Floating vs Sinking ✅

How to find the tension in a rope holding a buoy under

Submerged-mass scale problems

Submerged Free-Body Diagram

For an object hanging or held underwater, three vertical forces:

$\sum F_y = F_b - W + T = m a$

where $T$ can be tension (up if rope pulls up, down if rope holds object down), normal force from a surface, or applied force.

Apparent Weight

If you weigh a submerged object on a spring scale, the scale reads:

$W_{app} = W - F_b = mg - \rho_{fluid}\, g\, V$

Object Tied DOWN (more dense than fluid would float, but is held under)

$T_{down} = F_b - W$ (rope pulls down to keep it under)

Object Tied UP (heavier than fluid; would sink, but is held)

$T_{up} = W - F_b$ (rope holds it up against gravity, helped by buoyancy)

Object Resting on Bottom

$N = W - F_b$ (normal force from bottom)

If $W < F_b$ → impossible to rest (would float up).

Submerged Object Concepts 🎯

Submerged Calculations 🧮 (g = 10, $\rho_w = 1000$ )

A solid metal ball of mass 4.0 kg and volume $5.0\times10^{-4}$ m³ is fully submerged. Apparent weight on a spring scale (N)?
A foam buoy of mass 2.0 kg and volume 0.020 m³ is held UNDER water by a rope. Tension in rope (N)?
A 50 N weight rests on a pool bottom. It displaces $2.0\times10^{-3}$ m³. Normal force from the pool bottom (N)?

Submerged Reasoning 🔍

Exit Quiz — Submerged Objects ✅

ρ_{f l u i d} g V_{s u b} = m_{t o t a l} g

$\rho_{fluid}\, V_{sub} = m_{total}$

Draft Depth (rectangular hull)

For a rectangular bottom of area $A$ and submerged depth $d$ :

$V_{sub} = A \cdot d \quad \Rightarrow \quad d = \frac{m_{total}}{\rho_{fluid}\, A}$

When cargo is added, the boat sinks lower. The CHANGE in submerged depth:

$\Delta d = \frac{m_{cargo}}{\rho_{fluid}\, A_{waterline}}$

A boat sinks when its hull edge reaches the waterline. The max cargo before sinking:

$m_{max\ cargo} = \rho_{fluid}\, V_{hull} - m_{boat}$

where $V_{hull}$ is the hull's displaceable internal volume.

Floating Equilibrium Concepts 🎯

Floating Calculations 🧮 (g = 10, $\rho_w = 1000$ , $\rho_{sw} = 1030$ )

A barge has a flat bottom of area $50$ m² and total mass $30{,}000$ kg. Draft depth in fresh water (m)?
Same barge in seawater. Draft depth (m, 4 decimal places)?
A wooden raft (density 600 kg/m³) is shaped as a $2\text{ m} \times 2\text{ m} \times 0.30\text{ m}$ slab. Submerged depth in water (m)?

Floating Object Reasoning 🔍

Exit Quiz — Floating Calculations ✅

Apply

\sum F_y = ma

Watch SIGN of tension — does the rope pull up or down?

Six Standard Problem Patterns

Pattern	Setup	Key Equation
Spring scale (in air)	Object hanging	$F_{scale} = W$
Spring scale (submerged)	Object hanging in water	$F_{scale} = W - F_b$
Floating	Object on surface	$F_b = W$ ⇒ $V_{sub} = (\rho_{obj}/\rho_f)V_{obj}$
Tied UP (sink-prone)	Rope holds heavy object	$T = W - F_b$
Tied DOWN (float-prone)	Rope holds light object under	$T = F_b - W$
Resting on bottom	Heavy object on pool floor	$N = W - F_b$

Tip: Three Force Equations Are All Variants Of...

$\sum F_y = F_b - W + T_{net} = ma$

Where $T_{net}$ is positive if the extra force pushes UP, negative if DOWN. In static problems, set $a = 0$ and solve.

Workshop MC 🎯

Workshop Calculations 🧮 (g = 10, $\rho_w = 1000$ )

A 12 kg block of volume $5.0\times10^{-3}$ m³ is fully submerged. Apparent weight on a scale (N)?
A 2.0 kg piece of foam (volume 0.020 m³) is held UNDER water by a string. String tension (N)?
A 1.5 kg wooden block (density 750 kg/m³) floats. Volume submerged (m³)?

Workshop Reasoning 🔍

Exit Quiz — Workshop ✅

Floating ⇒

F_b = W

⇒

V_{sub}/V_{obj} = \rho_{obj}/\rho_{fluid}

Submerged ⇒ apparent weight

= W - F_b

Buoyancy depends on FLUID density and DISPLACED volume only

AP Buoyancy Cheat Sheet

Quantity	Equation
Buoyant force	$F_b = \rho_{fluid}\, g\, V_{disp}$
Floating (equilibrium)	$\rho_{fluid} V_{sub} = m_{obj}$
Fraction submerged	$V_{sub}/V_{obj} = \rho_{obj}/\rho_{fluid}$
Apparent weight	$W_{app} = W - F_b$
Tension (object held UP)	$T = W - F_b$
Tension (object held DOWN)	$T = F_b - W$
Normal on bottom	$N = W - F_b$

AP-Style Question Patterns

"Will it float?" → compare $\rho_{obj}$ vs $\rho_{fluid}$ .
"Spring scale in air vs water" → (gives volume).

AP Synthesis MC 🎯

AP Synthesis Calculations 🧮 (g = 10, $\rho_w = 1000$ )

An iceberg has total volume 800 m³ ( $\rho_{ice} = 917$ ). Volume above water in fresh water (m³, 1 decimal)?
A 50-kg log floats with 70% submerged. Volume of the log (m³, 4 decimals)?
A 4 kg ball with $V = 0.0040$ m³ is held by a string at the BOTTOM (submerged). Tension (N)?

AP Synthesis Reasoning 🔍

Exit Quiz — AP Synthesis ✅

F_b = W_{air} - W_{water}

F_{b} = W_{ai r} - W_{w a t er}

"How much load can a boat carry?" →

m_{cargo,max} = \rho_{fluid} V_{hull} - m_{boat}

"Submarine neutral buoyancy" →

\rho_{sub,avg} = \rho_{fluid}

"Iceberg fraction below water" →

\rho_{ice}/\rho_{water} ≈ 92\%

Buoyancy and Archimedes' Principle

Archimedes' Principle

The Three Scenarios

Why "Displaced Volume" — Not Object Volume

Quick Numerical Anchor

The Density Rule

Fraction Submerged Formula (floating object)

Examples

Why a Steel Ship Floats

Submerged Free-Body Diagram

Apparent Weight

Object Tied DOWN (more dense than fluid would float, but is held under)

Object Tied UP (heavier than fluid; would sink, but is held)

Object Resting on Bottom

Draft Depth (rectangular hull)

Adding Cargo

Maximum Load

Six Standard Problem Patterns

Tip: Three Force Equations Are All Variants Of...

AP Buoyancy Cheat Sheet

AP-Style Question Patterns