Buoyancy and Archimedes' Principle

🛟 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 $F_{bot} = P_{bot} A$, with $P_{bot} > P_{top}$.

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)

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, $\rho_{water} = 1000$)

1. A fully submerged object displaces 0.020 $m^{3}$ of water. Buoyant force (N)?

2. A block, half-submerged, displaces 0.0050 $m^{3}$. Buoyant force (N)?

3. A balloon of volume 1.0 $m^{3}$ in air ($\rho_{air} = 1.2$). Buoyant force from air (N)?

Buoyancy Reasoning 🔍

Exit Quiz — Intro to Buoyancy ✅

🏛 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 $F_b = \rho_{fluid}\, g\, V_{disp}$
• 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 $(kg/m^{3})$
• $V_{disp}$: volume of fluid displaced $(m^{3})$
• $g = 9.8$ $m/s^{2}$ (or 10 in many AP problems)

The Three Scenarios

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 $m^{3}$ of water weighs ~10,000 N (using $g \approx 10$, $\rho = 1000$).
• So 1 $m^{3}$ submerged in water → buoyant force ~10,000 N.

Archimedes' Principle Concepts 🎯

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

1. A 0.30 $m^{3}$ object is fully submerged in fresh water. Buoyant force (N)?

2. Same object, this time fully submerged in seawater. Buoyant force (N)?

3. A boat displaces 8.0 $m^{3}$ of water when floating. Its weight (N)?

Archimedes Reasoning 🔍

Exit Quiz — Archimedes' Principle ✅

🚢 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.

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:

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

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, $\rho_w = 1000$, $\rho_{sw} = 1030$)

1. An object of density 600 $kg/m^{3}$ floats in water. Fraction submerged (decimal)?

2. A 0.20 $m^{3}$ block of density 800 $kg/m^{3}$ floats in water. Volume submerged $(m^{3})$?

3. Same block in seawater. Fraction submerged (decimal, 4 sig figs)?

Density vs Buoyancy 🔍

Exit Quiz — Floating vs Sinking ✅

🪨 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: $W_{app} = W - F_b$
• 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, $\rho_w = 1000$)

1. A solid metal ball of mass 4.0 kg and volume $5.0\times10^{-4}$ $m^{3}$ is fully submerged. Apparent weight on a spring scale (N)?

2. A foam buoy of mass 2.0 kg and volume 0.020 $m^{3}$ is held under water by a rope. Tension in rope (N)?

3. A 50 N weight rests on a pool bottom. It displaces $2.0\times10^{-3}$ $m^{3}$. Normal force from the pool bottom (N)?

Submerged Reasoning 🔍

Exit Quiz — Submerged Objects ✅

🛶 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

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

Cancelling $g$:

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

Draft Depth (rectangular hull)

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

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

Adding Cargo

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

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

Maximum Load

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, $\rho_w = 1000$, $\rho_{sw} = 1030$)

1. A barge has a flat bottom of area $50$ $m^{2}$ and total mass $30{,}000$ kg. Draft depth in fresh water (m)?

2. Same barge in seawater. Draft depth (m, 4 decimal places)?

3. A wooden raft (density 600 $kg/m^{3}$) 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 ✅

🛠 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:

1. Sketch the FBD: weight (down), buoyancy (up), other forces.
2. Identify $V_{disp}$: full $V_{obj}$ if fully submerged, or only the part below the waterline.
3. Apply $\sum F_y = ma$.
4. Watch SIGN of tension — does the rope pull up or down?

Six Standard Problem Patterns

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, $\rho_w = 1000$)

1. A 12 kg block of volume $5.0\times10^{-3}$ $m^{3}$ is fully submerged. Apparent weight on a scale (N)?

2. A 2.0 kg piece of foam (volume 0.020 $m^{3}$) is held under water by a string. String tension (N)?

3. A 1.5 kg wooden block (density 750 $kg/m^{3}$) floats. Volume submerged $(m^{3})$?

Workshop Reasoning 🔍

Exit Quiz — Workshop ✅

🎯 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)
• 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

AP-Style Question Patterns

1. "Will it float?" → compare $\rho_{obj}$ vs $\rho_{fluid}$.
2. "Spring scale in air vs water" → $F_b = W_{air} - W_{water}$ (gives volume).
3. "How much load can a boat carry?" → $m_{cargo,max} = \rho_{fluid} V_{hull} - m_{boat}$.
4. "Submarine neutral buoyancy" → $\rho_{sub,avg} = \rho_{fluid}$.
5. "Iceberg fraction below water" → $\rho_{ice}/\rho_{water} ≈ 92\%$.

AP Synthesis MC 🎯

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

1. An iceberg has total volume 800 $m^{3}$ ($\rho_{ice} = 917$). Volume above water in fresh water ($m^{3}$, 1 decimal)?

2. A 50-kg log floats with 70% submerged. Volume of the log ($m^{3}$, 4 decimals)?

3. A 4 kg ball with $V = 0.0040$ $m^{3}$ is held by a string at the bottom (submerged). Tension (N)?

AP Synthesis Reasoning 🔍

Exit Quiz — AP Synthesis ✅