Q1 Buoyancy, Oscillation, Density, Conduction, and Energy

Floating Bodies, SHM, Density Anomaly of Water, and Thermal Physics

IB Physics HL • Paper 2 • Long-Answer Question • Buoyancy, Oscillation, Density, Conduction, and Energy

Lesson Overview

In this Paper 2 long-answer question, we bring together several important IB Physics ideas in one realistic context. The question begins with a floating cork and uses equilibrium and buoyancy to derive a relationship between densities. It then develops into simple harmonic motion, the density anomaly of water, the floating depth of an iceberg, momentum in a collision, and thermal energy transfer through ice.

This type of question rewards clear scientific reasoning, careful algebra, correct use of units, and concise explanations in words. In the worked solutions below, each part is written in an IB-style format with method marks, answer marks, and clear physical interpretation.

⭐ Key Concepts

A floating object is in equilibrium when its weight equals the upthrust.
Buoyant force equals the weight of fluid displaced.
Simple harmonic motion occurs when acceleration is proportional to displacement and directed towards equilibrium.
Density is given by

$\rho=\frac{m}{V}$

Most solids are denser than their liquids because the particles are more closely packed.
Water is unusual because its maximum density occurs at about $4^\circ\text{C}$ .
Momentum is conserved in collisions when external forces are negligible.
The rate of thermal energy transfer by conduction through a layer is

$\frac{Q}{t}=\frac{kA\Delta T}{x}$

Q/t is the rate of heat transfer (Watts).
k is the thermal conductivity of the material.
A is the cross-sectional area perpendicular to the heat flow.
T₁ – T₂ is the temperature difference across the slab.
x (or L, d) is the thickness of the slab.

Thermal energy required to cool or freeze a substance is found using

$Q=mc\Delta T$

$Q=mL_f$

📘 Clear IB-Style Explanation

Paper 2 questions like this often move across several areas of the syllabus, so the best approach is to treat each part as a small physics argument.

For equilibrium, identify all forces first.
For derivations, begin with a physical principle before substituting expressions.
For explanation questions, describe both the cause and the effect.
For thermal physics, keep track of whether the question asks for a rate of transfer, an energy per unit area, or a total energy.
For density anomaly questions, connect the graph to the survival of aquatic life.

A strong IB answer is not just a number. It must show the correct method, the correct physics, and a final statement that answers the question directly.

📌 Worked Example 1 — Floating Cork in Equilibrium

A cylindrical cork of height $H$ and cross-sectional area $A$ floats stationary in water. Its depth below the water surface is $D$ .

(a.i) Draw and label the forces acting on the cork.

(a.ii) Show that

$\frac{D}{H}=\frac{\rho_c}{\rho_w}$

where $\rho_c$ is the density of the cork and $\rho_w$ is the density of water.

Solution

(a.i) Forces on the cork

Since the cork is floating stationary, the forces on it are:

weight, acting vertically downward
buoyant force (upthrust), acting vertically upward

These two forces balance because the cork is in equilibrium.

📊 Marks: A1

(a.ii) Derivation of the density relationship

For a floating object in equilibrium,

$\text{weight}=\text{buoyant force}$

The weight of the cork is

$W=mg$

The mass of the cork is

$m=\rho_c \times \text{volume}=\rho_c AH$

So the weight is

$W=\rho_c AH g$

The buoyant force equals the weight of water displaced. The submerged volume is $AD$ , so

$F_B=\rho_w AD g$

Equating weight and upthrust:

$\rho_c AH g=\rho_w AD g$

Cancel $Ag$ :

$\rho_c H=\rho_w D$

Rearranging:

$\frac{D}{H}=\frac{\rho_c}{\rho_w}$

Hence the required expression is shown.

📊 Marks: M1 M1 A1

📌 Worked Example 2 — Why the Cork Oscillates in SHM

The cork is pushed downward and released. It then oscillates.

(b) Outline why the cork undergoes simple harmonic motion.

Solution

When the cork is pushed down, a greater volume of the cork is submerged. This increases the buoyant force.

The weight of the cork remains constant, but the upthrust becomes larger than the weight, so there is a resultant force upward.

As the cork rises back towards equilibrium, this restoring force decreases. The force is always directed towards the equilibrium position.

Therefore, the acceleration is proportional to the displacement from equilibrium and opposite in direction:

$a\propto -x$

So the cork undergoes simple harmonic motion.

📊 Marks: M1 A1

📌 Worked Example 3 — Density of Solids and Water’s Anomaly

(c.i) Explain why the density of most substances in a solid state is larger than its density in a liquid state.

(c.ii) Identify the temperature at which water has its maximum density.

Solution

(c.i) Why most solids are denser

For most substances, particles in the solid state are packed more closely together than in the liquid state.

This means that the same mass occupies a smaller volume in the solid state.

Since density is mass per unit volume,

$\rho=\frac{m}{V}$

a smaller volume for the same mass gives a larger density.

📊 Marks: A1 A1

(c.ii) Maximum density of water

Water has its maximum density at

$4^\circ\text{C}$

📊 Marks: A1

📌 Worked Example 4 — Iceberg Mass and Collision Speed

An iceberg is modeled as a cylinder of cross-sectional area $4200\text{ m}^2$ . Its height above sea level is $32\text{ m}$ .

Given:

$\rho_{\text{ice}}=920\text{ kg m}^{-3}$

$\rho_{\text{sea water}}=1030\text{ kg m}^{-3}$

(d.i) Show that the mass of the iceberg is about $1.2\times10^9\text{ kg}$ .

The designers assume that the mass of the ship is about $\frac{1}{40}$ of the mass of the iceberg and that the ship is moving at $12\text{ m s}^{-1}$ before the collision. The ship and iceberg stick together.

(d.ii) Calculate the speed of the ship after the collision.

Solution

(d.i) Mass of the iceberg

For a floating object,

$\frac{D}{H}=\frac{\rho_{\text{ice}}}{\rho_{\text{sea water}}}$

The total height of the iceberg is $H$ , and the part above sea level is $32\text{ m}$ , so the submerged depth is

$D=H-32$

Therefore,

$\frac{H-32}{H}=\frac{920}{1030}$

$1-\frac{32}{H}=0.893$

$\frac{32}{H}=0.107$

$H\approx299\text{ m}$

So the total height is about

$H\approx300\text{ m}$

The volume of the iceberg is

$V=AH=(4200)(300)=1.26\times10^6\text{ m}^3$

Its mass is

$m=\rho V=(920)(1.26\times10^6)$

$m=1.1592\times10^9\text{ kg}$

So the mass is approximately

$1.2\times10^9\text{ kg}$

📊 Marks: M1 A1

(d.ii) Speed after collision

Let the mass of the iceberg be $M$ . Then the mass of the ship is

$\frac{M}{40}$

Initially, only the ship is moving, so initial momentum is

$p_i=\frac{M}{40}\times12$

After the collision, the total mass is

$M+\frac{M}{40}=\frac{41M}{40}$

If the common speed after collision is $v$ , conservation of momentum gives

$\frac{M}{40}\times12=\frac{41M}{40}v$

Cancel $\frac{M}{40}$ :

$12=41v$

$v=\frac{12}{41}$

$v=0.293\text{ m s}^{-1}$

Therefore, the speed after the collision is

$0.29\text{ m s}^{-1}$

📊 Marks: M1 A1

📌 Worked Example 5 — Conduction Through Ice and Freezing a Lake

A lake is covered by ice of thickness $1.9\text{ cm}$ . The air above the ice is at $-6.0^\circ\text{C}$ .

The thermal conductivity of ice is

$k=2.3\text{ W m}^{-1}\text{K}^{-1}$

(e.i) Calculate the rate per unit area at which thermal energy leaves the lake by conduction through the ice layer.

The water below the ice has depth $22\text{ m}$ and average initial temperature $2.0^\circ\text{C}$ .

Given:

$c=4.2\times10^3\text{ J kg}^{-1}\text{K}^{-1}$

$L_f=3.3\times10^5\text{ J kg}^{-1}$

$\rho=1000\text{ kg m}^{-3}$

(e.ii) Estimate the minimum thermal energy per unit area that must be removed to freeze all the water in the lake.

(e.iii) Explain how the rate calculated in (e.i) changes as the ice layer grows thicker.

(e.iv) Discuss why the anomaly in the value of the density of water supports life in water on Earth.

Solution

(e.i) Rate per unit area of thermal energy transfer

For conduction through a flat layer,

$\frac{Q}{t}=\frac{kA\Delta T}{x}$

So the rate per unit area is

$\frac{1}{A}\frac{Q}{t}=\frac{k\Delta T}{x}$

The temperature difference is

$\Delta T=6.0\text{ K}$

The ice thickness is

$x=1.9\text{ cm}=0.019\text{ m}$

Substitute:

$\frac{1}{A}\frac{Q}{t}=\frac{(2.3)(6.0)}{0.019}$

$\frac{1}{A}\frac{Q}{t}=726.3$

So the rate per unit area is approximately

$7.3\times10^2\text{ W m}^{-2}$

📊 Marks: M1 A1

(e.ii) Minimum thermal energy per unit area to freeze all the water

Take unit area $A=1\text{ m}^2$ .

The volume of water is

$V=22\times1=22\text{ m}^3$

So the mass is

$m=\rho V=(1000)(22)=2.2\times10^4\text{ kg}$

First, cool the water from $2.0^\circ\text{C}$ to $0^\circ\text{C}$ :

$Q_1=mc\Delta T$

$Q_1=(2.2\times10^4)(4.2\times10^3)(2.0)$

$Q_1=1.848\times10^8\text{ J}$

Then freeze the water:

$Q_2=mL_f$

$Q_2=(2.2\times10^4)(3.3\times10^5)$

$Q_2=7.26\times10^9\text{ J}$

Total energy per unit area:

$Q=Q_1+Q_2$

$Q=1.848\times10^8+7.26\times10^9$

$Q=7.4448\times10^9\text{ J}$

Therefore, the minimum thermal energy per unit area that must be removed is

$7.4\times10^9\text{ J m}^{-2}$

📊 Marks: M1 M1 A1

(e.iii) Effect of increasing ice thickness

From

$\frac{Q}{t}=\frac{kA\Delta T}{x}$

the rate of thermal energy transfer is inversely proportional to the thickness $x$ .

So as the ice layer gets thicker, the rate of thermal energy transfer becomes smaller.

📊 Marks: A1

(e.iv) Why this anomaly supports life

Water is most dense at about $4^\circ\text{C}$ , so water at this temperature sinks to the bottom of a lake.

When the surface water cools below $4^\circ\text{C}$ , it becomes less dense and remains near the surface, where it can freeze.

The ice then forms a layer on top rather than sinking. This surface ice insulates the water below and slows further heat loss.

As a result, the lake does not usually freeze solid, allowing aquatic life to survive beneath the ice.

📊 Marks: A1 A1

⚠ Common Mistakes

Writing “gravity” instead of weight when labelling the force on the cork.
Using the full volume of displaced water instead of only the submerged volume.
Stating SHM without explaining that the restoring force or acceleration is directed towards equilibrium.
Forgetting that water’s maximum density is at $4^\circ\text{C}$ , not $0^\circ\text{C}$ .
Using momentum conservation incorrectly by forgetting that the ship and iceberg move together after collision.
Not converting $1.9\text{ cm}$ into $0.019\text{ m}$ in the conduction calculation.
For part (e.ii), calculating only the cooling energy or only the latent heat instead of both.
Confusing total energy with energy per unit area.

📘 IB Exam Tips

For floating-body questions, always begin with weight = upthrust if the object is in equilibrium.
When asked to “outline why” motion is SHM, mention both proportionality and opposite direction.
For explanation questions, do not stop at a definition. Link particle arrangement to density explicitly.
In energy questions, decide whether the process includes cooling, heating, melting, freezing, or a combination.
Write every final answer with units, even when the markscheme seems obvious.
For discussion questions, make a clear physical point and then explain why it matters in context.

🧪 Challenge Problem

A wooden block of height $0.30\text{ m}$ and cross-sectional area $0.020\text{ m}^2$ floats in water with $0.18\text{ m}$ below the surface.

(a) Determine the density of the wood.

(b) The block is pushed down slightly and released. State one reason why it undergoes SHM.

(c) A layer of ice of thickness $2.5\text{ cm}$ covers a lake. The temperature difference across the ice is $8.0\text{ K}$ . The thermal conductivity of ice is $2.3\text{ W m}^{-1}\text{K}^{-1}$ . Calculate the rate per unit area of thermal energy transfer through the ice.

(d) Explain why floating ice on the surface helps aquatic life survive in winter.

✅ Self-Practice

An iceberg floats in seawater. The density of ice is $920\text{ kg m}^{-3}$ and the density of seawater is $1025\text{ kg m}^{-3}$ .

(a) Show that the fraction of the iceberg submerged is approximately $0.90$ .

(b) A ship of mass $2.0\times10^7\text{ kg}$ moving at $10\text{ m s}^{-1}$ collides with the iceberg of mass $8.0\times10^8\text{ kg}$ . They stick together. Calculate their common speed after collision.

(c) A frozen lake has an ice layer of thickness $0.030\text{ m}$ . The thermal conductivity of ice is $2.3\text{ W m}^{-1}\text{K}^{-1}$ and the temperature difference across the ice is $5.0\text{ K}$ . Calculate the rate per unit area of heat transfer.

(d) State why the rate of energy transfer decreases as the ice gets thicker.

Show Solutions

Challenge Problem

(a)

For a floating object,

$\frac{D}{H}=\frac{\rho_{\text{wood}}}{\rho_{\text{water}}}$

$\frac{0.18}{0.30}=\frac{\rho_{\text{wood}}}{1000}$

$0.60=\frac{\rho_{\text{wood}}}{1000}$

$\rho_{\text{wood}}=600\text{ kg m}^{-3}$

(b)

When the block is pushed down, more water is displaced, so the upthrust increases. This creates a restoring force towards equilibrium, and the acceleration is proportional to displacement and opposite in direction.

(c)

$\frac{1}{A}\frac{Q}{t}=\frac{k\Delta T}{x}$

$\frac{1}{A}\frac{Q}{t}=\frac{(2.3)(8.0)}{0.025}$

$\frac{1}{A}\frac{Q}{t}=736\text{ W m}^{-2}$

(d)

Ice floats on the surface and forms an insulating layer. This reduces heat loss from the water below and prevents the whole lake from freezing solid, allowing life to survive underneath.

Self-Practice

(a)

$\frac{D}{H}=\frac{\rho_{\text{ice}}}{\rho_{\text{sea water}}}$

$\frac{D}{H}=\frac{920}{1025}=0.898$

So the fraction submerged is approximately

$0.90$

(b)

Using conservation of momentum:

$(2.0\times10^7)(10)=(8.0\times10^8+2.0\times10^7)v$

$2.0\times10^8=(8.2\times10^8)v$

$v=0.244\text{ m s}^{-1}$

So the common speed is

$0.24\text{ m s}^{-1}$

(c)

$\frac{1}{A}\frac{Q}{t}=\frac{k\Delta T}{x}$

$\frac{1}{A}\frac{Q}{t}=\frac{(2.3)(5.0)}{0.030}$

$\frac{1}{A}\frac{Q}{t}=383.3\text{ W m}^{-2}$

Therefore,

$3.8\times10^2\text{ W m}^{-2}$

(d)

Because the thickness $x$ is in the denominator of

$\frac{Q}{t}=\frac{kA\Delta T}{x}$

a larger thickness gives a smaller rate of thermal energy transfer.

Floating Bodies, SHM, Density Anomaly of Water, and Thermal Physics

Lesson Overview

⭐ Key Concepts

📘 Clear IB-Style Explanation

📌 Worked Example 1 — Floating Cork in Equilibrium

Solution

📌 Worked Example 2 — Why the Cork Oscillates in SHM

Solution

📌 Worked Example 3 — Density of Solids and Water’s Anomaly

Solution

📌 Worked Example 4 — Iceberg Mass and Collision Speed

Solution

📌 Worked Example 5 — Conduction Through Ice and Freezing a Lake

Solution

⚠ Common Mistakes

📘 IB Exam Tips

🧪 Challenge Problem

✅ Self-Practice

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Q1 Buoyancy, Oscillation, Density, Conduction, and Energy

Floating Bodies, SHM, Density Anomaly of Water, and Thermal Physics

Lesson Overview

⭐ Key Concepts

📘 Clear IB-Style Explanation

📌 Worked Example 1 — Floating Cork in Equilibrium

Solution

📌 Worked Example 2 — Why the Cork Oscillates in SHM

Solution

📌 Worked Example 3 — Density of Solids and Water’s Anomaly

Solution

📌 Worked Example 4 — Iceberg Mass and Collision Speed

Solution

📌 Worked Example 5 — Conduction Through Ice and Freezing a Lake

Solution

⚠ Common Mistakes

📘 IB Exam Tips

🧪 Challenge Problem

✅ Self-Practice

Related

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