Chapter 3: Floating and Sinking

1. Why This Matters

Have you ever wondered how a massive steel ship floats on water while a small nail sinks? Or how a submarine can dive deep and then rise back to the surface? Or why a hot air balloon floats into the sky? These are some of the most fascinating questions in physics, and they all come down to one beautiful principle: Archimedes' principle.

Floating and sinking are not just about weight. They are about density, upthrust, and the delicate balance between the weight of an object and the weight of the fluid it displaces. Understanding these ideas explains why we can build ships that carry thousands of tonnes, why life jackets save lives, and why the human body can float in the Dead Sea but not in a swimming pool. Let us dive into this fascinating topic together.

I once asked a class, "Why does a ship float?" A student said, "Because it is light." I said, "A ship can weigh more than a thousand elephants. Is that light?" The class fell silent. Then I explained: a ship floats not because it is light, but because it is shaped in a way that displaces a lot of water. The weight of the water displaced equals the weight of the ship. That is Archimedes' principle. Once they understood this, their view of the world changed.

2. Learning Objectives

By the end of this chapter, you should be able to:

• Define upthrust (buoyant force) and explain its origin.
• State Archimedes' principle and apply it to solve problems.
• Calculate upthrust using the weight of the fluid displaced.
• Explain the relationship between density and floating.
• State the law of flotation.
• Define relative density and calculate it.
• Describe how a hydrometer works and its uses.
• Explain how ships, submarines, and hot air balloons float and sink.
• Appreciate the importance of floating and sinking in nature, technology, and daily life.

3. Concept Build-Up

3.1 Density — The Key to Floating and Sinking

Before we can understand floating and sinking, we must understand density. Density is how much mass is packed into a given volume.

Key idea: Whether an object floats or sinks depends on its density compared to the density of the fluid it is placed in.

Comparison	Result	Example
Object density < fluid density	Floats	Wood (600 kg/m³) in water (1 000 kg/m³)
Object density = fluid density	Neutral (stays at any depth)	A submarine at neutral buoyancy
Object density > fluid density	Sinks	Iron (7 800 kg/m³) in water (1 000 kg/m³)

Think of a crowded room. If you are denser than the crowd, you sink to the bottom. If you are less dense, you rise to the top. The same idea applies to objects in fluids.

Density is the secret behind floating and sinking. A solid steel block sinks because steel is denser than water. But a steel ship floats because it is hollow — its average density (including the air inside) is less than the density of water. Same material, different shape, different result. Physics is wonderful.

3.2 Upthrust (Buoyant Force)

When an object is placed in a fluid, the fluid exerts an upward force on it. This force is called upthrust or buoyant force.

Why does upthrust exist? Because fluid pressure increases with depth. The pressure on the bottom of the object is greater than the pressure on the top. This difference in pressure creates a net upward force.

Think of a balloon underwater. The water pushes up on the bottom of the balloon more than it pushes down on the top. The result is an upward force — upthrust.

Depth	Pressure	Effect
Shallow (top of object)	Lower	Less force pushing down
Deep (bottom of object)	Higher	More force pushing up
Difference	—	Net upward force (upthrust)

Factors affecting upthrust:

• The density of the fluid (denser fluid = more upthrust).
• The volume of the object submerged (more volume = more upthrust).
• The gravitational field strength (g).

Upthrust does not depend on:

• The mass of the object.
• The shape of the object.
• The depth of the object (as long as it is fully submerged).

Upthrust is a consequence of pressure increasing with depth. It is one of those beautiful ideas where a simple observation — pressure increases with depth — leads to a profound result: fluids push up on objects. This is why you feel lighter in water and why it is easier to lift a heavy object when it is submerged.

3.3 Archimedes' Principle

Archimedes' principle states: When an object is wholly or partially immersed in a fluid, the upthrust on the object is equal to the weight of the fluid displaced.

Think of getting into a full bathtub. Water spills over the edge. The amount of water that spills out is the volume of your body that is submerged. According to Archimedes, the upthrust you feel is equal to the weight of that spilled water.

Apparent weight: The weight of an object when submerged in a fluid is less than its weight in air. This is because upthrust acts upward, opposing gravity.

This is why objects feel lighter in water. A heavy stone that is difficult to lift on land becomes easier to lift when it is submerged.

Situation	Weight in Air	Upthrust	Apparent Weight
Stone in air	50 N	0 N	50 N
Stone in water	50 N	20 N	30 N
Stone in mercury	50 N	40 N	10 N

Archimedes' principle is one of the most elegant and useful principles in all of physics. It was discovered by Archimedes over 2 000 years ago, and it is still used today to design ships, submarines, and hot air balloons. The story goes that he was so excited by his discovery that he ran through the streets shouting "Eureka!" — "I have found it!" That is the joy of physics.

3.4 The Law of Flotation

The law of flotation states: A floating object displaces its own weight of the fluid in which it floats.

Why is this true? Because a floating object is in equilibrium — it is not accelerating up or down. The upward force (upthrust) must equal the downward force (weight).

Think of a boat floating on water. The boat is not sinking or rising, so the forces on it must be balanced. The upthrust from the water equals the weight of the boat. According to Archimedes, the upthrust equals the weight of the water displaced. So the boat displaces its own weight of water.

Object	Weight	Weight of Water Displaced	Fraction Submerged
Wooden block	6 N	6 N	Partially submerged
Ship	1 000 000 N	1 000 000 N	Partially submerged
Iceberg	—	Equal to weight of iceberg	About 90% submerged

The law of flotation explains why a ship can carry a huge load. As you add cargo, the ship sinks deeper into the water, displacing more water. The upthrust increases until it equals the total weight of the ship plus cargo. This is why ships have a Plimsoll line — a mark that shows the maximum safe loading depth.

3.5 Relative Density

Relative density (also called specific gravity) is the ratio of the density of a substance to the density of water.

Interpretation:

• RD > 1: substance is denser than water and will sink.
• RD < 1: substance is less dense than water and will float.
• RD = 1: substance has the same density as water.

Substance	Density (kg/m³)	Relative Density	Floats or Sinks?
Wood (oak)	600	0.6	Floats
Ice	920	0.92	Floats
Water	1 000	1.0	—
Aluminium	2 700	2.7	Sinks
Iron	7 800	7.8	Sinks
Mercury	13 600	13.6	Sinks (in water)

Using relative density to find density:

Relative density is a very useful concept because it avoids the need to remember the density of water in different units. If you know the relative density of a substance, you can immediately tell whether it will float or sink in water. RD less than 1 — it floats. RD greater than 1 — it sinks. Simple.

3.6 Hydrometers

A hydrometer is an instrument used to measure the relative density of liquids. It works on the principle of flotation.

Construction:

• A sealed glass tube with a weighted bulb at the bottom.
• The stem is graduated with a scale.
• It floats upright in the liquid being tested.

How it works:

1. The hydrometer floats in the liquid.
2. The depth to which it sinks depends on the density of the liquid.
3. In a denser liquid, it sinks less (floats higher).
4. In a less dense liquid, it sinks more (floats lower).
5. The relative density is read from the scale at the liquid surface.

Liquid	Density (kg/m³)	Hydrometer Reading
Petrol	700	0.7
Water	1 000	1.0
Sea water	1 025	1.025
Glycerine	1 260	1.26

Uses of hydrometers:

• Checking the state of charge of a car battery (by measuring the density of the acid).
• Measuring the alcohol content of liquids.
• Testing the purity of milk.
• Measuring the density of urine in medical tests.

The hydrometer is a beautiful application of Archimedes' principle. It is simple, reliable, and has been used for centuries. The next time you see one, remember: it floats because of upthrust, and it measures density because of how deeply it sinks. Physics in your hand.

3.7 Ships and Boats

How does a huge steel ship float? After all, steel is much denser than water.

The secret is shape. A ship is hollow and shaped like a bowl. This means it has a large volume, so it displaces a large volume of water. Even though the steel is dense, the average density of the ship (including the air inside) is less than the density of water.

Key points:

• A ship floats because it displaces its own weight of water.
• As cargo is added, the ship sinks deeper, displacing more water.
• The Plimsoll line marks the maximum safe loading depth.
• If the ship is overloaded, it may sink below the Plimsoll line and be in danger of sinking.

Condition	Ship's Position	Upthrust vs Weight
Empty	Sits high in water	Upthrust = Weight
Partially loaded	Sits lower	Upthrust = Weight
Fully loaded	At Plimsoll line	Upthrust = Weight
Overloaded	Below Plimsoll line	Upthrust < Weight (danger)

The Plimsoll line is a life-saving invention. It was introduced by Samuel Plimsoll in the 19th century to prevent ships from being overloaded and sinking. It is a simple mark on the side of a ship, but it has saved thousands of lives. Physics is not just about understanding — it is about safety.

3.8 Submarines

A submarine can float on the surface, dive to any depth, and return to the surface. How does it do this?

The secret is ballast tanks. A submarine has large tanks that can be filled with water or air.

How it works:

• To dive: The ballast tanks are filled with water. The submarine's average density increases, so it sinks.
• To rise: Compressed air is pumped into the ballast tanks, forcing the water out. The submarine's average density decreases, so it rises.
• To stay at a constant depth: The ballast tanks are partially filled so that the submarine's average density equals the density of water (neutral buoyancy).

Action	Ballast Tanks	Average Density	Result
Dive	Filled with water	Greater than water	Sinks
Rise	Filled with air	Less than water	Floats up
Neutral	Partially filled	Equal to water	Stays at depth

The submarine is a brilliant example of how controlling density allows an object to move up and down in a fluid. It is the same principle that allows a hot air balloon to rise and fall — by changing its average density. Physics gives us mastery over the vertical dimension.

3.9 Hot Air Balloons

A hot air balloon rises because the air inside the balloon is less dense than the air outside.

How it works:

1. The burner heats the air inside the balloon.
2. Hot air expands and becomes less dense.
3. The less dense air inside the balloon creates an upthrust greater than the weight of the balloon and its payload.
4. The balloon rises.
5. To descend, the air inside is allowed to cool, increasing its density.

Think of a hot air balloon as a bubble of warm, light air in a sea of cooler, denser air. The warm air rises because it is less dense, just as a bubble of air rises in water.

Action	Air Inside Balloon	Density	Result
Heat air	Expands	Less than outside air	Rises
Cool air	Contracts	More than outside air	Descends
Neutral	Same temperature	Equal to outside air	Stays at altitude

The hot air balloon is one of the oldest and most beautiful applications of Archimedes' principle. It was invented in 1783 by the Montgolfier brothers, and it still captures our imagination today. The next time you see one, remember: it floats because of physics.

3.10 Floating and Sinking in Nature

Floating and sinking are not just human concerns — they are vital to life on Earth.

Natural Phenomenon	How It Works
Ice floats on water	Ice is less dense than water (0.92 g/cm³ vs 1.0 g/cm³)
Fish control depth	Swim bladder adjusts gas content to change density
Plankton floats	Lipids and gas bubbles reduce density
Birds float on water	Feathers trap air, reducing average density
Dead Sea floating	High salt content increases water density
Continents float on mantle	Less dense continental crust floats on denser mantle

Why ice floats: Water is unusual — it expands when it freezes. This makes ice less dense than liquid water. If ice sank, lakes and oceans would freeze from the bottom up, killing aquatic life. Because ice floats, it forms an insulating layer on top, allowing life to survive beneath.

The fact that ice floats is one of the reasons life exists on Earth. If ice sank, the oceans would freeze solid, and life as we know it would be impossible. This is a beautiful example of how a simple physical property — the density of ice — has profound consequences for the entire planet.

3.11 Applications of Floating and Sinking

Application	How It Works
Ships	Hollow shape displaces large volume of water
Submarines	Ballast tanks control average density
Hot air balloons	Heated air is less dense than surrounding air
Hydrometers	Float at different depths depending on liquid density
Life jackets	Increase volume, reducing average density
Swim bladders in fish	Adjust gas content to control depth
Plimsoll line	Marks maximum safe loading depth for ships
Car battery testing	Hydrometer measures acid density

Floating and sinking are everywhere in technology and nature. Every time you see a ship, a submarine, or a hot air balloon, you are seeing Archimedes' principle in action. It is one of the most practical and beautiful topics in all of physics.

4. Formula Box

5. Worked Examples

Example 1: Calculating Upthrust

Question: A stone of volume 0.002 m³ is fully submerged in water. Calculate the upthrust on the stone. Density of water = 1 000 kg/m³, g = 10 N/kg.

Answer:

Formula: Upthrust = ρ_fluid × V_displaced × g

Substitution: Upthrust = 1 000 × 0.002 × 10

Answer: Upthrust = 20 N

Units check: kg/m³ × m³ × N/kg = N. The answer is in newtons, correct for force. Notice that the upthrust depends only on the volume of water displaced and the density of water, not on the mass of the stone.

Example 2: Apparent Weight

Question: A stone weighs 50 N in air. When fully submerged in water, it weighs 30 N. Calculate the upthrust and the volume of the stone. Density of water = 1 000 kg/m³, g = 10 N/kg.

Answer:

Upthrust = W_actual − W_apparent = 50 − 30 = 20 N

Volume: Upthrust = ρ × V × g → 20 = 1 000 × V × 10 → V = 20 / 10 000 = 0.002 m³

Answer: Upthrust = 20 N, Volume = 0.002 m³

Units check: the answer is in N and m³. Notice that we used the apparent weight to find the upthrust. This is a common experimental method for finding the volume of an irregular object.

Example 3: Relative Density

Question: A metal block has a mass of 540 g and a volume of 200 cm³. Calculate its density and relative density. Will it float or sink in water?

Answer:

Density = m / V = 540 g / 200 cm³ = 2.7 g/cm³ = 2 700 kg/m³

Relative density = density of substance / density of water = 2.7 / 1 = 2.7

Since RD > 1, the metal will sink in water.

Units check: the answer is in g/cm³ and kg/m³. Notice that the relative density has no units. Does 2.7 make sense? Yes, this is the relative density of aluminium.

Example 4: Law of Flotation

Question: A wooden block floats in water with 60% of its volume submerged. The density of water is 1 000 kg/m³. Calculate the density of the wood.

Answer:

Let V be the total volume of the wood. Volume submerged = 0.6V.

By the law of flotation: Weight of wood = Weight of water displaced

ρ_wood × V × g = ρ_water × 0.6V × g

ρ_wood = 0.6 × ρ_water = 0.6 × 1 000 = 600 kg/m³

Answer: The density of the wood is 600 kg/m³.

Units check: the answer is in kg/m³. Notice that the fraction submerged equals the relative density of the wood. This is a useful shortcut: for a floating object, the fraction submerged = RD of the object.

Example 5: Hydrometer

Question: A hydrometer floats in water with 8 cm of its stem above the surface. When placed in another liquid, 6 cm of its stem is above the surface. Calculate the relative density of the liquid.

Answer:

Let the total length of the stem be L and the volume of the hydrometer be V.

In water: Volume submerged = V − A × 0.08 (where A is the cross-sectional area of the stem)

In liquid: Volume submerged = V − A × 0.06

By the law of flotation: Weight of hydrometer = Weight of fluid displaced

In water: W = ρ_water × (V − 0.08A) × g

In liquid: W = ρ_liquid × (V − 0.06A) × g

Equating: ρ_water × (V − 0.08A) = ρ_liquid × (V − 0.06A)

This is complex without knowing V and A. However, in a typical problem, the hydrometer is designed so that the stem is uniform and the volume submerged is approximately equal to the total volume minus the exposed stem. A simpler approach: the relative density is the ratio of the submerged lengths (if the stem is uniform). RD = (L − 0.06) / (L − 0.08). Without L, we cannot solve numerically. Let us assume the hydrometer has a total length of 0.2 m. Then RD = (0.2 − 0.06) / (0.2 − 0.08) = 0.14 / 0.12 = 1.17.

Answer: The relative density of the liquid is approximately 1.17.

Units check: relative density has no units. This problem shows how a hydrometer works — the depth to which it sinks depends on the density of the liquid. In a denser liquid, it sinks less. In a less dense liquid, it sinks more.

6. Graph Work

7. Practical / Experiment

8. Common Mistakes

• Thinking that heavy objects always sink. A ship is heavy but floats because of its shape.
• Confusing mass and density. A large object can be light if its density is low.
• Forgetting that upthrust depends on the volume displaced, not the mass of the object.
• Thinking that upthrust increases with depth for a fully submerged object. It remains constant.
• Forgetting to convert units. Volume in m³, density in kg/m³, mass in kg.
• Confusing relative density with density. Relative density has no units.
• Thinking that a floating object has no weight. It has weight, but the upthrust balances it.
• Forgetting that the law of flotation applies only to floating objects. For sinking objects, upthrust is less than weight.

9. Checkpoint Questions

10. Structured Exam-Style Questions

11. Challenge Questions

12. Real-World Applications

13. Closing Thought

Floating and sinking are not about weight alone — they are about the delicate balance between an object's weight and the weight of the fluid it displaces. In that balance, we find ships, submarines, and the beauty of Archimedes' principle.