Core Concept
Definition of principle: Archimedes’ Principle tells us that when you place an object in a fluid—like water or air—the fluid pushes up on the object. This push is called the buoyant force. The object could be fully under the fluid or just partly in it, but either way, the fluid creates an upward force that acts against the object’s weight.
Displaced fluid weight: This upward force (the buoyant force) is equal to the weight of the fluid that the object pushes aside, or displaces. For example, if a boat pushes aside 10 kg of water, the water pushes back up with the same force as 10 kg of weight.
Fluid types: Archimedes’ Principle doesn’t just apply to water. It works with all fluids, which include both liquids and gases. So this principle explains why boats float in water and also why hot air balloons rise in air.
Understanding Buoyant Force
Origin of buoyant force: The buoyant force is caused by the fact that pressure in a fluid increases with depth. So when an object is under the fluid, the bottom part of the object feels more pressure than the top. This difference creates an upward push.
Pressure depth relationship: In fluids, the deeper you go, the higher the pressure. That means the part of the object that is lower down will feel a stronger push than the part near the surface. This is why you feel more pressure on your ears the deeper you swim underwater.
Net upward force: Because the pressure at the bottom of the object is greater than the pressure at the top, the difference in force creates a net (total) force pushing the object upward. This net force is what we call the buoyant force.
Formula for Buoyant Force
Buoyant force formula: We can calculate how strong the buoyant force is by using a simple formula: 𝐹ᴮ = 𝑉ρg. This formula tells us that the upward force depends on how much fluid the object displaces, how dense the fluid is, and how strong gravity is.
Variable definitions: Let’s break down what the symbols mean:
- 𝐹ᴮ is the buoyant force, which is the upward force from the fluid.
- 𝑉 is the volume of fluid that the object pushes aside.
- ρ (rho) is the density of the fluid, or how heavy it is for its size.
- g is gravitational acceleration, which is about 9.8 m/s² on Earth and shows how strongly gravity pulls things.
Weight of displaced fluid: Since we know mass equals density times volume (𝑚 = ρ𝑉), and weight equals mass times gravity (𝑊 = 𝑚g), we can combine these two to say 𝐹ᴮ = ρ𝑉g. That means the buoyant force equals the weight of the fluid that’s been displaced.
Conditions for Floating and Sinking
Sinking condition: An object will sink in a fluid if the weight of the object is greater than the buoyant force. In other words, if 𝐹ᴮ < 𝑊, then the object cannot be supported by the fluid and it sinks.
Floating condition: If the weight of the object is equal to the buoyant force (𝐹ᴮ = 𝑊), then the object floats in the fluid or stays suspended without rising or sinking. It is in balance.
Rising condition: If the buoyant force is greater than the object’s weight (𝐹ᴮ > 𝑊), then the object will rise through the fluid until it floats at the top or until the forces are balanced.
Density and Buoyancy Relationship
Sinking by density: If the object is denser than the fluid around it (ρₒᵦⱼₑ꜀ₜ > ρfₗᵤᵢd), it will sink. Density is how much mass is packed into a given space. Heavy things for their size tend to sink.
Floating by density: If the object’s density is less than or equal to the fluid’s density (ρₒᵦⱼₑ꜀ₜ ≤ ρfₗᵤᵢd), the object will float. That’s why wood floats in water—it’s less dense.
Displacement relationship: A floating object pushes aside (displaces) just enough fluid so that the fluid’s weight equals the object’s weight. This keeps it afloat. That relationship is written as:
ρfₗᵤᵢd × 𝑉dᵢₛₚₗₐcₑd × g = ρₒᵦⱼₑ꜀ₜ × 𝑉ₒᵦⱼₑ꜀ₜ × g
If we remove g from both sides, we get:
𝑉dᵢₛₚₗₐcₑd = (ρₒᵦⱼₑ꜀ₜ / ρfₗᵤᵢd) × 𝑉ₒᵦⱼₑ꜀ₜ
This tells us how much fluid needs to be displaced for the object to float.
Applications of Archimedes’ Principle
Ships and Submarines
Ship buoyancy: Even though ships are made of metal, they float because they have a lot of empty space inside. This makes their overall density less than water, allowing them to float.
Diving submarines: Submarines take in water into their ballast tanks, which makes them heavier and increases their density. This causes them to sink to deeper levels.
Surfacing submarines: When submarines want to rise, they pump out the water from their ballast tanks and replace it with air. This reduces their weight and makes them float upward.
Neutral buoyancy: Submarines can stay at the same depth without rising or sinking by adjusting their weight so that the buoyant force exactly equals their weight. This is called neutral buoyancy.
Hydrometers
Hydrometer function: A hydrometer is a tool that floats in different liquids to measure how dense the liquids are. It floats deeper in lighter liquids and higher in heavier ones.
Density reading: The scale printed on the hydrometer stem shows the liquid’s density depending on how deep the hydrometer floats. It tells us how concentrated or dilute a liquid is.
Hot Air Balloons
Buoyant lift in air: Hot air balloons float in air just like boats float in water. When the air inside the balloon is heated, it becomes less dense than the cooler air outside, which makes the balloon rise.
Rising mechanism: The balloon lifts off the ground if the buoyant force from the surrounding air is greater than the weight of the balloon system (the balloon, basket, and people).
Descent mechanism: If the air inside the balloon cools down or is released, it becomes denser and heavier, which reduces the buoyant force and causes the balloon to slowly come down.
Floating altitude: A hot air balloon can stay at a certain height when the buoyant force is exactly equal to its weight. At this point, the balloon hovers steadily in the air.
Plimsoll Line
Plimsoll purpose: The Plimsoll line is a special mark on the side of a ship that shows how much load the ship can safely carry. If the water reaches above the line, the ship is overloaded.
Density variation effect: Ships float differently in different water types. They float higher in cold saltwater (which is denser) and lower in warm freshwater (which is less dense). The Plimsoll line adjusts for these changes.
Safety control: The Plimsoll line helps make sure ships are not carrying too much weight. It protects the ship from sinking by keeping the underwater part of the ship small enough for the buoyant force to match the ship’s total weight.