Archimedes' Principle: Why Things Float or Sink
The Story of a Eureka Moment
The story begins over 2000 years ago in the ancient city of Syracuse with a brilliant thinker named Archimedes. The king, Hiero II, suspected that a new crown he had commissioned was not made of pure gold but was adulterated with silver. He needed to prove this without damaging the crown, so he turned to Archimedes. The problem tormented Archimedes until one day, as he stepped into a full bath, he noticed water spilling over the sides. He realized that the volume of water displaced was equal to the volume of the part of his body he had submerged. In a flash of insight, he understood he could use this to solve the king's problem. He was so excited that he reportedly ran through the streets naked, shouting "Eureka! Eureka!" which means "I have found it!" in Greek.
Defining the Principle and Its Components
Archimedes' Principle can be formally stated as:
Let's break down the key terms in this definition:
- Fluid: A substance that can flow and take the shape of its container. Liquids (like water, oil) and gases (like air, helium) are both fluids.
- Upthrust (or Buoyant Force, $F_b$): This is the upward push exerted by the fluid on the object. It's the reason why you feel lighter when you're in a swimming pool.
- Displaced Fluid: When an object enters a fluid, it pushes some of the fluid aside. This "moved" fluid is called the displaced fluid.
- Weight of Displaced Fluid: This is the force of gravity acting on the mass of the displaced fluid. If you know the volume of fluid displaced ($V_{disp}$) and the density of the fluid ($\rho_{fluid}$), you can calculate it. The mass of displaced fluid is $m_{fluid} = \rho_{fluid} \times V_{disp}$. Its weight is $W_{fluid} = m_{fluid} \times g = \rho_{fluid} \times V_{disp} \times g$, where $g$ is the acceleration due to gravity (~9.8 m/s²).
Therefore, the mathematical formula for Archimedes' Principle is:
$F_b = W_{fluid displaced} = \rho_{fluid} \times V_{disp} \times g$
The Crucial Role of Density
Density is the secret ingredient that determines whether an object will float or sink. Density ($\rho$) is defined as mass per unit volume: $\rho = m / V$. It tells us how much "stuff" is packed into a given space.
When you place an object in a fluid, two forces are at war: the weight of the object ($W_{object}$) pulling it down, and the buoyant force ($F_b$) pushing it up. What happens next depends on the density of the object compared to the density of the fluid.
| Condition | Forces at Play | Result | Example |
|---|---|---|---|
| Object's Density > Fluid's Density ($\rho_{object} > \rho_{fluid}$) | $W_{object} > F_b$ | The object sinks. | A metal coin in water. |
| Object's Density < Fluid's Density ($\rho_{object} < \rho_{fluid}$) | $W_{object} < F_b$ | The object floats. It will rise to the surface and displace a volume of fluid whose weight equals its own weight. | A wooden block in water. |
| Object's Density = Fluid's Density ($\rho_{object} = \rho_{fluid}$) | $W_{object} = F_b$ | The object is neutrally buoyant. It will remain at rest at whatever depth it is placed, neither sinking nor rising. | A submarine at a controlled depth, or a fish with a swim bladder properly adjusted. |
Archimedes' Principle in Action: From Ships to Balloons
Let's explore how this principle operates in various real-world scenarios.
How a Giant Ship Floats: A solid steel bar will sink immediately in water because steel is denser than water. So, how does a ship made of thousands of tons of steel float? The answer lies in its shape. A ship is not a solid block of metal; it's a hollow shell. This design encloses a large volume of air, which is much less dense than water. The total volume of the ship (steel + air) is immense. When the ship is in the water, it displaces a volume of water that weighs as much as the entire ship. The weight of this displaced water creates a huge buoyant force that counteracts the ship's weight, allowing it to float. The "waterline" on a ship's hull marks the point where the weight of the displaced water equals the weight of the ship.
Hot Air Balloons and Helium Balloons: Archimedes' Principle applies to gases as well as liquids. The surrounding air is a fluid. A hot air balloon floats because the air inside its envelope is heated, making it less dense than the cooler air outside. The weight of the cold air displaced by the balloon is greater than the total weight of the balloon (envelope, basket, burner, and the hot air inside). This creates a net upward force, causing the balloon to rise. Similarly, a helium balloon floats because helium gas is much less dense than the nitrogen-oxygen mixture that makes up our atmosphere.
Swimming and Submarines: When you're swimming, you feel lighter. This is because the water is providing a buoyant force that opposes your weight. A submarine uses Archimedes' Principle to control its depth. It has ballast tanks that can be filled with water or air. To dive, the tanks are flooded with water, increasing the submarine's overall density until it becomes greater than that of the surrounding water. To surface, compressed air is pumped into the tanks, forcing the water out. This decreases the submarine's density, making it less than the water's density, and the submarine rises.
The Apparent Weight Loss: If you hang a weight from a spring scale and note its reading, that is its true weight in air. Now, if you submerge the weight in water, the scale reading decreases. The new, lower reading is called the apparent weight. The difference between the true weight and the apparent weight is exactly the buoyant force. You can verify Archimedes' Principle by collecting the displaced water and weighing it; its weight will match the calculated buoyant force.
Common Mistakes and Important Questions
Q: Does the buoyant force depend on the depth or the weight of the object?
No, and no. The buoyant force depends only on the density of the fluid and the volume of the object that is submerged ($F_b = \rho_{fluid} \times V_{submerged} \times g$). For a fully submerged object, the buoyant force is constant at any depth because the fluid density is essentially constant for liquids, and the submerged volume doesn't change. The weight of the object itself is irrelevant for calculating the buoyant force; it only determines whether the object will sink or float based on the comparison of forces.
Q: If two objects have the same weight but different shapes, will they experience the same buoyant force?
Not necessarily. The buoyant force depends on the submerged volume, not the weight. Imagine a 1 kg block of steel and a 1 kg block of Styrofoam. They weigh the same. The solid steel block is very dense, so its volume is small. When fully submerged, it displaces a small volume of water, resulting in a small buoyant force. The Styrofoam block is very light and has a large volume. When fully submerged, it displaces a large volume of water, resulting in a large buoyant force. This is why the Styrofoam floats easily while the steel sinks.
Q: Does Archimedes' Principle work in a gravity-free environment, like space?
No. The principle relies on the concept of weight, which is the force due to gravity. In microgravity, objects are weightless, and fluids do not have weight. Therefore, the concept of "weight of the displaced fluid" becomes zero, and there is no buoyant force. Objects in a fluid in space would not float or sink in the way they do on Earth.
Footnote
1 Buoyant Force ($F_b$): The upward force exerted by a fluid that opposes the weight of an object immersed in it.
2 Density ($\rho$): A measure of mass per unit volume, calculated as $\rho = m/V$. It is typically measured in kg/m³ or g/cm³.
3 Displaced Fluid: The volume of fluid that is pushed out of the way when an object is placed in it.
4 Neutral Buoyancy: A state where an object's average density is equal to the density of the fluid it is in, causing it to neither sink nor float.