General Science Unit 4: The Physics of Fluids - Archimedes' Principle

Foundations of Buoyancy and Archimedes’ Principle

Across centuries and cultures, the ability of objects to float has influenced the way humans navigate, transport goods, and survive in aquatic environments. Whether it is a child playing with a plastic toy in a bathtub or engineers designing massive cargo ships, the principle behind floating remains the same. Archimedes’ Principle, which explains how fluids exert upward forces on submerged objects, is foundational in both physics and engineering. This principle offers a powerful explanation for many natural and engineered phenomena, from floating boats and hot air balloons to submerged submarines and floating sensors. Understanding this concept deepens insight into how physical properties govern whether objects float, sink, or hover in a fluid.

Archimedes’ Principle states that any object wholly or partially immersed in a fluid experiences an upward force, known as the buoyant force, which is equal to the weight of the fluid it displaces. This discovery occurred over 2,000 years ago when the Greek mathematician Archimedes realized that when he stepped into a bathtub, the water level rose. This indicated that his body had displaced a certain volume of fluid. Legend describes Archimedes shouting "Eureka!" upon this realization. This displacement is the fundamental key to understanding why objects behave differently in fluids.

Experimental Verification and Quantitative Measurement

Archimedes’ Principle can be verified through a practical experiment using a spring balance, a solid object, an overflow can, and a collection container. When an object is weighed in air, a spring balance shows its full weight, such as $7.00\,kg$. However, when the object is fully submerged in water, the balance shows a reduced reading. If the reading drops to $5.00\,kg$, it indicates that the object experiences an upward buoyant force equivalent to the weight of $2.00\,kg$, which is approximately $19.6\,N$. Simultaneously, the water displaced by the submerged object overflows into a container. If that displaced water is weighed, it will weigh exactly $2.00\,kg$, confirming that the buoyant force is equal to the weight of the displaced fluid.

The buoyant force is not an abstract concept but a measurable and predictable quantity calculated using the following general equation:

$F_b = \rho \times V \times g$

In this formula, $F_b$ is the buoyant force measured in newtons ($N$). The variable $\rho$ (rho) represents the density of the fluid, such as $1000\,kg/m^3$ for freshwater. The variable $V$ represents the volume of the fluid displaced, measured in cubic meters ($m^3$). Finally, $g$ represents the acceleration due to gravity, which is constant at $9.8\,m/s^2$. This formula shows that the invisible upward force depends entirely on the volume of fluid displaced and the density of that fluid.

Core Physical Properties: Mass, Volume, and Density

To understand why objects float or sink, three physical properties must be defined and understood:

Mass is the amount of matter in an object, traditionally measured in kilograms ($kg$) or grams ($g$). For instance, a 1-liter bottle of water has a mass of approximately $1,000\,g$ or $1\,kg$.

Volume refers to the amount of space an object occupies, measured in cubic meters ($m^3$), cubic centimeters ($cm^3$), or liters ($L$). A 1-liter bottle has a volume of $1,000\,cm^3$.

Density is the ratio of mass to volume, indicating how compact matter is within a specific space. The relationship between these properties determines buoyancy. For example, if four blocks have the same volume of $5.00\,L$ but different masses, they will submerge to different depths. A block with a mass of $2.00\,kg$ will float significantly, while a block with a mass of $8.00\,kg$ will sink completely. This demonstrates that as mass increases for a fixed volume, density increases, causing the object to sink deeper.

Comparative Analysis of Buoyant Behavior

By comparing the density of objects to the density of water ($1,000\,kg/m^3$), one can predict flotation. Consider four blocks (A, B, C, and D) all possessing a volume of $5.00\,L$ or $0.005\,m^3$:

Block A has a mass of $8.00\,kg$. Its density is calculated as $\frac{8.00\,kg}{0.005\,m^3} = 1,600\,kg/m^3$. Because its density is higher than water, it sinks deepest.

Block B has a mass of $2.00\,kg$. Its density is $\frac{2.00\,kg}{0.005\,m^3} = 400\,kg/m^3$. Since it is less dense than water, it floats high.

Block C sinks because its mass makes it denser than water.

Block D partially submerges because it remains less dense than water, though it is denser than Block B.

The specific conditions for movement in a fluid are as follows:

If the buoyant force is greater than the object's weight, the object will rise and float ($F_b > W$).

If the buoyant force is equal to the object's weight, it remains suspended, a state known as neutral buoyancy ($F_b = W$).

If the object's weight is greater than the buoyant force, the object sinks ($W > F_b$).

Floating vs. Sinking: Mathematical Distinctions

There is a crucial distinction between submerged and floating objects regarding displaced volume. When an object is completely submerged, such as a rock underwater, the volume of the fluid displaced ($V_{displaced}$) is exactly equal to the volume of the object ($V_{object}$). When an object is floating, like a boat, the volume of the fluid displaced is less than the total volume of the object ($V_{displaced} < V_{object}$) because only a portion of the object is beneath the fluid surface.

For Block A (Sinking): With a mass of $8.00\,kg$ and volume of $0.005\,m^3$, its weight is $W = 8.00\,kg \times 9.8\,m/s^2 = 78.4\,N$. The buoyant force it experiences while submerged is $F_b = 1000\,kg/m^3 \times 0.005\,m^3 \times 9.8\,m/s^2 = 49.0\,N$. Since $78.4\,N > 49.0\,N$, the block sinks. The volume displaced is the full $0.005\,m^3$.

For Block B (Floating): With a mass of $2.00\,kg$ and volume of $0.005\,m^3$, we find the submerged volume by setting the buoyant force equal to the weight: $F_b = Weight$. This allows us to solve for volume: $V_{displaced} = \frac{m_{object}}{\rho_{fluid}}$. Plugging in values, $V_{displaced} = \frac{2.00\,kg}{1000\,kg/m^3} = 0.002\,m^3$. Since $0.002\,m^3$ equals $2.00\,L$, we find that only $2.00\,L$ of the $5.00\,L$ block is submerged.

Engineering and Design Considerations for Buoyancy

Several physical properties are manipulated by designers to optimize buoyancy:

Mass: Higher mass typically increases weight, which increases the likelihood of sinking unless it is balanced by a large volume.

Volume: A larger volume increases the amount of fluid displaced, which in turn increases the buoyant force. This is why wide, hollow structures are effective at floating.

Shape: Shape determines how fluid is displaced. A flat, broad shape will displace more water than a compact, dense shape of the same mass. Ship designers utilize wide hulls to maximize displacement and minimize sinking risks.

In the Philippines, coastal fishing communities utilize bangkas, which are outrigger canoes made from fiberglass or wood. These boats are designed with wide hulls to displace enough water to support the weight of fishers, equipment, and catch. This demonstrates an empirical application of Archimedes' Principle through generations of craftsmanship. Similarly, students at a Philippine science high school developed a prototype rescue floatation vest by using lightweight, recyclable plastic bottles. They calculated the volume each bottle could displace to ensure a sufficient buoyant force, showing how the principle can be applied for community-centered innovation and disaster preparedness.

Practical Applications of Archimedes’ Principle

Archimedes' Principle is ubiquitous in various settings:

Home and Personal Use: Bathtubs and sinks demonstrate displacement through overflowing. Toy boats and floating furniture during floods are common examples.

Transportation and Business: Cargo ships are specifically engineered to distribute immense loads over a wide area to displace enough water to remain afloat while carrying tons of goods.

Industrial and Specialized Vehicles: Submarines utilize ballast tanks to manipulate buoyancy. By taking in water, the ballast tanks increase the submarine’s average density, causing it to sink. By expelling water and replacing it with air, the density decreases, allowing the vessel to rise. This controlled adjustment of density allows machines to move vertically within fluid environments. This principle is also fundamental to the operation of hydrometers and life-saving devices.

Practice Examples and Solutions

Example 1: A wooden block with a mass of $4.0\,kg$ floats in freshwater. To find the volume of water displaced, we acknowledge that for a floating object, the buoyant force equals the weight ($F_b = W$). Since $F_b = \rho \times V \times g$ and $W = m \times g$, we equate them: $\rho \times V \times g = m \times g$. This simplifies to $V = \frac{m}{\rho}$. Solving with the given values: $V = \frac{4.0\,kg}{1000\,kg/m^3} = 0.004\,m^3$, which equivalent to $4.0\,L$.

Example 2: A metal block with a volume of $0.005\,m^3$ is fully submerged in water, and it weighs $60.0\,N$ in air. To find the reading on a spring balance (apparent weight), we first compute the buoyant force: $F_b = 1000\,kg/m^3 \times 0.005\,m^3 \times 9.8\,m/s^2 = 49.0\,N$. We then subtract the buoyant force from the actual weight: $60.0\,N - 49.0\,N = 11.0\,N$. The spring balance reading is $11.0\,N$.

Example 3: A block has a mass of $6.0\,kg$ and a volume of $0.004\,m^3$. To determine if it floats or sinks, we calculate its density: $\text{Density} = \frac{6.0\,kg}{0.004\,m^3} = 1,500\,kg/m^3$. Since this is greater than the density of water ($1000\,kg/m^3$), it should sink. To justify with force: the weight is $W = 6.0\,kg \times 9.8\,m/s^2 = 58.8\,N$, and the buoyant force when submerged is $F_b = 1000\,kg/m^3 \times 0.004\,m^3 \times 9.8\,m/s^2 = 39.2\,N$. Because weight exceeds buoyant force ($58.8\,N > 39.2\,N$), the block sinks.

Practice Exercises

Scenario 1: A plastic container floats in water with $75\%$ of its volume submerged. If its total volume is $8.0\,L$, what is the weight of the container?

Scenario 2: A student drops a solid object into an overflow can. It displaces $0.003\,m^3$ of water. What is the magnitude of the buoyant force acting on it?

Scenario 3: A rock weighs $40.0\,N$ in air and $25.0\,N$ when submerged in water. If water has a density of $1,000\,kg/m^3$, what is the volume of the rock?