Buoyant Force and Archimedes' Principle

Fluid Mechanics

Buoyant Force

When an object is immersed in a fluid, it experiences an upward force known as the buoyant force. This force is responsible for the floating or sinking behavior of objects in fluids.

• Definition: The buoyant force is the upward force exerted by a fluid on a submerged or partially submerged object.

• Cause: The pressure in a fluid increases with depth, so the pressure on the bottom of an object is greater than at the top, resulting in a net upward force.

• Formula: The magnitude of the buoyant force $F_B$ is given by: $F_B = \rho_{fluid} V_{disp} g$ where:

  • $\rho_{fluid}$ = density of the fluid

  • $V_{disp}$ = volume of fluid displaced by the object

  • $g$ = acceleration due to gravity

• Example: A beach ball pushed underwater experiences a strong upward force due to the large volume of water it displaces.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.

• Statement: A fluid exerts an upward buoyant force on an object immersed in or floating on the fluid. The magnitude of the buoyant force equals the weight of the fluid displaced by the object.

• Mathematical Expression: $F_B = \text{weight of displaced fluid} = m_{fluid} g = \rho_{fluid} V_{disp} g$

• Application: Used to determine whether an object will float, sink, or remain neutrally buoyant in a fluid.

Float or Sink? (Density Comparison)

Whether an object floats or sinks depends on the comparison between the object's density and the fluid's density.

Object's Density	Compared to Fluid	Behavior
Lower	< Fluid	Floats
Higher	> Fluid	Sinks
Equal	= Fluid	Neutral Buoyancy ("hangs" in fluid)

Worked Example: Floating Block

Problem: A block of plastic floats in water. 20% of the block’s volume is above the waterline, the rest is submerged. What is the density of the block?

• Given: 20% above water, 80% submerged; $\rho_{water} = 1000\ \text{kg/m}^3$

• Solution:

  1. By Archimedes' principle, the weight of the floating object equals the weight of the water displaced.

  2. Let $\rho_0$ be the density of the block, $V_0$ its total volume. The volume of water displaced is $V_{disp} = 0.8 V_0$.

  3. Set up the equation: $\rho_0 V_0 g = \rho_{water} (0.8 V_0) g$

  4. Cancel $V_0$ and $g$: $\rho_0 = 0.8 \rho_{water}$

  5. Substitute values: $\rho_0 = 0.8 \times 1000\ \text{kg/m}^3 = 800\ \text{kg/m}^3$

• Conclusion: The block's density is 800 kg/m3.

Apparent Weight

The apparent weight of an object is its weight as measured when submerged in a fluid. It is less than its actual weight due to the upward buoyant force.

• Definition: Apparent weight is the reading on a scale when the object is in a fluid.

• Relationship: $w' = w - F_B$ where $w'$ is the apparent weight, $w$ is the true weight, and $F_B$ is the buoyant force.

• Application: Measuring apparent weight in different fluids can help determine the object's volume or density.

• Example: A casserole dish has a mass of 1555 g. When immersed in water, its apparent mass is 880 g. The difference is due to the buoyant force.

Buoyancy in Air

Buoyant force is present in all fluids, including air. Objects less dense than air (such as helium balloons) rise due to this force, though the effect is much smaller than in water because air's density is much lower.

• Example: Helium balloons rise in air because helium is less dense than the surrounding atmosphere.

• Biological Application: Birds have hollow bones to reduce their density, making flight easier by maximizing the small buoyant force of air.

Summary Table: Buoyant Force Concepts

Concept	Formula	Description
Buoyant Force	$F_B = \rho_{fluid} V_{disp} g$	Upward force exerted by a fluid
Archimedes' Principle	$F_B = \text{weight of displaced fluid}$	Buoyant force equals weight of displaced fluid
Apparent Weight	$w' = w - F_B$	Measured weight in a fluid

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