Buoyancy and Buoyant Force

Introduction to Buoyancy

Buoyancy is the upward force exerted by a fluid on an object immersed in it. This force arises due to the pressure difference between the top and bottom surfaces of the submerged object.

• Buoyant Force (FB): The net upward force on an object in a fluid, equal to the weight of the fluid displaced.

• Archimedes' Principle: States that the buoyant force is equal to the weight of the fluid displaced by the object.

Key Equations:

• Pressure at depth:

• Buoyant force:

• Buoyant force equals the weight of displaced fluid:

Example: If a block is submerged in water, the upward force it experiences is equal to the weight of the water displaced by the block.

Apparent Mass and Weight

When an object is submerged in a fluid, its apparent mass is less than its true mass due to the buoyant force.

• Apparent mass:

• Apparent weight:

Example: A spring scale measures a lower weight for an object submerged in water compared to its weight in air.

Applications of Buoyancy

Problem 1: Human Buoyancy

A man with mass 80 kg and density 955 kg/m3:

• Volume:

• Buoyant force:

• Weight:

• Ratio:

Conclusion: The buoyant force from air is much smaller than the person's weight.

Problem 2: Bird Bones and Buoyancy

Bird bones have air pockets, reducing their average density. When weighed in air and water:

• Mass in air:

• Apparent mass in water:

• Mass of water displaced:

• Volume:

• Average density:

Conclusion: Bird bones are less dense than other animal bones due to air pockets.

Hydrodynamics: Fluid Flow

Types of Fluid Flow

Fluid flow can be classified as either laminar or turbulent:

• Laminar flow: Smooth, orderly flow where fluid moves in parallel layers.

• Turbulent flow: Chaotic, irregular flow with mixing and eddies.

Example: Water flowing slowly through a pipe is laminar; at high speeds, it becomes turbulent.

Equation of Continuity

The equation of continuity expresses the conservation of mass in fluid flow. For an incompressible fluid:

• Where is cross-sectional area and is velocity.

• Volume rate of flow:

Example: If a pipe narrows, the fluid speed increases to conserve flow rate.

Bernoulli's Principle and Equation

Conservation of Energy in Fluids

Bernoulli's equation is derived from the conservation of energy for a flowing fluid:

• Where is pressure, is density, is velocity, and is height.

• Energy per unit volume:

Bernoulli's Theorem: In a streamline flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant.

Applications of Bernoulli's Equation

• Fluid speed from height difference:

• Lift force: Pressure difference across a wing creates lift, as shown by streamlines and Bernoulli's principle.

Example: The pressure is lower above an airplane wing than below, resulting in an upward lift force.

Problem Solving in Fluid Dynamics

Problem 3: Heart Pumping Blood

The left ventricle of a heart pumps blood at a flow rate of 83.0 cm3/s, increasing pressure by 110 mm Hg, speed from zero to 30.0 cm/s, and height by 5.00 cm.

• Flow rate:

• Pressure increase:

• Power output:

• Calculation involves kinetic, potential, and pressure energy changes.

Conclusion: Most of the heart's power output is used to increase blood pressure.

Summary Table: Key Fluid Equations

Additional info: These notes cover core concepts from Chapter 10: Fluids, including buoyancy, fluid statics, and fluid dynamics, as well as applications of Bernoulli's equation in biological and engineering contexts.