Archimedes' principle is the key to understanding buoyancy in fluid mechanics. It states that the upward force on an object in a fluid equals the weight of fluid it displaces. This principle explains why ships float and submarines dive.

Buoyant force calculations depend on whether an object is fully or partially submerged. The relationship between an object's weight and buoyant force determines if it floats, sinks, or achieves neutral buoyancy. Understanding these concepts is crucial for designing marine vessels and predicting object behavior in fluids.

Archimedes' Principle and Buoyancy

Archimedes' principle in fluid mechanics

• States that buoyant force acting on an object immersed in a fluid equals weight of fluid displaced by object
  • Buoyant force ($F_b$) is an upward force exerted by fluid on object (ship hull, submarine)
  • Weight of displaced fluid equals volume of object multiplied by fluid density and acceleration due to gravity ($\rho V g$)
• Fundamental in understanding behavior of objects submerged in fluids
  • Helps determine stability and equilibrium of floating objects (boats, icebergs)
  • Used in designing and analyzing hydraulic systems (pumps, valves, hydraulic lifts)
• Applies to both liquids and gases, as they are both fluids
  • Objects in air experience buoyant force, although it is usually negligible compared to their weight (helium balloons)

Calculation of buoyant force

• For a fully submerged object:
  1. Buoyant force ($F_b$) equals weight of displaced fluid
  2. $F_b = \rho V g$, where $\rho$ is fluid density, $V$ is object volume, and $g$ is acceleration due to gravity
• For a partially submerged object:
  1. Buoyant force ($F_b$) equals weight of displaced fluid
  2. $F_b = \rho V_{submerged} g$, where $V_{submerged}$ is volume of object submerged in fluid
• Buoyant force acts through center of buoyancy, which is centroid of displaced fluid volume
  • Location of center of buoyancy affects stability of floating objects (ships, offshore platforms)

Conditions for floating vs sinking

• Floating occurs when buoyant force equals object's weight
  • Happens when object density is less than fluid density (wood in water, oil in water)
• Sinking occurs when object's weight is greater than buoyant force
  • Happens when object density is greater than fluid density (stone in water, steel in mercury)
• Neutral buoyancy occurs when object's weight equals buoyant force
  • Happens when object density equals fluid density (fish in water, submarines at certain depths)
  • Object remains suspended at any depth in fluid

Equilibrium of floating objects

• For a floating object in equilibrium:
  1. Buoyant force equals weight of object
  2. $F_b = W_{object}$, where $W_{object}$ is weight of object
  3. $\rho_{fluid} V_{displaced} g = m_{object} g$, where $\rho_{fluid}$ is fluid density, $V_{displaced}$ is volume of displaced fluid, $m_{object}$ is object mass, and $g$ is acceleration due to gravity
• To find volume of fluid displaced by floating object:
  • $V_{displaced} = \frac{m_{object}}{\rho_{fluid}}$
  • This volume equals volume of object submerged in fluid (iceberg, ship hull)
• Stability of floating objects depends on relative positions of center of gravity and center of buoyancy
  • Metacenter is point where lines of action of buoyant force intersect when object is tilted (ships, boats)
  • Object is stable if metacenter is above center of gravity, unstable if below (capsizing)