Fluid statics: Pressure in Fluids

• Fluid statics studies fluids at rest.
• Fluid statics is a branch of fluid mechanics that addresses equilibrium conditions relating to floating and submerged bodies.
• Fluids at rest do not experience shear stress.
• The only stress in a fluid at rest is normal stress, known as pressure.

Pressure in a Fluid

• Pressure refers to the normal force a fluid exerts per unit area.
• Pressure is a scalar quantity, possessing magnitude but lacking direction.
• Pressure at a given point in a fluid remains uniform across all directions.
• Pressure in a fluid escalates with depth due to the weight of the overlying fluid.
• Pressure at depth h in a fluid with density ρ is calculated by: P = P₀ + ρgh, where P₀ represents surface pressure and g stands for gravitational acceleration.
• Gauge pressure is the differentiation between absolute pressure and atmospheric pressure: P_gauge = P - P_atm.
• Absolute pressure is the total pressure a fluid exerts, which includes atmospheric pressure.

Pascal's Law

• Pascal's Law: pressure change on an enclosed fluid transmits undiminished to all points in the fluid and the container walls.
• This principle underlies hydraulic systems.
• Hydraulic systems multiply force using pistons of varying sizes: F₁/A₁ = F₂/A₂, with F as force and A as area.
• Applying a small force on a small area results in a larger force on a larger area.

Buoyancy and Archimedes' Principle

• Buoyancy is the upward force exerted by a fluid opposing the weight of an immersed object.
• Archimedes' Principle: the buoyant force is equal to the weight of the fluid displaced by the object: F_B = ρ_fluid * V_displaced * g.
• An object floats if the buoyant force exceeds its weight.
• An object sinks if the buoyant force is less than its weight.
• An object is neutrally buoyant if buoyant force equals its weight.
• The apparent weight of a submerged object is the difference between its actual weight and the buoyant force.
• For floating objects, weight equals buoyant force: W = F_B.
• For floating objects, the weight of the displaced fluid equals the weight of the object.
• The fraction of a submerged object equals the ratio of the object's density to the fluid's density.

Stability of Floating Objects

• Stability is an objects ability to return to its original orientation after disruption.
• The center of gravity (CG) is the point where the entire object's weight is considered to act.
• The center of buoyancy (CB) is the center of gravity of the displaced fluid.
• A floating object remains stable if its center of buoyancy is above its center of gravity, or its metacenter is above its center of gravity.
• The metacenter is where a vertical line through the tilted object's center of buoyancy intersects the original line containing the center of buoyancy
• Metacentric height (GM) is the distance separating the center of gravity and the metacenter.
• A larger metacentric height indicates greater stability, whereas too large of a metacentric height can cause uncomfortable motions.

Pressure Measurement

• Barometers measure atmospheric pressure.
• Mercury barometers use the height of a mercury column to measure pressure: P = ρgh.
• Manometers measure the pressure difference between two points.
• Simple manometers use a U-shaped tube filled with fluid.
• Differential manometers measure small pressure differences.
• Piezometers measure the pressure of a liquid at rest.
• Pressure transducers convert pressure into an electrical signal.

Applications of Fluid Statics

• Utilized in hydraulic systems: brakes, lifts, and jacks.
• Used in the design of ships and submarines.
• Used for buoyancy calculations for balloons including other floating objects.
• Used for pressure measurement in various applications.
• Used for dam design, where water pressure increases with depth.
• Used in atmospheric pressure understanding, including variations with altitude.