Topic 8: Fluids

Fluids are substances that flow and take the shape of their container, including both liquids and gases. This unit treats fluids using familiar conservation laws, describing how pressure, density, and flow speed are connected. Mastering a handful of relationships lets you predict floating, sinking, hydraulic lifts, and the behavior of moving streams.

Density and Pressure

Density measures how much mass is packed into a given volume, written rho = m/V with units of kg/m^3. Water sits near 1000 kg/m^3, while air at sea level is roughly 1.2 kg/m^3, which is why objects behave so differently in each. Pressure is force spread over an area, P = F/A, measured in pascals where 1 Pa equals 1 N/m^2. Because the same force concentrated on a smaller area produces a larger pressure, a sharp knife cuts more easily than a dull one. Fluid pressure acts equally in all directions at a point and always pushes perpendicular to any surface it contacts.

Pressure as a Function of Depth

Inside a fluid at rest, pressure increases as you go deeper because more weight of fluid presses down from above. The relationship is P = P0 + rho g h, where P0 is the pressure at the top surface, rho is fluid density, g is the gravitational field strength, and h is the depth below that surface. The term rho g h is often called the gauge pressure, the amount added on top of the surface pressure. Notice that depth, not horizontal position or container shape, sets the pressure, so two connected containers of different widths show the same fluid level. At a lake surface P0 is atmospheric pressure, about 1.0 x 10^5 Pa, and every 10 m of water adds roughly another full atmosphere.

Buoyant Force and Archimedes' Principle

Because pressure grows with depth, the upward push on the bottom of a submerged object exceeds the downward push on its top, producing a net upward buoyant force. Archimedes' principle states that this buoyant force equals the weight of the fluid displaced: F_b = rho_fluid g V_displaced. An object floats when the buoyant force balances its weight and sinks when its weight wins. Worked example: a solid block of volume 2.0 x 10^-3 m^3 is fully submerged in water (rho = 1000 kg/m^3) with g = 10 N/kg. The buoyant force is F_b = (1000)(10)(2.0 x 10^-3) = 20 N upward. If the block weighs 30 N, the net force is 30 - 20 = 10 N downward, so it sinks; a scale attached to it would read an apparent weight of 10 N.

Pascal's Principle and Hydraulics

Pascal's principle says that a pressure change applied to an enclosed fluid is transmitted undiminished to every part of that fluid. Hydraulic systems exploit this: a small force on a narrow piston creates a pressure that acts on a wide piston to produce a much larger force. Since pressure is shared, F1/A1 = F2/A2, so the wide piston multiplies force in proportion to its larger area. This is not free energy, though, because the wide piston moves a shorter distance; the work done on each side is equal, honoring conservation of energy. Car brakes and lift jacks rely on exactly this trade of force for distance.

The Continuity Equation

When an incompressible fluid flows steadily through a pipe, the same volume must pass every cross section each second, since fluid cannot pile up or vanish. This gives the continuity equation A1 v1 = A2 v2, where A is cross-sectional area and v is flow speed. The product A times v is the volume flow rate, measured in m^3/s and held constant along the pipe. The practical consequence is that fluid speeds up where the pipe narrows and slows where it widens, which is why pinching a garden hose makes the water shoot out faster. Continuity is really a statement of conservation of mass for a constant-density fluid.

Bernoulli's Equation and Applications

Bernoulli's equation applies energy conservation to a flowing ideal fluid, balancing pressure, height, and speed: P + rho g y + (1/2) rho v^2 stays constant along a streamline. Each term is an energy per unit volume, so this is the work-energy theorem written for a parcel of fluid. A key result is that where a fluid moves faster, its pressure drops, and where it slows, its pressure rises. This explains lift on an airplane wing, the curve of a spinning ball, and why a shower curtain pulls inward when the water runs. Bernoulli's equation assumes steady, nonviscous, incompressible flow, so it is an idealization rather than an exact description of real plumbing.

Conservation Laws Applied to Fluid Flow

The fluid relationships in this unit are not separate rules but applications of the same conservation laws used throughout physics. Continuity is conservation of mass, ensuring flow rate is preserved as area changes. Bernoulli's equation is conservation of energy, tracking how pressure energy, kinetic energy, and gravitational potential energy convert into one another along a streamline. Hydraulic systems combine conservation of energy with Pascal's principle, trading force for distance while keeping total work constant. Recognizing which conservation idea governs a problem is often the fastest route to a solution, so before plugging numbers, ask whether mass, energy, or pressure transmission is the central concept.