$\rho = \frac{m}{V}$

• Mass per unit volume
• Units: kg/m3
• Characterizes materials
• Important for buoyancy calculations

$P = \frac{F}{A}$

• Force per unit area
• Units: Pa (Pascal) = N/m2
• Also use: atm, mm Hg, bar
• 1 atm = 101.3 kPa = 760 mm Hg

• Absolute pressure: Total pressure (including atmosphere)
• Gauge pressure: Pressure above atmospheric pressure
• $P_{abs} = P_{gauge} + P_{atm}$
• Tire gauges and blood pressure gauges typically read gauge pressure

$P = P_0 + \rho gh$

• Pressure increases with depth in fluid
• $P_0$: pressure at surface
• $\rho$: fluid density
• $h$: depth below surface
• Explains water pressure increases with depth

• Linear increase in pressure with depth
• Deeper = higher pressure
• Dependent on fluid density
• Independent of container shape
• Same horizontal level: same pressure (connected vessels)

• Pressure applied to enclosed fluid transmitted undiminished
• $\frac{F_1}{A_1} = \frac{F_2}{A_2}$
• Hydraulic lifts operate on this principle
• Used in hydraulic brakes, jacks, presses

• Buoyant force equals weight of fluid displaced
• $F_B = \rho_{fluid} V_{displaced}g$
• Upward force on submerged object
• Explains why objects float or sink

$F_B = \rho_{fluid}V_{submerged}g$

• Depends on fluid density and submerged volume
• Not equal to object's weight
• Can support objects that would otherwise sink

• Floats: $\rho_{object} < \rho_{fluid}$
• Partially submerged: $V_{sub} = \frac{\rho_{obj}}{\rho_{fluid}}V_{total}$
• Sinks: $\rho_{object} > \rho_{fluid}$
• Entirely submerged (if completely submerged)
• Accelerates downward if not supported
• Neutrally buoyant: $\rho_{object} = \rho_{fluid}$

• Apparent weight = actual weight - buoyant force
• $W_{app} = mg - \rho_{fluid}V_{sub}g$
• Object feels lighter in fluid
• Explains why lifting in water is easier
• Basis for buoyancy devices

• Incompressible (constant density)
• Non-viscous (no internal friction)
• Irrotational (no rotation)
• Steady flow (velocity at each point constant over time)
• Simplified model for real fluids

$A_1v_1 = A_2v_2 = \text{constant}$

• Flow rate constant throughout pipe
• Product of area and velocity constant
• Narrower section -> higher velocity
• Wider section -> lower velocity
• Volume flow rate: $Q = Av$ (m3/s)

$Q = Av$

• Volume of fluid passing point per unit time
• Units: m3/s
• Constant in steady flow (incompressible fluid)
• Same flow rate at all pipe cross-sections

$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

• Conservation of energy per unit volume
• Pressure + kinetic energy + potential energy constant along streamline
• Higher velocity -> lower pressure
• Height differences affect pressure

• Energy conservation for ideal fluids
• Pressure energy + kinetic energy + potential energy
• Explains: venturi effect, airplane lift
• Basis for flow measurement devices

$v = \sqrt{2gh}$

• Speed of fluid exiting tank
• Same as free-fall speed from height h
• Torricelli's law
• Assumes no friction/viscosity