Fluids & Pressure Cheat Sheet

A printable reference covering density, pressure, Pascal's principle, buoyancy, continuity, Bernoulli's equation, and viscosity for grades 10-12.

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Fluids include liquids and gases, and they are described by how they flow, push, and support objects. This cheat sheet helps students connect density, pressure, buoyancy, and fluid motion in one organized reference. These ideas are important for solving problems about hydraulics, floating objects, pipes, blood flow, aircraft, and weather systems. It is especially useful when deciding which formula matches a physical situation. The core idea is that pressure comes from force spread over area and from the weight of fluid above a point. Buoyancy depends on the weight of displaced fluid, not just the size or weight of the object. Moving fluids follow conservation ideas such as continuity and Bernoulli's equation. Viscosity adds real-world resistance, so not every fluid behaves like an ideal fluid.

Key Facts

Density is mass per unit volume, given by $\rho = \frac{m}{V}$ .
Pressure is force per unit area, given by $P = \frac{F}{A}$ .
Hydrostatic pressure increases with depth according to $P = P_0 + \rho g h$ .
Pascal's principle gives hydraulic force multiplication as $\frac{F_1}{A_1} = \frac{F_2}{A_2}$ .
The buoyant force on an object is the weight of displaced fluid, $F_B = \rho_{\text{fluid}} g V_{\text{disp}}$ .
For steady incompressible flow, the continuity equation is $A_1 v_1 = A_2 v_2$ .
Bernoulli's equation for ideal steady flow is $P + \frac{1}{2}\rho v^2 + \rho g y = \text{constant}$ .
For laminar flow through a pipe, Poiseuille's law is $Q = \frac{\pi r^4 \Delta P}{8 \eta L}$ .

Vocabulary

Density: Density is the mass per unit volume of a substance, calculated by $\rho = \frac{m}{V}$ .
Pressure: Pressure is the normal force applied per unit area, calculated by $P = \frac{F}{A}$ .
Gauge Pressure: Gauge pressure is pressure measured relative to atmospheric pressure, so $P_{\text{gauge}} = P_{\text{absolute}} - P_{\text{atm}}$ .
Buoyant Force: Buoyant force is the upward force a fluid exerts on an object equal to the weight of the displaced fluid.
Continuity: Continuity means mass flow is conserved, so an incompressible fluid speeds up when it moves through a smaller cross-sectional area.
Viscosity: Viscosity is a measure of a fluid's internal resistance to flow, represented by $\eta$ .

Common Mistakes to Avoid

Using object density instead of fluid density in $F_B = \rho_{\text{fluid}} g V_{\text{disp}}$ is wrong because buoyancy depends on the fluid displaced by the object.
Forgetting atmospheric pressure in $P = P_0 + \rho g h$ is wrong when absolute pressure is required, because the surface pressure still acts on the fluid.
Assuming pressure depends on container shape is wrong because hydrostatic pressure at rest depends on $\rho$ , $g$ , and $h$ , not on the container's width or shape.
Using $A_1 v_1 = A_2 v_2$ for compressible gases without checking conditions is wrong because that simple form assumes constant density.
Applying Bernoulli's equation across a pump, turbine, or highly viscous region is wrong because $P + \frac{1}{2}\rho v^2 + \rho g y$ is conserved only for ideal flow without energy added or lost.

Practice Questions

1 A hydraulic lift has a small piston area of $0.020\,\text{m}^2$ and a large piston area of $0.50\,\text{m}^2$ . If $120\,\text{N}$ is applied to the small piston, what force acts on the large piston?
2 What is the absolute pressure at a depth of $8.0\,\text{m}$ in water if $\rho = 1000\,\text{kg/m}^3$ , $g = 9.8\,\text{m/s}^2$ , and $P_0 = 1.01 \times 10^5\,\text{Pa}$ ?
3 A pipe narrows from an area of $0.060\,\text{m}^2$ to $0.015\,\text{m}^2$ . If water moves at $2.0\,\text{m/s}$ in the wider section, what is its speed in the narrow section?
4 Explain why a ship made of steel can float even though solid steel is denser than water.