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 .
- Pressure is force per unit area, given by .
- Hydrostatic pressure increases with depth according to .
- Pascal's principle gives hydraulic force multiplication as .
- The buoyant force on an object is the weight of displaced fluid, .
- For steady incompressible flow, the continuity equation is .
- Bernoulli's equation for ideal steady flow is .
- For laminar flow through a pipe, Poiseuille's law is .
Vocabulary
- Density
- Density is the mass per unit volume of a substance, calculated by .
- Pressure
- Pressure is the normal force applied per unit area, calculated by .
- Gauge Pressure
- Gauge pressure is pressure measured relative to atmospheric pressure, so .
- 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 .
Common Mistakes to Avoid
- Using object density instead of fluid density in is wrong because buoyancy depends on the fluid displaced by the object.
- Forgetting atmospheric pressure in 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 , , and , not on the container's width or shape.
- Using 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 is conserved only for ideal flow without energy added or lost.
Practice Questions
- 1 A hydraulic lift has a small piston area of and a large piston area of . If is applied to the small piston, what force acts on the large piston?
- 2 What is the absolute pressure at a depth of in water if , , and ?
- 3 A pipe narrows from an area of to . If water moves at 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.
Understanding Fluids & Pressure
At the particle level, pressure comes from countless collisions. Molecules in a gas strike the walls of a container. Particles in a liquid push on nearby particles and on the container walls.
A liquid changes volume very little when squeezed, so a pressure change can travel through it effectively. This is why a hydraulic brake can transfer a push from a foot pedal to the brake pads. The larger output piston can produce a larger force, but it moves a shorter distance.
Energy is not created. The input work is traded for force over distance.
Gases need more care because they compress noticeably. A bicycle pump works because the same amount of air is forced into a smaller volume, raising its pressure.
Pressure in a resting fluid has a direction property that students often miss. At one location, it pushes equally in every direction. The pressure does not point downward just because gravity points downward.
Gravity creates a pressure difference between heights, while the local pressure pushes on every surface. This explains why a dam is built thicker near its base. It also explains why a diver feels stronger squeezing at greater depth.
When studying floating objects, compare average density with the density of the surrounding fluid. An object can be heavy yet float if it displaces enough water before it becomes fully submerged. A steel ship floats because its hollow shape gives it a low average density.
Stability depends on shape too. If a tilted floating object develops a restoring turning effect, it tends to right itself.
Fluid flow problems become clearer when each part of the path is compared. In a narrowing pipe, fluid must move faster if the flow is steady and the fluid is nearly incompressible. The faster speed does not mean the fluid gains energy from nowhere.
Pressure energy or gravitational potential energy can change into kinetic energy. Bernoulli reasoning tracks these exchanges along a streamline. It works best when the flow is smooth, steady, and has little friction.
Real flows can contain turbulence, swirling motion, and energy loss. A wide river can have slow average motion while fast local currents occur near bends or obstacles.
In an airplane wing problem, pressure differences matter, but the full explanation must include the way the wing turns air downward. A single rule about fast air and low pressure is incomplete.
Viscosity describes internal friction within a fluid. Honey resists flow more than water because its layers do not slide past each other as easily. Viscosity usually decreases as a liquid warms, which is why warm oil pours more readily than cold oil.
In narrow tubes, radius has an especially large effect on flow rate. A small decrease in radius can greatly reduce flow, since the relevant dependence involves the fourth power of radius. This matters in blood vessels.
Narrowing caused by plaque makes the heart work harder to maintain flow. It matters in drinking straws, medical syringes, plumbing, and lubrication in engines.
When solving problems, state the assumptions before choosing a model. Check units carefully, distinguish volume flow rate from speed, and decide whether the situation is static, smoothly flowing, or strongly affected by viscosity.