AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/8/8-2)
Unit 8 · Topic 8.2
8.2 Pressure
Pressure is the perpendicular force per unit area on a surface, P = F/A, measured in pascals, and it's a scalar. In a fluid at rest, pressure grows with depth: the absolute pressure at depth h is P = P₀ + ρgh, where P₀ is the pressure at the surface and ρgh is the gauge pressure from the fluid above.
Key terms
- pressure
- pascal
- absolute pressure
- gauge pressure
- atmospheric pressure
Pressure is force spread over area
Pressure is the size of the perpendicular force on a surface divided by the area it's spread over: P = F⊥/A. The unit is the pascal: 1 Pa = 1 N/m². A pascal is tiny, so you'll often see kilopascals (1 kPa = 1000 Pa).
The same force can make very different pressures. Snowshoes spread your weight over a large area so you don't sink, and a sharp knife focuses a small force onto a tiny edge.
Pressure is a scalar: it has no direction of its own. The force a fluid exerts because of its pressure always pushes perpendicular to whatever surface it touches, whether that surface faces up, down or sideways.
Where fluid pressure comes from
A fluid's pressure is the combined effect of countless particles colliding with a surface. Each collision is a tiny push; together they add up to a steady force on every square meter.
The air around you does this too. Atmospheric pressure at sea level is about 1.01 × 10⁵ Pa. That's about 100,000 N on every square meter, but you don't notice because the fluids in your body push back just as hard.
Pressure increases with depth
Picture a column of liquid with cross-sectional area A and height h. Its weight is mg = ρ(Ah)g. For the column to sit still, the pressure at its bottom must hold up that weight, on top of whatever is pushing down on the surface. Dividing by A gives the pressure at depth h:
P = P₀ + ρgh. P₀ is the pressure at the surface, usually atmospheric pressure for an open container. The extra part, ρgh, is the gauge pressure: the amount above the surface pressure. A tire gauge or a blood pressure cuff reads gauge pressure. P itself is the absolute pressure.
Pressure depends only on depth and the fluid's density, not on the container's shape or width. All points at the same depth in the same connected, still fluid are at the same pressure.
Graphs of pressure against depth
A graph of absolute pressure against depth is a straight line. Its vertical intercept is P₀ and its slope is ρg. A denser liquid gives a steeper line. A graph of gauge pressure against depth goes through the origin.
| Depth in fresh water | Gauge pressure ρgh | Absolute pressure |
|---|---|---|
| 0 m | 0 | 1.01 × 10⁵ Pa |
| 10 m | 0.98 × 10⁵ Pa | 1.99 × 10⁵ Pa |
| 20 m | 1.96 × 10⁵ Pa | 2.97 × 10⁵ Pa |
Incompressible fluids
Because an ideal fluid is incompressible, its density doesn't change even though the pressure deep down is much higher. That's what makes ρgh a straight line. (Air is compressible, so the atmosphere thins out with height, but you won't need to model that.)
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Same force, different area
A crate weighing 1200 N rests on the floor on its 0.50 m × 0.80 m face. (a) What pressure does it exert on the floor? (b) It's tipped onto its 0.30 m × 0.50 m face. What's the pressure now?
Show the solutionHide the solution
- Step 1: (a) A = 0.50 × 0.80 = 0.40 m². P = F/A = 1200 N / 0.40 m² = 3000 Pa.
- Step 2: (b) A = 0.30 × 0.50 = 0.15 m². P = 1200 / 0.15 = 8000 Pa.
- Step 3: The force on the floor is the same both times; only the area changed.
Answer: (a) 3.0 × 10³ Pa (b) 8.0 × 10³ Pa
- Example 2Calculator allowed
A scuba diver
A diver is 12 m below the surface of the sea (ρ = 1025 kg/m³). Atmospheric pressure is 1.01 × 10⁵ Pa; use g = 9.8 m/s². Find the gauge pressure and absolute pressure on the diver.
Show the solutionHide the solution
- Step 1: Gauge pressure = ρgh = (1025)(9.8)(12) ≈ 1.21 × 10⁵ Pa.
- Step 2: Absolute pressure = P₀ + ρgh = 1.01 × 10⁵ + 1.21 × 10⁵ ≈ 2.22 × 10⁵ Pa.
- Step 3: That's a bit over twice atmospheric pressure. Every 10 m or so of water adds roughly one more atmosphere.
Answer: Gauge ≈ 1.2 × 10⁵ Pa; absolute ≈ 2.2 × 10⁵ Pa
- Example 3Calculator allowed
Finding density from two readings (classic trap)
A pressure sensor lowered into an open tank of an unknown liquid reads 101,800 Pa at a depth of 0.10 m and 104,270 Pa at a depth of 0.30 m. Use g = 9.8 m/s². Find the liquid's density.
Show the solutionHide the solution
- Step 1: The trap is to divide one reading by gh, for example 104,270/(9.8 × 0.30) ≈ 35,000 kg/m³. That's wrong because the reading is absolute pressure, which includes the atmosphere.
- Step 2: Use the change instead. Between the two depths, P₀ cancels: ΔP = ρgΔh.
- Step 3: ΔP = 104,270 − 101,800 = 2470 Pa, and Δh = 0.20 m.
- Step 4: ρ = ΔP/(gΔh) = 2470/(9.8 × 0.20) ≈ 1260 kg/m³. (This is the slope of a P against depth graph, divided by g.)
Answer: About 1260 kg/m³
Common mistakes
- Giving pressure a direction. Pressure is a scalar; only the force it causes on a surface has a direction (perpendicular to that surface).
- Mixing up absolute and gauge pressure. Absolute includes P₀; gauge is just ρgh.
- Thinking a wider or differently shaped container has a different pressure at the same depth. Only depth and density matter.
- Measuring h from the bottom of the container instead of down from the fluid's surface.
On the exam
- Ranking questions often show several containers of different shapes, or points at different depths, and ask you to rank the pressures. Rank by depth below the surface (and by density if the fluids differ).
- Graphs of pressure against depth are a natural experimental question: the slope is ρg and the intercept is the surface pressure.
Connected topics
Videos
Check yourself
4 questions on 8.2 Pressure. Pick an answer to see if you got it, and why.
A person who weighs 600 N stands still on both feet. Each foot is in contact with the floor over an area of 0.015 m². What is the average pressure the person's feet exert on the floor?
A 50 N force pushes on a flat panel with an area of 0.010 m². The force is directed at 30° to the surface of the panel. What pressure does this force exert on the panel?
Which statement about the pressure at a point inside a fluid at rest is correct?
Which statement best explains, at the level of particles, why water exerts a pressure on the walls of its container?
0 of 4 answered