AP Physics 1 · Lesson 1

Density and pressure

Know Know it

Density is mass per volume, ρ=m/V\rho = m/V, in kg/m³. It belongs to the material: cut a block in half and each half has the same density.

Pressure is the perpendicular force on a surface divided by its area, P=F/AP = F/A, in pascals (1 Pa = 1 N/m²). The same force on a smaller area makes a larger pressure.

In a fluid at rest, the pressure grows with depth because each layer holds up the fluid above it: P=P0+ρghP = P_0 + \rho g h, where P0P_0 is the pressure at the surface (about 1.01 × 10⁵ Pa for open air at sea level). The part ρgh\rho g h is the gauge pressure; the total is the absolute pressure. At one depth in a connected fluid at rest the pressure is the same everywhere, whatever the container’s shape.

Pressure in fresh water at a few depths (surface at 1.01 × 10⁵ Pa)
DepthGauge pressureAbsolute pressure
1.0 m9.8 × 10³ Pa1.11 × 10⁵ Pa
5.0 m4.9 × 10⁴ Pa1.50 × 10⁵ Pa
10 m9.8 × 10⁴ Pa1.99 × 10⁵ Pa
20 m1.96 × 10⁵ Pa2.97 × 10⁵ Pa

Calculator On the exam

On the exam you may use a four-function, scientific or graphing calculator on both sections, and the testing app also has scientific and graphing calculators built in. You also get an equation sheet with reference tables, but it will not tell you which equation fits a situation or what each symbol stands for there.

Worked example Easy

A 72 kg hiker stands on snow in snowshoes. Together the two snowshoes touch the snow over an area of 0.36 m². Use g = 9.8 m/s². What average pressure do the snowshoes put on the snow?

  1. 2.0 × 10³ PaAnswer
  2. 2.0 × 10² Pa
  3. 2.5 × 10² Pa
  4. 5.1 × 10⁻⁴ Pa

How to solve it

  1. Force on the snow = weight = mg = (72)(9.8) = 706 N.
  2. P = F/A = 706 N / 0.36 m² = 1960 Pa, about 2.0 × 10³ Pa.

Why each choice is right or wrong

  • A. Correct. The force is the hiker’s weight, (72)(9.8) = 706 N, so P = 706 N / 0.36 m² = 2.0 × 10³ Pa.
  • B. Incorrect. 2.0 × 10² Pa divides the mass, 72 kg, by the area; pressure needs the weight in newtons.
  • C. Incorrect. 2.5 × 10² Pa multiplies the weight by the area; pressure is force divided by area.
  • D. Incorrect. 5.1 × 10⁻⁴ Pa divides the area by the weight, which turns the formula upside down.

Trap Gauge or absolute: read the question

A tire gauge or a depth gauge reads only the pressure above atmospheric. If a question asks for the total (absolute) pressure, add P0P_0; if it asks for the extra pressure due to the liquid, or the gauge pressure, do not.

Explain Explain it

Task verbs: explain, calculate

Why ρgh\rho g h: picture a column of liquid with cross section A and height h. Its weight is ρ(Ah)g\rho(Ah)g, and the liquid below must push up on it with that much extra force, so the extra pressure is ρ(Ah)g/A=ρgh\rho(Ah)g/A = \rho g h. The area cancels, which is why the container’s shape does not matter.

Pascal’s principle: extra pressure added to a sealed-in fluid shows up throughout it, at full strength. That is how a hydraulic jack lets a small force on a small piston hold up a large force on a large piston.

Worked example Medium

A snorkeler dives to 2.5 m below the surface of the sea. Seawater has a density of 1025 kg/m³ and the air pressure at the surface is 1.01 × 10⁵ Pa. Use g = 9.8 m/s². What is the absolute pressure at her depth?

  1. 2.51 × 10⁴ Pa
  2. 1.04 × 10⁵ Pa
  3. 1.26 × 10⁵ PaAnswer
  4. 7.59 × 10⁴ Pa

How to solve it

  1. Gauge part: ρgh=(1025)(9.8)(2.5)=2.51×104\rho g h = (1025)(9.8)(2.5) = 2.51 \times 10^4 Pa.
  2. Absolute: add the surface pressure, 1.01×105+0.251×105=1.26×1051.01 \times 10^5 + 0.251 \times 10^5 = 1.26 \times 10^5 Pa.

Why each choice is right or wrong

  • A. Incorrect. 2.51 × 10⁴ Pa is the gauge pressure, the water’s part alone; the absolute pressure also includes the air pressure at the surface.
  • B. Incorrect. 1.04 × 10⁵ Pa leaves g out of the water term, adding (1025)(2.5) instead of (1025)(9.8)(2.5).
  • C. Correct. P=P0+ρgh=1.01×105+(1025)(9.8)(2.5)=1.01×105+2.51×104=1.26×105P = P_0 + \rho g h = 1.01 \times 10^5 + (1025)(9.8)(2.5) = 1.01 \times 10^5 + 2.51 \times 10^4 = 1.26 \times 10^5 Pa.
  • D. Incorrect. 7.59 × 10⁴ Pa subtracts the water term from the surface pressure; pressure rises with depth, so it must be added.

Explain Your turn

When you drink through a straw, the drink rises up the straw into your mouth. Using pressure, explain what pushes the liquid up the straw.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Says sucking lowers the air pressure inside the straw and mouth
  • Says the higher atmospheric pressure on the drink’s open surface pushes the liquid up the straw

Model answer

Sucking enlarges your mouth and lowers the air pressure inside it and inside the straw, below atmospheric pressure. The air in the room still presses down on the open surface of the drink at full atmospheric pressure. That higher outside pressure pushes the liquid up the straw toward the low-pressure region; the drink is pushed up, not pulled.

Apply Apply it

Practice this topic. AP Practice questions on exactly what this lesson covers, with an explanation for every answer.