Unit 8 · Topic 8.2 Beta

Pressure

Pressure is perpendicular force per area, a scalar in pascals; in a fluid at rest the absolute pressure is P = P₀ + ρgh, set only by depth, and the gauge pressure (what gauges read) is ρgh.

Practice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

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Question set for this topic

Part 1 · Hook

Why this matters

Dive to the bottom of a swimming pool and your ears ache. Push a thumbtack and it slides into a wall, though you could never push your thumb in. Both come down to the same quantity, pressure: how much force is spread over how much area, and how it grows as you go deeper in a fluid.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which is a scalar?

  1. Mass
  2. Force
  3. Velocity
  4. Displacement
Show the answer

Mass has a size but no direction.

  • Correct: Mass:
  • Force:
  • Velocity:
  • Displacement:

2. A 2.0 kg book rests on a table. Near Earth's surface (g = 9.8 N/kg), the force of the table on it is

  1. 19.6 N upward
  2. 2.0 N upward
  3. 19.6 N downward
  4. 9.8 N upward
Show the answer

Equilibrium: the normal force balances the weight, mg = 19.6 N.

  • Correct: 19.6 N upward:
  • 2.0 N upward:
  • 19.6 N downward:
  • 9.8 N upward:

3. Density is

  1. mass per volume
  2. volume per mass
  3. weight per area
  4. mass times volume
Show the answer

ρ = m/V.

  • Correct: mass per volume:
  • volume per mass:
  • weight per area:
  • mass times volume:

Part 4 · See it

See it first

An open tank of water with points A and B at the same depth and point C deeper. At each point equal arrows push from all four sides, longer at C. Absolute pressure P = P₀ + ρgh; gauge pressure P_gauge = ρgh.
Pressure at a point acts equally in every direction and grows with depth. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Fluid particles collide with any surface they touchthey push on it, perpendicular to the surface, with a pressure P = F⊥/A
  2. Each layer of a fluid at rest holds up the weight of the fluid above itpressure increases with depth
  3. A column of height h of a fluid of density ρ has weight ρghA on area Athe pressure grows by ρgh: P = P₀ + ρgh
  4. A gauge reads zero in open airit measures gauge pressure, ρgh, not the absolute pressure P₀ + ρgh

Part 6 · Key ideas

Key ideas

  • Pressure is perpendicular force per area, P = F⊥/A, in pascals (1 Pa = 1 N/m²). It is a scalar: the force it makes on any surface points straight into that surface.
  • In a fluid at rest, P = P₀ + ρgh. Pressure depends only on depth, not on the container's shape or the amount of fluid.
  • Absolute pressure P includes the atmosphere's P₀ ≈ 1.0 × 10⁵ Pa. Gauge pressure P − P₀ = ρgh is what a gauge or tire gauge reads.
  • About 10 m of water adds one atmosphere of pressure.

Part 7 · Misconception

A common mistake

The wrong idea: Pressure is a force, so it points down, and more water in a container means more pressure at the bottom.

What actually happens: Pressure is force per area, a scalar. It acts equally in every direction at a point, and at the bottom of a container it depends only on the depth: P = P₀ + ρgh.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

Three points in a tank

JKL1.5 m4.5 mopen to the air

The water has density 1000 kg/m³. Atmospheric pressure at the surface is 1.0 × 10⁵ Pa.

1. How does the pressure at J compare with the pressure at K?

  1. It is the same, because J and K are at the same depth
  2. It is larger at J, because J is nearer the wall
  3. It is larger at K, because more water is above K
  4. It is smaller at J, because the water at J pushes sideways
Show the answer

In a fluid at rest, pressure depends only on depth: P = P₀ + ρgh. J and K are both 1.5 m down, so their pressures are equal, whatever the tank's shape.

  • Correct: It is the same, because J and K are at the same depth: Right: equal depth, equal pressure.
  • It is larger at J, because J is nearer the wall: Distance from a wall does not appear in P = P₀ + ρgh.
  • It is larger at K, because more water is above K: The same depth of water lies above both points.
  • It is smaller at J, because the water at J pushes sideways: At a point, a fluid pushes equally in every direction; that does not change the pressure from J to K.

2. What is the absolute pressure at L? Use g = 9.8 m/s². Give the value and its unit.

Type a number and its unit.

Show the answer

P = P₀ + ρgh = 1.0 × 10⁵ Pa + (1000 kg/m³)(9.8 m/s²)(4.5 m) = 1.0 × 10⁵ Pa + 4.41 × 10⁴ Pa = 1.4 × 10⁵ Pa.

  • Answer: 1.4 × 105 Pa

3. What is the gauge pressure at L? Use g = 9.8 m/s². Give the value and its unit.

Type a number and its unit.

Show the answer

P_gauge = ρgh = (1000 kg/m³)(9.8 m/s²)(4.5 m) = 4.4 × 10⁴ Pa. This is what a pressure gauge, which reads zero in open air, would show.

  • Answer: 4.4 × 104 Pa

4. L is 3 times as deep as J. How does the absolute pressure at L compare with that at J?

  1. Larger, but less than 3 times as large
  2. Exactly 3 times as large
  3. Exactly 9 times as large
  4. The same, because the water is incompressible
Show the answer

The gauge pressure ρgh triples (1.47 × 10⁴ Pa to 4.41 × 10⁴ Pa), but the absolute pressure also includes P₀ = 1.0 × 10⁵ Pa, which is the same at both. So P goes from 1.15 × 10⁵ Pa to 1.44 × 10⁵ Pa, a factor of about 1.3.

  • Correct: Larger, but less than 3 times as large: Right: only the ρgh part triples; P₀ does not.
  • Exactly 3 times as large: This would be true of the gauge pressure, not the absolute pressure.
  • Exactly 9 times as large: Pressure grows linearly with depth, not with depth squared.
  • The same, because the water is incompressible: Constant density is what makes P grow steadily with depth; it does not make P constant.

5. A student says, "Pressure is a force, so it has a direction." What is the best correction?

  1. Pressure is force per area, a scalar; its force on a surface is perpendicular to it
  2. Pressure is a force, but its direction is straight down, toward Earth
  3. Pressure has a direction: in a fluid it points toward the bottom of the container
  4. Pressure is the same as weight, so it points down and is measured in newtons
Show the answer

P = F⊥/A is a scalar measured in pascals (N/m²). Pressure has no direction of its own; the force it exerts on any surface is perpendicular to that surface, which is why water pushes sideways on a dam.

  • Correct: Pressure is force per area, a scalar; its force on a surface is perpendicular to it: Right: scalar pressure, perpendicular force.
  • Pressure is a force, but its direction is straight down, toward Earth: Water pushes sideways on walls and up on the bottom of a submerged object, so the force is not always down.
  • Pressure has a direction: in a fluid it points toward the bottom of the container: At a point in a fluid, the push is the same in every direction.
  • Pressure is the same as weight, so it points down and is measured in newtons: Pressure is force per area (Pa), not force (N), and not weight.

6. A tire gauge reads 2.2 × 10⁵ Pa. Atmospheric pressure is 1.0 × 10⁵ Pa. What is the absolute pressure of the air inside the tire? Give the value and its unit.

Type a number and its unit.

Show the answer

P = P₀ + P_gauge = 1.0 × 10⁵ Pa + 2.2 × 10⁵ Pa = 3.2 × 10⁵ Pa. A flat tire reads 0 on the gauge but still holds air at 1.0 × 10⁵ Pa.

  • Answer: 3.2 × 105 Pa

Part 9 · Summary

Summary

Pressure is perpendicular force per area, a scalar in pascals; in a fluid at rest the absolute pressure is P = P₀ + ρgh, set only by depth, and the gauge pressure (what gauges read) is ρgh.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections