--- title: "Unit 8.1 | Density and Pressure" description: "Learn density and pressure in AP Physics 1: ρ = m/V, P = F/A, gauge vs. absolute pressure, P = P₀ + ρgh, U-tubes and straws, with practice." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133452.jpg" url: "https://nerd-notes.com/unit-8-1-density-and-pressure/" date_modified: "2026-10-31T09:35:22+00:00" --- # Unit 8.1 | Density and Pressure ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-8-1-density-and-pressure/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-1-density-and-pressure/ Every fluids problem starts with two ideas: how packed a fluid is and how hard it pushes. This lesson covers density, pressure and how pressure grows with depth. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 8 Breakdown You are on Lesson 1 of 4 1. [Unit 8.1 | Density and Pressure](https://nerd-notes.com/unit-8-1-density-and-pressure/) 2. [Unit 8.2 | Buoyancy and Archimedes’ Principle](https://nerd-notes.com/unit-8-2-buoyancy-and-archimedes-principle/) 3. [Unit 8.3 | Fluid Flow and the Continuity Equation](https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/) 4. [Unit 8.4 | Bernoulli’s Equation](https://nerd-notes.com/unit-8-4-bernoullis-equation/) #### In this lesson:  - What a fluid is, and what makes a fluid “ideal” on the AP exam - Density, [katex] \rho = \frac{m}{V} [/katex], and why a piece of an object has the same density as the whole - Pressure, [katex] P = \frac{F_\perp}{A} [/katex], and how it grows with depth: [katex] P = P_0 + \rho g h [/katex] - The 4 problem types: density, pressure from a force, pressure at depth, and containers, U-tubes and straws [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=top) ### Introduction This lesson covers **density and pressure**, the two ideas every fluids problem is built on. It starts Unit 8, the last unit of AP Physics 1. A fluid is anything that flows: liquids and gases. A fluid has no fixed shape, so it pushes on every surface it touches. Two numbers describe that push. **Density** says how much mass is packed into the fluid, and **pressure** says how hard it pushes on each square meter. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Fluids and the Ideal Fluid The particles in a fluid move freely and bump into everything around them. Those collisions are what we feel as pressure. In AP Physics 1 you treat every liquid as an **ideal fluid**: - **Incompressible:** its density stays the same everywhere, at any depth and any pressure. - **No viscosity:** no internal friction, so no mechanical energy turns into thermal energy as it flows. Real gases are compressible, which is why air thins out as you climb a mountain. Unless a question says otherwise, treat liquids as ideal. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Density [katex] \rho = \frac{m}{V} [/katex] Density [katex] \rho [/katex] (the Greek letter rho) is mass per volume, in kg/m³. Water is about [katex] 1000\,\text{kg/m}^3 [/katex], which is the same as [katex] 1\,\text{g/cm}^3 [/katex]. A substance that is the same all the way through has one density, so any piece of it, big or small, has that same density. | **Substance** | **Density (kg/m³)** | | --- | --- | | Air (sea level) | 1.2 | | Wood (pine) | about 500 | | Oil | about 800–900 | | Water | 1000 | | Seawater | 1025 | | Aluminum | 2700 | | Steel | 7800 | ### Pressure [katex] P = \frac{F_\perp}{A} [/katex] Pressure is the perpendicular force on a surface divided by its area. Its unit is the pascal: [katex] 1 \text{ Pa} = 1 \text{ N/m}^2 [/katex]. The air around you pushes with about [katex] 1.0 \times 10^5 \text{ Pa} [/katex], called 1 atmosphere ([katex] 1\,\text{atm} [/katex]). [Remove Ads](https://nerd-notes.com/pricing/?for=students) ▶ **Watch:** density and pressure basics (starts at 1:49). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=109&rel=0) Pressure is a **scalar**. At any point in a fluid at rest it pushes the same amount in every direction, and the force it makes on a surface always points straight into that surface. ### Pressure at a Depth Go deeper and the fluid above you weighs more, so the pressure grows. Here is where the formula comes from: 1. Picture a column of liquid with top area [katex] A [/katex] and height [katex] h [/katex]. Its mass is [katex] m = \rho V = \rho A h [/katex]. 2. Its weight pushes on the area at the bottom: [katex] \frac{mg}{A} = \frac{\rho A h g}{A} = \rho g h [/katex]. 3. The air above the surface pushes down too, with [katex] P_0 [/katex]. Add the two. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] P = P_0 + \rho g h [/katex] \(P_0\) \(h\) \(P = P_0 + \rho g h\) \(P_{gauge} = P - P_0 = \rho g h\) \(P_0 \approx 1.0 \times 10^5 \text{ Pa}\) *At depth \(h\) the pressure is the air pressure on the surface plus the weight of the liquid above, per area.* [katex] P [/katex] is the **absolute pressure**, the total. The part from the liquid alone, [katex] P_{gauge} = P – P_0 = \rho g h [/katex], is the **gauge pressure**: what a tire gauge reads. Only depth matters, not how wide the container is. Pressure grows linearly with depth nerd-notes.com *The vertical intercept is \(P_0\). The slope is \(\rho g\), so the denser liquid has the steeper line.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) About every [katex] 10\,\text{m} [/katex] of water adds [katex] 1\,\text{atm} [/katex]. At [katex] 10\,\text{m} [/katex] down the absolute pressure is about [katex] 2\,\text{atm} [/katex]; at [katex] 20\,\text{m} [/katex], about [katex] 3\,\text{atm} [/katex]. Every density and pressure question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Density | mass and volume, “cut in half”, a displaced volume | [katex] \rho = \frac{m}{V} [/katex] | | 2. Pressure from a force | a force pressing on an area: feet, tires, pistons | [katex] P = \frac{F_\perp}{A} [/katex] | | 3. Pressure at depth | “below the surface”, “absolute”, “gauge” | [katex] P = P_0 + \rho g h [/katex] | | 4. Containers, U-tubes and straws | odd-shaped containers, two liquids in a U-tube, a straw or barometer | Same fluid, same level, same pressure | [Remove Ads](https://nerd-notes.com/pricing/?for=students) 2 Key Fluid Properties Official course · step-by-step video walkthrough [Watch · 25 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=video-mid#course=94339&module=98826&lesson=106562) ### Problem Type 1: Density **How to spot it:** A mass and a volume, a shape whose volume you can work out, an object cut into pieces, or a volume measured by how much water rises. **Setup:** Find the volume first (a box is [katex] l \times w \times h [/katex], a sphere is [katex] \tfrac{4}{3}\pi r^3 [/katex]), then [katex] \rho = \frac{m}{V} [/katex]. A rise in water level equals the volume of what sank into it. A solid aluminum block measures [katex] 0.50\,\text{m} [/katex] × [katex] 0.20\,\text{m} [/katex] × [katex] 0.10\,\text{m} [/katex] and has a mass of [katex] 27\,\text{kg} [/katex]. Find its density. The block is then cut in half. What is the density of each half? [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] V = (0.50)(0.20)(0.10) = 0.010 \text{ m}^3 [/katex], so [katex] \rho = \frac{27}{0.010} = 2700 \text{ kg/m}^3 [/katex]. Each half has half the mass and half the volume: [katex] \frac{13.5}{0.0050} = 2700 \text{ kg/m}^3 [/katex]. Same material, same density. **AP trap:** Thinking a bigger or heavier piece is denser. Density belongs to the material, not to the amount of it. [Open in UBQ](https://nerd-notes.com/ubq/107045/) Question 13 A certain object has a density of ρ_obj. If you cut the object exactly in half, what is the density of one of the remaining pieces? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107048/) Question 14 A block of uniform material is cut into two pieces, where one piece is cut to have a volume that is 3 times the volume of the other. How does the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/87425/) Question 15 Diamond has a density of 3500 kg/m³. During a physics lab, a diamond drops out of Virginia's necklace and falls into her graduated cylinder filled… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=mid) ### Problem Type 2: Pressure from a force **How to spot it:** A force spread over an area: a person standing, a tire on the road, a nail or a heel, a book on a table. **Setup:** Find the force perpendicular to the surface (often a weight, [katex] mg [/katex]), convert the area to m², then [katex] P = \frac{F_\perp}{A} [/katex]. A [katex] 60\,\text{kg} [/katex] skater stands on one foot. Her shoe touches the ice over [katex] 0.015\,\text{m}^2 [/katex]. On her skate the blade touches only [katex] 4.0 \times 10^{-4} \text{ m}^2 [/katex]. Find the pressure on the ice each way. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Her weight is [katex] (60)(9.8) = 588 \text{ N} [/katex]. Shoe: [katex] P = \frac{588}{0.015} \approx 3.9 \times 10^4 \text{ Pa} [/katex]. Blade: [katex] P = \frac{588}{4.0 \times 10^{-4}} \approx 1.5 \times 10^6 \text{ Pa} [/katex]. Same force, about 38 times less area, so about 38 times the pressure. **AP trap:** Forgetting to convert cm² to m². [katex] 1 \text{ cm}^2 = 10^{-4} \text{ m}^2 [/katex], not [katex] 10^{-2} [/katex]. Also, double the force and quadruple the area and the pressure is halved, not doubled. [Open in UBQ](https://nerd-notes.com/ubq/107038/) Question 16 If a perpendicular force F is applied to a surface area A, resulting in a pressure P, what is the new pressure if the force is doubled and the area… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/87423/) Question 17 Caleb is filling up water balloons for the Physics Olympics balloon toss competition. Caleb sets a 0.50 kg spherical water balloon on the kitchen… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) Fluid Statics Part 1 – Pascal, Open Containers, and U-tubes Official course · step-by-step video walkthrough [Watch · 22 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=video-mid2#course=94339&module=98826&lesson=106564) ### Problem Type 3: Pressure at depth: absolute and gauge **How to spot it:** “At a depth of”, “below the surface”, “absolute pressure”, “gauge pressure”, divers, submarines, dams. **Setup:** Measure [katex] h [/katex] down from the free surface. Gauge: [katex] \rho g h [/katex]. Absolute: add [katex] P_0 \approx 1.0 \times 10^5 \text{ Pa} [/katex]. Read which one the question wants. A diver is [katex] 12\,\text{m} [/katex] below the surface of a lake. Find the gauge pressure and the absolute pressure on her. Use [katex] P_0 = 1.0 \times 10^5 \text{ Pa} [/katex]. Gauge: [katex] \rho g h = (1000)(9.8)(12) \approx 1.18 \times 10^5 \text{ Pa} [/katex]. Absolute: [katex] 1.0 \times 10^5 + 1.18 \times 10^5 \approx 2.2 \times 10^5 \text{ Pa} [/katex], a bit over [katex] 2\,\text{atm} [/katex]. If she dives to [katex] 24\,\text{m} [/katex], the gauge pressure doubles, but the absolute pressure does not. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ▶ **Watch:** pressure at depth, worked from scratch (starts at 5:51). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=351&rel=0) **AP trap:** Doubling the absolute pressure when the depth doubles. Only the [katex] \rho g h [/katex] part doubles; [katex] P_0 [/katex] stays the same. And changing the air pressure changes the absolute pressure but not the gauge pressure. [Open in UBQ](https://nerd-notes.com/ubq/107033/) Question 18 If the absolute pressure at a depth of 10 m in a lake is 2 atm, what would the absolute pressure be at a depth of 20 m in the same lake? (Assume 1… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107056/) Question 19 A student measures the pressure at the bottom of a vertical column of liquid to be P_total. If the atmospheric pressure Pₐₜₘ doubles, what happens to… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/91161/) Question 20 A cube of side length s rests on the bottom surface of a container of fluid. The fluid is at a height y above the bottom of the tank. The fluid has… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/87424/) Question 21 Wanda watches the fish in her fish tank and notices that the angelfish like to feed at the water’s surface, while the catfish feed 0.300 m below at… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Problem Type 4: Containers, U-tubes and straws **How to spot it:** Containers of different shapes, a U-tube with two liquids, a piston on one arm, a straw, or a barometer. **Setup:** In one connected fluid at rest, points at the same level have the same pressure, whatever the shape. Pick a level, write the pressure there from each side, and set them equal. \(h\) \(P\) \(P\) \(P\) \(\text{same depth } h \;\Rightarrow\; \text{same } P = P_0 + \rho g h\) *Same liquid, same depth, same pressure at the bottom. The shape of the container does not matter.* \(L\) \(d\) \(\rho_o g L = \rho_w g d\) \(\rho_o = \rho_w \frac{d}{L}\) *Below the oil, the two arms hold the same water. At the level of the interface the pressures match: \(\rho_o g L = \rho_w g d\).* [Remove Ads](https://nerd-notes.com/pricing/?for=students) Oil is poured into the left arm of a U-tube that holds water. The oil column is [katex] 20\,\text{cm} [/katex] tall, and the oil–water interface is [katex] 16\,\text{cm} [/katex] below the water surface in the right arm. Find the density of the oil. Pick the level of the interface. Left: [katex] P_0 + \rho_o g (0.20) [/katex]. Right: [katex] P_0 + (1000) g (0.16) [/katex]. Set them equal; [katex] P_0 [/katex] and [katex] g [/katex] cancel: [katex] \rho_o = 1000 \cdot \frac{0.16}{0.20} = 800 \text{ kg/m}^3 [/katex]. The lighter oil needs a taller column to make the same pressure. Straws work the same way. Sucking lowers the pressure in the straw, and the air pushing on the drink outside pushes it up. Even a perfect vacuum can lift water only [katex] h = \frac{P_0}{\rho g} = \frac{1.0 \times 10^5}{(1000)(9.8)} \approx 10 \text{ m} [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ▶ **Watch:** how air pressure holds up a column of liquid (starts at 2:52). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=172&rel=0) **Pascal’s principle** is the same idea in a closed container: a pressure added at one point reaches every point of the fluid. A hydraulic lift uses it, [katex] \frac{F_1}{A_1} = \frac{F_2}{A_2} [/katex], so a small force on a small piston holds up a car on a big one. Pascal’s principle is not in the AP Physics 1 course framework, so it is good to know but not required. If a lift does show up, [katex] P = \frac{F}{A} [/katex] and “same level, same pressure” solve it. **AP trap:** Thinking a wider container, or one with more water, has more pressure at the bottom. Pressure depends only on the depth and the liquid. Also, a straw does not “pull” the drink up: the air outside pushes it. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Open in UBQ](https://nerd-notes.com/ubq/107132/) Question 22 Three containers have the same base area A and are filled with water to the same height h, but they have different shapes: one is cylindrical, one… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107125/) Question 23 A U-tube open to the atmosphere at both ends is partially filled with water of density ρ_w. Oil of density ρₒ (where ρₒ < ρ_w) is poured into the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107162/) Question 24 A person uses a straw to drink water from a cup. Which of the following best describes the physics of how the water moves up the straw? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) Applying Fluid Fundamentals Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=practice-practice#course=94339&module=98826&lesson=106563) Quiz Lab Density and Pressure How deep you are sets the pressure, not the shape of the container. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=185) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-1-density-and-pressure&utm_content=end) ### Lesson 8.2 Preview In the [next lesson](https://nerd-notes.com/unit-8-2-buoyancy-and-archimedes-principle/) the pressure difference between the top and bottom of a submerged object becomes a force: the **buoyant force**. It explains why ships float, why rocks feel lighter underwater, and how much of an iceberg hides below the surface. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-8-1-density-and-pressure/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-1-density-and-pressure/