--- title: "Unit 3.2 | Work and the Work-Energy Theorem" description: "Learn work in physics: W = Fd cos(theta), positive and negative work, work from graphs, and the work-energy theorem, with worked AP Physics 1 examples." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/pulling-a-sled-across-snow-work-energy-theorem.jpg" url: "https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/" date_modified: "2026-10-31T09:35:16+00:00" --- # Unit 3.2 | Work and the Work-Energy Theorem ## 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-3-2-work-and-the-work-energy-theorem/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/ Work is how forces add energy to an object or take it away. This lesson connects everything you learned about forces to the energy ideas in this unit. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 3 Breakdown You are on Lesson 2 of 4 1. [Unit 3.1 | Powering the Universe – Understanding Energy in Physics](https://nerd-notes.com/unit-3-1-understanding-energy-in-physics/) 2. [Unit 3.2 | Work and the Work-Energy Theorem](https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/) 3. [Unit 3.3 | Conservation of Energy, Systems, and Problem Solving](https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/) 4. [Unit 3.4 | Understanding and Applying Power](https://nerd-notes.com/unit-3-4-understanding-and-applying-power/) #### In this lesson:  - You will learn what work means in physics (it is not the same as effort) - How to calculate work, including forces applied at an angle - When work is positive, negative, or zero - Use the work-energy theorem to connect forces and energy - The 4 work problem types: the sign of work, forces at an angle, force vs. position graphs, and stopping distances [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-3-2-work-and-the-work-energy-theorem&utm_content=top) ### Introduction This lesson covers work and the **work-energy theorem**: how forces add energy to an object or take it away, and how the work-energy theorem links Unit 2 forces to energy. In [Unit 3.1](https://nerd-notes.com/unit-3-1-understanding-energy-in-physics/) you learned that objects can *have* energy: kinetic, gravitational, and spring. But how does an object get that energy in the first place? The answer is **work**. Work is how a force moves energy into an object or out of it. It is the bridge between the forces you mastered in Unit 2 and the energy ideas in this unit. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### What Is Work? In everyday language, “work” means effort. Holding a heavy backpack for an hour feels like a lot of work. In physics, you did **zero** work on that backpack, because it never moved. Work is only done when a force moves an object over a distance. [katex] W = Fd [/katex] Here [katex] F [/katex] is the force in newtons and [katex] d [/katex] is the displacement in meters. Work is measured in joules, the same unit as energy. That is not a coincidence: work *is* energy being transferred. Push a box across the floor with [katex] 50\,\text{N} [/katex] for [katex] 4\,\text{m} [/katex] and you do [katex] W = (50)(4) = 200 \, J [/katex] of work on the box. You moved [katex] 200\,\text{J} [/katex] of energy from your muscles into the box. [Remove Ads](https://nerd-notes.com/pricing/?for=students) #### Forces at an angle What if you pull a wagon by a handle tilted upward? Only the part of the force along the motion does work. The part pointing up is not moving the wagon forward. \(\vec{F}\) \(F\cos\theta \;\text{does work}\) \(F\sin\theta\) \(\theta\) \(\text{displacement } d\) \(W = Fd\cos\theta\) *Split the pull into components. Only \(F\cos\theta\), the part along the displacement, does work.* [katex] W = Fd\cos\theta [/katex] Here [katex] \theta [/katex] is the angle between the force and the direction of motion. A student pulls a wagon [katex] 5\,\text{m} [/katex] across a flat sidewalk. The handle makes a [katex] 37^\circ [/katex] angle with the ground and the student pulls with [katex] 50\,\text{N} [/katex]. How much work does the student do on the wagon? [Remove Ads](https://nerd-notes.com/pricing/?for=students) 1. Identify the force and displacement: [katex] F = 50 \, N [/katex], [katex] d = 5 \, m [/katex], [katex] \theta = 37^\circ [/katex]. 2. Apply the equation: [katex] W = (50)(5)\cos 37^\circ [/katex]. 3. Solve: [katex] W = (50)(5)(0.8) = 200 \, J [/katex]. Notice the wagon’s mass was never needed. Work only cares about the force, the distance, and the angle between them. ### Positive, Negative, and Zero Work Work is a scalar, but it can still be positive or negative. The sign tells you which way the energy flows. *A force along the motion adds energy, a force against it removes energy, and a force perpendicular to it transfers none.* | **Work** | **When** | **Example** | | --- | --- | --- | | Positive | Force has a part along the motion ([katex] \theta \lt 90^\circ [/katex]) | Pushing a stalled car forward | | Negative | Force has a part against the motion ([katex] \theta \gt 90^\circ [/katex]) | Friction on a sliding box | | Zero | Force is perpendicular to the motion ([katex] \theta = 90^\circ [/katex]), or nothing moves | Normal force on a box sliding across a floor | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Friction almost always does negative work. It pulls energy out of the object and turns it into heat. That’s the “lost” energy from Unit 3.1. ### Work From a Force vs. Position Graph In [Unit 1.3](https://nerd-notes.com/unit-1-2-mastering-motion-graphing-kinematic-variables/) you found displacement as the area under a velocity vs. time graph. Force graphs work the same way. The work done by a force equals the **area under a force vs. position graph**. This is how you find work when the force changes as the object moves. The spring force is the classic example. It grows as you stretch the spring ([katex] F = kx [/katex]), so its graph is a straight line and the area under it is a triangle: \(F\) \(x\) \(kx\) \(\tfrac{1}{2}(x)(kx) = \tfrac{1}{2}kx^2\) \(F = kx\) *The triangle under \(F = kx\) has area \(\tfrac{1}{2}(x)(kx) = \tfrac{1}{2}kx^2\): the spring potential energy from Unit 3.1.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] W = \frac{1}{2}(x)(kx) = \frac{1}{2}kx^2 [/katex] That is exactly the spring potential energy from Unit 3.1. The work you do stretching a spring is the energy the spring stores. 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-3-2-work-and-the-work-energy-theorem&utm_content=mid) ### The Work-Energy Theorem Add up the work done by *every* force on an object (the net work) and you get the change in its kinetic energy: [katex] W_{net} = \Delta KE = KE_f – KE_i [/katex] Net positive work speeds an object up, net negative work slows it down, and zero net work leaves its speed unchanged. In Unit 2, a net force caused an acceleration. Now a net force acting over a distance causes a change in kinetic energy. Same physics, different lens. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Keep this idea in your back pocket: work is how energy *enters or leaves*. In [Unit 3.3](https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/) it becomes the [katex] W [/katex] term in the conservation of energy framework, [katex] E_i + W = E_f [/katex], the one equation you will use for almost every energy problem. Every work question on the AP exam is one of these four types: | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. The sign of work | “positive, negative or zero work?” | Compare the force with the motion | | 2. Forces at an angle | a handle, rope or push at an angle | [katex] W = Fd\cos\theta [/katex] | | 3. Force vs. position graphs | a graph of [katex] F [/katex] against [katex] x [/katex] | Area = work | | 4. Stopping distances | “how far does it slide?” | [katex] W_{net} = \Delta KE [/katex] | [Remove Ads](https://nerd-notes.com/pricing/?for=students) \(E_i = E_f\), Work, and Systems Official course · step-by-step video walkthrough [Watch · 31 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-3-2-work-and-the-work-energy-theorem&utm_content=video-mid#course=94339&module=94419&lesson=102322) ### Problem Type 1: The sign of work **How to spot it:** Which force does positive, negative or zero work? Often no numbers at all. **Setup:** Compare each force with the direction of motion: along it is positive, against it is negative, perpendicular is zero. A ball on a string moves in a horizontal circle at constant speed (like in circular motion). How much work does the tension do on the ball? Answer: zero. Tension points into the center of the circle while the ball moves along the circle, so the force is always perpendicular to the motion and [katex] \cos 90^\circ = 0 [/katex]. That is why the ball’s speed never changes. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Saying a force does work just because it is there. The normal force on a box sliding across the floor is perpendicular to the motion, so it does zero work. [Open in UBQ](https://nerd-notes.com/ubq/21723/) Question 1 The box in the diagram is sliding to the right across a horizontal table, under the influence of the forces shown. Which force(s) is doing negative… 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/22707/) Question 2 You kick a ball straight up. Compare the sign of the work done by gravity on the ball while it goes up with the sign of the work done by gravity… 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/122461/) Question 3 A small bob attached to a light string revolves at constant speed in a horizontal circle, forming a conical pendulum. An observer notes that the work… 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 2: Forces at an angle **How to spot it:** A handle, rope or push tilted at an angle to the motion. **Setup:** [katex] W = Fd\cos\theta [/katex], with [katex] \theta [/katex] the angle between the force and the motion. The wagon above is this type. **AP trap:** Using the angle from the vertical, or sine instead of cosine. Measure [katex] \theta [/katex] from the direction of motion. [Open in UBQ](https://nerd-notes.com/ubq/109311/) Question 4 A child pulls a toy wagon across a level floor by exerting a constant tension force of 20 N on a handle. The handle is oriented at an angle of 60°… 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/125198/) Question 5 A crate moves a distance d to the right along a horizontal surface while an applied force of constant magnitude F is exerted on it in three separate… 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 3: Work from a force vs. position graph **How to spot it:** A graph of force against position. **Setup:** The area under the graph is the work. Split it into rectangles and triangles. **AP trap:** Mixing up the graphs. Area under force vs. *position* is work; area under force vs. *time* is impulse (Unit 4). [Open in UBQ](https://nerd-notes.com/ubq/109303/) Question 6 A particle is moved along the x-axis by a net force Fₓ that varies with the particle’s position x as shown in the graph. The particle starts at rest… 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: Stopping distances (the work-energy theorem) **How to spot it:** A force speeds up or stops an object over a distance: “how far does it skid?”, “how fast is it going after?” **Setup:** Find [katex] \Delta KE [/katex], write the net work as force times distance with its sign, and set [katex] W_{net} = \Delta KE [/katex]. A [katex] 1000\,\text{kg} [/katex] car is traveling at [katex] 20\,\text{m/s} [/katex] when the driver slams on the brakes. Friction exerts a constant [katex] 8000\,\text{N} [/katex] force on the car. How far does the car skid before stopping? **1st step:** Find the change in kinetic energy ([katex] 20\,\text{m/s} [/katex] to [katex] 0\,\text{m/s} [/katex]): [katex] \Delta KE = 0 – \frac{1}{2}(1000)(20)^2 = -200{,}000 \, J [/katex] [Remove Ads](https://nerd-notes.com/pricing/?for=students) **2nd step:** Find the net work. Only friction does work (gravity and the normal force are perpendicular to the motion), and it points against the motion: [katex] W_{net} = -8000d [/katex] **3rd step:** Apply the work-energy theorem: [katex] -8000d = -200{,}000 \rightarrow d = 25 \, m [/katex] Answer: [katex] 25 \, m [/katex]. Show Answer: What if the car were going twice as fast? Kinetic energy depends on [katex] v^2 [/katex], so twice the speed means four times the kinetic energy ([katex] 800{,}000\,\text{J} [/katex]). With the same friction force, the car skids four times as far: [katex] 100 \, m [/katex]. **AP trap:** Dropping the minus sign on friction’s work. Friction points against the motion, so its work is negative, and so is the change in kinetic energy. [Remove Ads](https://nerd-notes.com/pricing/?for=students) This question is the “twice as fast” idea from above, with arrows. [Open in UBQ](https://nerd-notes.com/ubq/88148/) Question 7 Two identical arrows, one with 2 times the speed of the other, are fired into a bale of hay. Assuming the hay exerts a constant "frictional" force on… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) 14 Conceptual Questions To Understand Energy Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-3-2-work-and-the-work-energy-theorem&utm_content=practice-practice#course=94339&module=94419&lesson=102323) Quiz Lab Work and the Work-Energy Theorem Who does work, how much, and what it does to kinetic energy. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=175) 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-3-2-work-and-the-work-energy-theorem&utm_content=end) ### Lesson 3.3 Preview In the [next lesson](https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/) we put it all together with the **conservation of energy** framework: choose a system, mark two points, and solve roller coaster, pendulum, and spring problems without an FBD. ## 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-3-2-work-and-the-work-energy-theorem/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/