--- title: "Unit 7.3 | Representing and Analyzing SHM" description: "Learn SHM graphs and equations for AP Physics 1: x = A cos(2πft), reading amplitude and period, matching x, v and a graphs, and force vs. position graphs." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133460.jpg" url: "https://nerd-notes.com/unit-7-3-representing-and-analyzing-shm/" date_modified: "2026-10-31T09:35:34+00:00" --- # Unit 7.3 | Representing and Analyzing SHM ## 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-7-3-representing-and-analyzing-shm/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-7-3-representing-and-analyzing-shm/ Position is a cosine, velocity is its slope, acceleration is it flipped. This lesson shows how to read and match every SHM graph and equation on the AP exam. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 7 Breakdown You are on Lesson 3 of 4 1. [Unit 7.1 | Defining Simple Harmonic Motion](https://nerd-notes.com/unit-7-1-defining-simple-harmonic-motion/) 2. [Unit 7.2 | Period and Frequency of Springs and Pendulums](https://nerd-notes.com/unit-7-2-period-and-frequency-of-springs-and-pendulums/) 3. [Unit 7.3 | Representing and Analyzing SHM](https://nerd-notes.com/unit-7-3-representing-and-analyzing-shm/) 4. [Unit 7.4 | Energy in Simple Harmonic Motion](https://nerd-notes.com/unit-7-4-energy-in-simple-harmonic-motion/) #### In this lesson:  - Write position as a function of time: [katex] x = A\cos(2\pi f t) [/katex] and [katex] x = A\sin(2\pi f t) [/katex] - Read amplitude, period and frequency off a graph - Sketch velocity and acceleration graphs from a position graph, and the other way around - The 4 problem types: reading a position graph, using the equation, matching x, v and a, and force vs. position graphs [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-7-3-representing-and-analyzing-shm&utm_content=top) ### Introduction This lesson covers how to **represent simple harmonic motion**: with an equation, with SHM graphs of position, velocity and acceleration against time, and with a force vs. position graph. You know what SHM is ([Unit 7.1](https://nerd-notes.com/unit-7-1-defining-simple-harmonic-motion/)) and how long a cycle takes ([Unit 7.2](https://nerd-notes.com/unit-7-2-period-and-frequency-of-springs-and-pendulums/)). The graphs here use the same slope rules as the [motion graphs](https://nerd-notes.com/unit-1-2-mastering-motion-graphing-kinematic-variables/) from Unit 1, so nothing new to memorize, just a new shape. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### The Equation of SHM Position in SHM traces a cosine or a sine wave. Which one depends on where the motion starts. | **Starts at** | **Equation** | | --- | --- | | A turning point, [katex] x = +A [/katex], released from rest | [katex] x = A\cos(2\pi f t) [/katex] | | Equilibrium, [katex] x = 0 [/katex], moving in the + direction | [katex] x = A\sin(2\pi f t) [/katex] | Both are on the AP equation sheet. [katex] A [/katex] is the amplitude, [katex] f [/katex] is the frequency, and [katex] 2\pi f = \frac{2\pi}{T} [/katex]. The angle [katex] 2\pi f t [/katex] is in radians, so put your calculator in radian mode. One full cycle is [katex] 2\pi [/katex] radians, so the motion repeats every [katex] T [/katex] seconds. [Remove Ads](https://nerd-notes.com/pricing/?for=students) A pendulum or a block released from rest at [katex] x = -A [/katex] fits [katex] x = -A\cos(2\pi f t) [/katex]. Check any equation at [katex] t = 0 [/katex] before you use it. ▶ **Watch:** the SHM lines of the AP equation sheet, and how [katex] T = \frac{1}{f} [/katex] and [katex] \omega = 2\pi f [/katex] connect (starts at 18:42). [Embedded content](https://www.youtube-nocookie.com/embed/9f2UNm7taYU?start=1122&rel=0) ### Position, Velocity and Acceleration Graphs Here is a block released from rest at [katex] x = +0.20 \text{ m} [/katex], with a period of [katex] 2.0\,\text{s} [/katex]: Position of the block nerd-notes.com *Amplitude 0.20 m (the height of a peak), period 2.0 s (peak to peak).* The other two SHM graphs follow from rules you already know: - **Velocity** is the slope of the position graph. The slope is zero at every peak and trough (the turning points) and steepest where the graph crosses zero (equilibrium). - **Acceleration** is [katex] a = -\frac{k}{m}x [/katex], so its graph is the position graph flipped upside down. When [katex] x [/katex] is at its most positive, [katex] a [/katex] is at its most negative. [Remove Ads](https://nerd-notes.com/pricing/?for=students) x, v and a over one and a half cycles nerd-notes.com *Released from rest at +A. Velocity peaks a quarter cycle before position does; acceleration is exactly opposite to position.* | **Where** | **x** | **v** | **a** | | --- | --- | --- | --- | | Turning point [katex] +A [/katex] | [katex] +A [/katex] (max) | 0 | [katex] -a_{max} [/katex] | | Equilibrium | 0 | [katex] \pm v_{max} [/katex] | 0 | | Turning point [katex] -A [/katex] | [katex] -A [/katex] (min) | 0 | [katex] +a_{max} [/katex] | Every SHM representation question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Reading a position graph | an [katex] x [/katex] vs. [katex] t [/katex] graph: “amplitude”, “period”, “when is the speed greatest” | Peak height = [katex] A [/katex], peak to peak = [katex] T [/katex], zero crossings = fastest | | 2. Using the equation | an equation like [katex] x = (0.12)\cos(4\pi t) [/katex], or “write an equation” | Match it to [katex] x = A\cos(2\pi f t) [/katex], check [katex] t = 0 [/katex] | | 3. Matching x, v and a | pick the velocity or acceleration graph, or the signs at a given time | v = slope of x; a is x flipped | | 4. Force vs. position graphs | a graph of the restoring force against [katex] x [/katex] | Slope [katex] = -k [/katex]; area = stored energy | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Representing SHM via Energy, Forces, and Kinematics Official course · step-by-step video walkthrough [Watch · 25 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-7-3-representing-and-analyzing-shm&utm_content=video-mid#course=94339&module=98825&lesson=105403) ### Problem Type 1: Reading a position graph **How to spot it:** A graph of position against time for a block, glider or pendulum, with questions about amplitude, period, frequency, or when the speed or acceleration is largest. **Setup:** Amplitude is the height of a peak above the middle line, not peak to trough. Period is the time from one peak to the next. The object is fastest where the graph crosses the middle line and stopped at every peak and trough. Use the position graph above. Find the amplitude, period and frequency. At which times in the first [katex] 4\,\text{s} [/katex] is the block moving fastest? When does it have its largest positive acceleration? [Remove Ads](https://nerd-notes.com/pricing/?for=students) Amplitude [katex] 0.20\,\text{m} [/katex]. Period [katex] 2.0\,\text{s} [/katex] (peaks at 0, 2.0 and [katex] 4.0\,\text{s} [/katex]), so [katex] f = \frac{1}{2.0} = 0.50 \text{ Hz} [/katex]. Fastest where the graph crosses zero: 0.5, 1.5, 2.5 and [katex] 3.5\,\text{s} [/katex]. Largest positive acceleration at [katex] x = -A [/katex], the troughs: 1.0 and [katex] 3.0\,\text{s} [/katex]. **AP trap:** Reading the amplitude as the distance from peak to trough. That is [katex] 2A [/katex]. And the period is peak to peak, not peak to trough. [Open in UBQ](https://nerd-notes.com/ubq/109567/) Question 1 A block of mass m is attached to an ideal spring and oscillates with simple harmonic motion on a horizontal frictionless surface. The graph below… 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/88197/) Question 2 The graph represents the position x as a function of time t for an object undergoing simple harmonic motion. Which of the following equations could… 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-7-3-representing-and-analyzing-shm&utm_content=mid) ### Problem Type 2: Using the equation **How to spot it:** The question gives an equation for the position, asks you to write one, or asks where the object is at a given time. **Setup:** Line it up with [katex] x = A\cos(2\pi f t) [/katex]: the number in front is [katex] A [/katex], the number multiplying [katex] t [/katex] is [katex] 2\pi f [/katex]. Check the starting point at [katex] t = 0 [/katex]. Use radians. A block’s position is [katex] x = (0.12 \text{ m})\cos(4\pi t) [/katex]. Find its amplitude, frequency and period. A second block is released from rest at [katex] 5.0\,\text{cm} [/katex] from equilibrium and has a period of [katex] 0.80\,\text{s} [/katex]. Write its equation and find its position at [katex] 0.10\,\text{s} [/katex] and at [katex] 0.20\,\text{s} [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) First block: [katex] A = 0.12 \text{ m} [/katex], [katex] 2\pi f = 4\pi [/katex] so [katex] f = 2.0 \text{ Hz} [/katex] and [katex] T = 0.50 \text{ s} [/katex]. Second block: released from rest, so it is a cosine: [katex] x = (5.0 \text{ cm})\cos\left(\frac{2\pi t}{0.80}\right) [/katex]. At [katex] 0.10\,\text{s} [/katex]: [katex] 5.0\cos\left(\frac{\pi}{4}\right) = 3.5 \text{ cm} [/katex]. At [katex] 0.20\,\text{s} [/katex], a quarter period: [katex] 5.0\cos\left(\frac{\pi}{2}\right) = 0 [/katex], it is passing through equilibrium. Notice that at one eighth of a period the block is at [katex] 3.5\,\text{cm} [/katex], not halfway ([katex] 2.5\,\text{cm} [/katex]). It starts slow at the turning point, so it covers less than half the distance in the first half of the quarter cycle. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Leaving the calculator in degree mode. [katex] \cos(4\pi \cdot 0.10) [/katex] means the cosine of 1.26 radians, not 1.26 degrees. [Open in UBQ](https://nerd-notes.com/ubq/105430/) Question 3 At time t = 0, an object is released from rest at position x = +xₘₐₓ and undergoes simple harmonic motion along the x -axis about the equilibrium… 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/81714/) Question 4 An object undergoing simple harmonic motion has a maximum displacement of 6.2 m at t = 0.0 s. If the angular frequency of oscillation is 1.6 rad/s… 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: Matching position, velocity and acceleration **How to spot it:** Pick the velocity or acceleration graph that goes with a position graph, or give the signs of [katex] x [/katex], [katex] v [/katex] and [katex] a [/katex] at a certain time. **Setup:** Velocity: read the slope of the position graph. Acceleration: flip the position graph. At each turning point [katex] v = 0 [/katex] and [katex] a [/katex] is largest, pointing toward equilibrium. A pendulum is released from rest at [katex] x = +A [/katex] at [katex] t = 0 [/katex] and has period [katex] T [/katex]. Describe its position, velocity and acceleration at [katex] t = \frac{T}{4} [/katex] and at [katex] t = \frac{T}{2} [/katex]. At [katex] \frac{T}{4} [/katex] it is passing through equilibrium on its way to [katex] -A [/katex]: [katex] x = 0 [/katex], speed at its maximum and velocity negative, [katex] a = 0 [/katex]. At [katex] \frac{T}{2} [/katex] it has reached the other side: [katex] x = -A [/katex], [katex] v = 0 [/katex], and the acceleration is at its largest and positive, pointing back toward equilibrium. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Drawing the acceleration graph in step with the position graph. Acceleration always points toward equilibrium, so where [katex] x [/katex] is positive, [katex] a [/katex] is negative. [Open in UBQ](https://nerd-notes.com/ubq/111602/) Question 5 An object is attached to an ideal horizontal spring and is undergoing simple harmonic motion on a frictionless surface. A position-time graph for 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/30498/) Question 6 A simple pendulum oscillates with amplitude A and period T, as represented on the graph above. Which option best represents the pendulum’s velocity v… 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: Force vs. position graphs **How to spot it:** A graph of the force on the object (or the spring force) against its position, often from a lab with a force sensor. **Setup:** A straight line through the origin with a negative slope means SHM, and the slope is [katex] -k [/katex]. With [katex] k [/katex] you have the period. The area between the line and the axis is the energy stored in the spring, which is the next lesson. Force on a 0.40 kg block nerd-notes.com *A straight line through the origin: the force is restoring and proportional to x. Slope = \(-k\).* The graph shows the force on a [katex] 0.40\,\text{kg} [/katex] block on a spring. Find the spring constant and the period of the block. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Read one point on the line: [katex] F = -6.0 \text{ N} [/katex] at [katex] x = 15 \text{ cm} = 0.15 \text{ m} [/katex]. Slope: [katex] k = \frac{6.0}{0.15} = 40 \text{ N/m} [/katex]. Period: [katex] T = 2\pi\sqrt{\frac{0.40}{40}} = 2\pi(0.10) \approx 0.63 \text{ s} [/katex]. If the block oscillates with amplitude [katex] 0.15\,\text{m} [/katex], the triangle between the line and the axis from 0 to [katex] 0.15\,\text{m} [/katex] has area [katex] \tfrac{1}{2}(0.15)(6.0) = 0.45 \text{ J} [/katex]: the energy stored in the spring at the turning point. **AP trap:** Reading [katex] k [/katex] as the force at one point. The spring constant is the slope, force divided by displacement, in N/m. [Open in UBQ](https://nerd-notes.com/ubq/122838/) Question 7 A block of mass m on a horizontal frictionless surface is connected to a horizontal spring. An experimenter measures the magnitude of the restoring… 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/111708/) Question 8 A block of mass M is attached to an ideal spring and undergoes simple harmonic motion on a frictionless horizontal surface. The equilibrium position… 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) Practice Translating Between SHM Representations Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-7-3-representing-and-analyzing-shm&utm_content=practice-practice#course=94339&module=98825&lesson=105405) Quiz Lab Oscillations Complete all of these in 90 seconds per question or less 18 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=1) 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-7-3-representing-and-analyzing-shm&utm_content=end) ### Lesson 7.4 Preview The area under the force graph was energy. In the [last lesson of the unit](https://nerd-notes.com/unit-7-4-energy-in-simple-harmonic-motion/) we follow that energy around the cycle: kinetic at the middle, potential at the ends, and the same total the whole way. ## 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-7-3-representing-and-analyzing-shm/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-7-3-representing-and-analyzing-shm/