--- title: "Unit 7.2 | Period and Frequency of Springs and Pendulums" description: "Learn the period of a pendulum and a mass on a spring for AP Physics 1: T = 1/f, what changes the period, ratios, and finding g from a graph, with practice." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133459.jpg" url: "https://nerd-notes.com/unit-7-2-period-and-frequency-of-springs-and-pendulums/" date_modified: "2026-10-31T09:35:33+00:00" --- # Unit 7.2 | Period and Frequency of Springs and Pendulums ## 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-7-2-period-and-frequency-of-springs-and-pendulums/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-7-2-period-and-frequency-of-springs-and-pendulums/ Mass, length, gravity, amplitude: only some of them change the period. This lesson covers both period formulas and the four period and frequency problem types. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 7 Breakdown You are on Lesson 2 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:  - Period and frequency, and how they relate: [katex] T = \frac{1}{f} [/katex] - The period of a mass on a spring, [katex] T_s = 2\pi\sqrt{\frac{m}{k}} [/katex], and of a pendulum, [katex] T_p = 2\pi\sqrt{\frac{L}{g}} [/katex] - What changes the period, and what does not (amplitude, and the mass of a pendulum) - The 4 problem types: timing an oscillation, spring periods, pendulum periods, and comparisons and experiments [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-7-2-period-and-frequency-of-springs-and-pendulums&utm_content=top) ### Introduction This lesson covers the **period and frequency** of simple harmonic motion: how long one cycle takes for a mass on a spring, the period of a pendulum, and what does and does not change them. In [Unit 7.1](https://nerd-notes.com/unit-7-1-defining-simple-harmonic-motion/) you saw what makes motion simple harmonic: a restoring force proportional to the displacement. Now we put a clock on it. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Period and Frequency - **Period** [katex] T [/katex]: the time for one full cycle, in seconds. - **Frequency** [katex] f [/katex]: the number of cycles each second, in hertz ([katex] 1\,\text{Hz} [/katex] = 1 cycle per second). [katex] T = \frac{1}{f} [/katex] A long period means a low frequency. A pendulum that takes [katex] 2.0\,\text{s} [/katex] per swing back and forth has a frequency of [katex] 0.50\,\text{Hz} [/katex]. ### The Period of a Mass on a Spring [katex] T_s = 2\pi\sqrt{\frac{m}{k}} [/katex] [Remove Ads](https://nerd-notes.com/pricing/?for=students) Read it like a story. More mass is harder to speed up and slow down, so the cycle takes longer. A stiffer spring (bigger [katex] k [/katex]) pushes harder at every point, so the cycle is quicker. Because of the square root, it takes 4 times the mass to double the period. The amplitude is not in the formula. Pull the block twice as far and it has twice as far to go, but the force is twice as big too, so it moves twice as fast and the trip takes the same time. Gravity is not in the formula either: a vertical spring has the same period as a horizontal one with the same mass and spring. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### The Period of a Simple Pendulum [katex] T_p = 2\pi\sqrt{\frac{L}{g}} [/katex] The period of a pendulum depends only on its length and on [katex] g [/katex]. A longer string means a longer period. Stronger gravity pulls the bob back harder, so the period is shorter. This only holds for small angles, roughly [katex] 15^\circ [/katex] or less, where the pendulum is a simple harmonic oscillator. The mass is not in the formula. A heavier bob feels a bigger restoring force, [katex] mg\sin\theta [/katex], but it is harder to speed up by exactly the same factor, so the mass cancels. The amplitude is not in it either, as long as the angle stays small. [Remove Ads](https://nerd-notes.com/pricing/?for=students) \(m\) \(k\) \(T_s = 2\pi\sqrt{m/k}\) \(\text{depends on } m, k\) \(\text{not on } A \text{ or } g\) \(L\) \(T_p = 2\pi\sqrt{L/g}\) \(\text{depends on } L, g\) \(\text{not on } m \text{ or } A\) *Both formulas are on the AP equation sheet. What you need to know is what is in each one, and what is not.* ▶ **Watch:** the SHM section of the AP Physics 1 speed review: both period formulas and what affects them (starts at 1:12:52). [Embedded content](https://www.youtube-nocookie.com/embed/vyxFy63Ay9s?start=4372&rel=0) Every period and frequency question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Timing an oscillation | “completes 20 cycles in”, “swings from one side to the other in” | [katex] T = \frac{\text{time}}{\text{cycles}} [/katex], [katex] f = \frac{1}{T} [/katex] | | 2. Spring periods | a mass on a spring, “find k”, “what mass” | [katex] T_s = 2\pi\sqrt{\frac{m}{k}} [/katex] | | 3. Pendulum periods | a pendulum’s length, another planet, “find g” | [katex] T_p = 2\pi\sqrt{\frac{L}{g}} [/katex] | | 4. Comparisons and experiments | “what happens to the period if”, a graph of [katex] T^2 [/katex] or [katex] T [/katex] | Ratios, and a straight-line graph whose slope gives [katex] k [/katex] or [katex] g [/katex] | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Simple Harmonic Motion Official course · step-by-step video walkthrough [Watch · 19 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-7-2-period-and-frequency-of-springs-and-pendulums&utm_content=video-mid#course=94339&module=98825&lesson=105401) ### Problem Type 1: Timing an oscillation **How to spot it:** The question gives a number of cycles in a time, or the time for part of a cycle, and asks for the period or frequency. **Setup:** Count full cycles only, out and back. [katex] T = \frac{\text{total time}}{\text{number of cycles}} [/katex], then [katex] f = \frac{1}{T} [/katex]. A mass on a spring completes 30 full oscillations in [katex] 12\,\text{s} [/katex]. Find its period and frequency. A pendulum takes [katex] 0.75\,\text{s} [/katex] to swing from its far left to its far right. Find its period. Spring: [katex] T = \frac{12}{30} = 0.40 \text{ s} [/katex] and [katex] f = \frac{1}{0.40} = 2.5 \text{ Hz} [/katex]. Pendulum: left to right is half a cycle, so [katex] T = 2(0.75) = 1.5 \text{ s} [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) In a lab, time 10 or 20 cycles and divide. Your reaction time is the same whether you time 1 cycle or 20, so timing many cycles makes the error per cycle 20 times smaller. **AP trap:** Counting one swing from side to side as a full cycle. A full cycle ends where it started, moving the same way, so it is out *and* back. [Open in UBQ](https://nerd-notes.com/ubq/105735/) Question 9 What is the relationship between the period T and frequency f of an object in simple harmonic motion? 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/28678/) Question 10 A group of students must study the oscillatory motion of a pendulum. One end of a light string is attached to the ceiling, and the other end of the… 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-7-2-period-and-frequency-of-springs-and-pendulums&utm_content=mid) ### Problem Type 2: Spring periods **How to spot it:** A mass on a spring, horizontal or vertical. The question asks for the period, or uses a measured period to find the mass or the spring constant. **Setup:** Use [katex] T_s = 2\pi\sqrt{\frac{m}{k}} [/katex]. To solve for [katex] k [/katex] or [katex] m [/katex], square both sides first: [katex] T^2 = 4\pi^2\frac{m}{k} [/katex]. A [katex] 0.50\,\text{kg} [/katex] block oscillates on a spring with [katex] k = 200 \text{ N/m} [/katex]. Find its period and frequency. What mass would double the period? A different spring makes a [katex] 0.25\,\text{kg} [/katex] mass oscillate with a period of [katex] 0.50\,\text{s} [/katex]. Find its spring constant. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] T = 2\pi\sqrt{\frac{0.50}{200}} = 2\pi(0.050) = 0.31 \text{ s} [/katex] and [katex] f = \frac{1}{0.314} = 3.2 \text{ Hz} [/katex]. Doubling the period takes 4 times the mass, [katex] 2.0\,\text{kg} [/katex]. Second spring: [katex] k = \frac{4\pi^2 m}{T^2} = \frac{4\pi^2(0.25)}{(0.50)^2} \approx 39.5 \text{ N/m} [/katex]. **AP trap:** Changing the amplitude, or turning the spring vertical, and expecting a new period. Neither [katex] A [/katex] nor [katex] g [/katex] is in [katex] T_s = 2\pi\sqrt{\frac{m}{k}} [/katex]. [Open in UBQ](https://nerd-notes.com/ubq/122686/) Question 11 A block of mass m on a frictionless horizontal surface is attached to a uniform ideal spring of force constant k and oscillates with period T₀. 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/88176/) Question 12 A block with a mass of 4 kg is attached to a spring on the wall that oscillates back and forth with a frequency of 4 Hz and an amplitude of 3 m. What… 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: Pendulum periods **How to spot it:** A simple pendulum at small angles: find its period or length, or take it to the Moon, another planet or an elevator. **Setup:** Use [katex] T_p = 2\pi\sqrt{\frac{L}{g}} [/katex]. To solve for [katex] L [/katex] or [katex] g [/katex], square first: [katex] T^2 = 4\pi^2\frac{L}{g} [/katex]. Only [katex] L [/katex] and [katex] g [/katex] matter. How long must a pendulum be to have a period of [katex] 2.0\,\text{s} [/katex] on Earth? What would its period be on the Moon, where [katex] g = 1.6 \text{ m/s}^2 [/katex]? [katex] L = \frac{gT^2}{4\pi^2} = \frac{(9.8)(2.0)^2}{4\pi^2} = 0.99 \text{ m} [/katex]. On the Moon: [katex] T = 2\pi\sqrt{\frac{0.99}{1.6}} \approx 4.9 \text{ s} [/katex]. Weaker gravity pulls the bob back more gently, so each swing takes longer. The same pendulum is a bad clock on the Moon. A mass on a spring would keep the same period there. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Saying a heavier bob or a bigger swing changes a pendulum’s period. At small angles only the length and [katex] g [/katex] matter. [Open in UBQ](https://nerd-notes.com/ubq/28731/) Question 13 A pendulum has a period of 2.0 s on Earth. What is its length? 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/82867/) Question 14 A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. The… 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: Comparisons and experiments **How to spot it:** “What happens to the period if the mass is doubled”, two systems compared, or lab data of period against mass or length. **Setup:** For a comparison, write the ratio and cancel what does not change. For data, graph something that gives a straight line: [katex] T^2 [/katex] against [katex] m [/katex] (slope [katex] \frac{4\pi^2}{k} [/katex]) or [katex] T^2 [/katex] against [katex] L [/katex] (slope [katex] \frac{4\pi^2}{g} [/katex]). | **Change** | **Spring period** | **Pendulum period** | | --- | --- | --- | | Mass ×4 | ×2 | no change | | Spring constant ×4 | ×½ | (no spring) | | Length ×4 | (no length) | ×2 | | [katex] g [/katex] ÷4 (another planet) | no change | ×2 | | Amplitude ×2 | no change | no change (small angles) | ▶ **Watch:** a pendulum and a spring whose periods differ by a factor of 2, and which change makes them match (starts at 28:45). [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Embedded content](https://www.youtube-nocookie.com/embed/6ItRgoQqVNM?start=1725&rel=0) Spring system A is a block of mass [katex] m [/katex] on a spring of constant [katex] k [/katex]. System B is a block of mass [katex] 2m [/katex] on a spring of constant [katex] 8k [/katex]. Find [katex] \frac{T_B}{T_A} [/katex]. [katex] \frac{T_B}{T_A} = \sqrt{\frac{2m/8k}{m/k}} = \sqrt{\frac{2}{8}} = \frac{1}{2} [/katex]. The stiffer spring wins: B oscillates twice as fast. #### Finding g from a graph A student times a pendulum at five lengths and plots the period squared against the length. Why [katex] T^2 [/katex] and not [katex] T [/katex]? Squaring the period formula gives [katex] T^2 = \frac{4\pi^2}{g}\,L [/katex], which is the equation of a straight line through the origin. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Period squared against length nerd-notes.com *A straight line through the origin, as \(T^2 = \frac{4\pi^2}{g}L\) predicts. Its slope is about \(4.02\,\text{s}^2/\text{m}\).* The best-fit line runs from about [katex] (0,\ 0.01) [/katex] to [katex] (1.00,\ 4.03) [/katex], so its slope is [katex] \frac{4.03 – 0.01}{1.00 – 0} \approx 4.02 \text{ s}^2/\text{m} [/katex]. Since the slope is [katex] \frac{4\pi^2}{g} [/katex]: [katex] g = \frac{4\pi^2}{4.02} \approx 9.8 \text{ m/s}^2 [/katex]. Take the slope from two points on the line, not from your data points. **AP trap:** Plotting [katex] T [/katex] against [katex] L [/katex] and taking the slope of that curve. [katex] T [/katex] grows with [katex] \sqrt{L} [/katex], so the graph bends. Square the period first, then the slope means something. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Open in UBQ](https://nerd-notes.com/ubq/109383/) Question 15 On Earth, a simple pendulum of length L and a block-spring system with mass M and spring constant k both oscillate with a frequency f. The two… 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/88183/) Question 16 A linear spring of force constant k is used in a physics lab experiment. A block of mass m is attached to the spring and the resulting frequency, f… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) Finding Period and Frequency of SHM Systems Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-7-2-period-and-frequency-of-springs-and-pendulums&utm_content=practice-practice#course=94339&module=98825&lesson=105402) Quiz Lab Period of Springs and Pendulums What changes the period and what does not, plus how to measure it. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=182) 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-7-2-period-and-frequency-of-springs-and-pendulums&utm_content=end) ### Lesson 7.3 Preview You can time an oscillation. In the [next lesson](https://nerd-notes.com/unit-7-3-representing-and-analyzing-shm/) we draw it: graphs and equations of position, velocity and acceleration against time, and how to read one from another. ## 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-7-2-period-and-frequency-of-springs-and-pendulums/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-7-2-period-and-frequency-of-springs-and-pendulums/