--- title: "AP Physics 2 Mathematical Routines FRQ Practice" description: "77 Mathematical Routines questions written in the AP format, each scored part by part." canonical: "https://fiveable.me/ap-physics-2-revised/frqs/mathematical-routines" type: "frq-type-practice" subject: "AP Physics 2" --- # AP Physics 2 Mathematical Routines FRQ Practice ## What the Mathematical Routines FRQ asks The Mathematical Routines question is FRQ 1 on the AP Physics 2 exam. You analyze one physical scenario by completing a diagram and deriving an expression, then indicate an answer in part B and justify it. - **20–25 min**: suggested on the exam - **10 points**: across parts A and B - **95 min**: for all four FRQs | Part | What it asks | Points | | --- | --- | --- | | A: Analysis | Complete a diagram, then derive an expression | 7 | | B: Claim | Indicate an answer, such as whether a claim is correct, and justify it | 3 | ## How Mathematical Routines FRQ practice works 1. **Read the full question.** The scenario, its figures, and parts A and B, laid out the way the exam shows them. 2. **Write on a 23-minute timer.** Inside the 20 to 25 minutes the exam suggests for this question. Pause it or turn it off, and your response saves as you go. 3. **Submit for a score out of 10.** Your response is scored against the scoring guidelines written for that question. Your first score is included. ## All 77 questions ### Unit 9: Thermodynamics (6) - [Monatomic ideal gas isothermal expansion entropy change](/ap-physics-2-revised/frq-practice?id=698a9fb8e97825fc4c8a3bc2&type=frq-1-mathematical-routines) - [Thermal conduction through aluminum wall](/ap-physics-2-revised/frq-practice?id=698aa21ce97825fc4c8a3bd5&type=frq-1-mathematical-routines) - [Thermal conduction heating ideal monatomic gas](/ap-physics-2-revised/frq-practice?id=698aa640e97825fc4c8a3bf2&type=frq-1-mathematical-routines) - [Isobaric expansion of ideal gas entropy](/ap-physics-2-revised/frq-practice?id=698aa73be97825fc4c8a3bfb&type=frq-1-mathematical-routines) - [Ideal gas thermal equilibrium with reservoir](/ap-physics-2-revised/frq-practice?id=698aabe1e97825fc4c8a3c23&type=frq-1-mathematical-routines) - [Ideal gas thermal equilibration through movable piston](/ap-physics-2-revised/frq-practice?id=698ab232e97825fc4c8a3c59&type=frq-1-mathematical-routines) ### Unit 10: Electric Force, Field, and Potential (16) - [Charge magnitude on capacitor plates](/ap-physics-2-revised/frq-practice?id=6977f1cc1fb1cc32fbb25059&type=frq-1-mathematical-routines) - [Electrostatic potential from charged conducting spheres](/ap-physics-2-revised/frq-practice?id=698a9de4e97825fc4c8a3bb3&type=frq-1-mathematical-routines) - [Coulomb force on charged conducting spheres](/ap-physics-2-revised/frq-practice?id=698a9f52e97825fc4c8a3bbf&type=frq-1-mathematical-routines) - [Electric field from charged spheres with dielectric material](/ap-physics-2-revised/frq-practice?id=698aa052e97825fc4c8a3bc6&type=frq-1-mathematical-routines) - [Electrostatic forces on charged suspended sphere](/ap-physics-2-revised/frq-practice?id=698aa27ee97825fc4c8a3bd7&type=frq-1-mathematical-routines) - [Electric potential and force from fixed point charges](/ap-physics-2-revised/frq-practice?id=698aa5ede97825fc4c8a3bee&type=frq-1-mathematical-routines) - [Electrostatic forces between charged conducting spheres](/ap-physics-2-revised/frq-practice?id=698aa6f3e97825fc4c8a3bf9&type=frq-1-mathematical-routines) - [Charge redistribution and electrostatic forces](/ap-physics-2-revised/frq-practice?id=698aa807e97825fc4c8a3c05&type=frq-1-mathematical-routines) - [Electrostatic repulsion between charged conducting spheres](/ap-physics-2-revised/frq-practice?id=698aa9c0e97825fc4c8a3c11&type=frq-1-mathematical-routines) - [Electrostatic force between charged spheres in different media](/ap-physics-2-revised/frq-practice?id=698aaacbe97825fc4c8a3c1b&type=frq-1-mathematical-routines) - [Electric field from two point charges at specific location](/ap-physics-2-revised/frq-practice?id=698aabdae97825fc4c8a3c22&type=frq-1-mathematical-routines) - [Charged conducting spheres, electric field superposition](/ap-physics-2-revised/frq-practice?id=698aace5e97825fc4c8a3c2d&type=frq-1-mathematical-routines) - [Charged spheres in electrostatic equilibrium](/ap-physics-2-revised/frq-practice?id=698ab0bae97825fc4c8a3c4c&type=frq-1-mathematical-routines) - [Charged conducting spheres, electrostatic forces, equilibrium](/ap-physics-2-revised/frq-practice?id=698ab1c9e97825fc4c8a3c56&type=frq-1-mathematical-routines) - [Electrostatic force between charged conducting spheres](/ap-physics-2-revised/frq-practice?id=698ab450e97825fc4c8a3c6a&type=frq-1-mathematical-routines) - [Charged sphere equilibrium near conducting plate with dielectric](/ap-physics-2-revised/frq-practice?id=698ab546e97825fc4c8a3c77&type=frq-1-mathematical-routines) ### Unit 11: Electric Circuits (14) - [Power dissipation in resistor networks with switched configurations](/ap-physics-2-revised/frq-practice?id=6977fd964c3c2dafa90c5a3d&type=frq-1-mathematical-routines) - [Capacitor charging in resistor-battery circuit](/ap-physics-2-revised/frq-practice?id=698a9dfae97825fc4c8a3bb4&type=frq-1-mathematical-routines) - [Capacitor charging in series-parallel circuit](/ap-physics-2-revised/frq-practice?id=698a9f03e97825fc4c8a3bbb&type=frq-1-mathematical-routines) - [Capacitor charging and resistive power dissipation](/ap-physics-2-revised/frq-practice?id=698aa031e97825fc4c8a3bc5&type=frq-1-mathematical-routines) - [RC circuit capacitor charging behavior](/ap-physics-2-revised/frq-practice?id=698aa12ee97825fc4c8a3bcc&type=frq-1-mathematical-routines) - [RC circuit steady-state power dissipation](/ap-physics-2-revised/frq-practice?id=698aa38de97825fc4c8a3bdf&type=frq-1-mathematical-routines) - [Steady-state current distribution and resistor power dissipation](/ap-physics-2-revised/frq-practice?id=698aa7fee97825fc4c8a3c04&type=frq-1-mathematical-routines) - [Parallel RC circuit with capacitor charging behavior](/ap-physics-2-revised/frq-practice?id=698aaa89e97825fc4c8a3c18&type=frq-1-mathematical-routines) - [Resistor properties, circuit current direction, capacitor charging](/ap-physics-2-revised/frq-practice?id=698aab85e97825fc4c8a3c20&type=frq-1-mathematical-routines) - [Series and parallel capacitor combinations, steady-state circuits](/ap-physics-2-revised/frq-practice?id=698aac86e97825fc4c8a3c2a&type=frq-1-mathematical-routines) - [Capacitor charging and battery power delivery](/ap-physics-2-revised/frq-practice?id=698aad94e97825fc4c8a3c33&type=frq-1-mathematical-routines) - [Battery power in RC circuit at steady state](/ap-physics-2-revised/frq-practice?id=698aaf7ce97825fc4c8a3c3f&type=frq-1-mathematical-routines) - [Battery, internal resistance, capacitor charging circuits](/ap-physics-2-revised/frq-practice?id=698ab075e97825fc4c8a3c48&type=frq-1-mathematical-routines) - [RC circuits with capacitor charging behavior](/ap-physics-2-revised/frq-practice?id=698ab172e97825fc4c8a3c52&type=frq-1-mathematical-routines) ### Unit 12: Magnetism and Electromagnetism (17) - [Induced current direction in conductor loops](/ap-physics-2-revised/frq-practice?id=6977f2291fb1cc32fbb2505c&type=frq-1-mathematical-routines) - [Magnetic field and induced current in solenoid](/ap-physics-2-revised/frq-practice?id=698a9e11e97825fc4c8a3bb6&type=frq-1-mathematical-routines) - [Magnetic forces between parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698a9f3ae97825fc4c8a3bbe&type=frq-1-mathematical-routines) - [Magnetic field from current-carrying wire, electromagnetic induction](/ap-physics-2-revised/frq-practice?id=698aa07ee97825fc4c8a3bc7&type=frq-1-mathematical-routines) - [Magnetic force on current-carrying wire segment](/ap-physics-2-revised/frq-practice?id=698aa1cde97825fc4c8a3bd4&type=frq-1-mathematical-routines) - [Magnetic field and induced current between parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aa4b7e97825fc4c8a3be6&type=frq-1-mathematical-routines) - [Magnetic flux through loop with parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aa591e97825fc4c8a3beb&type=frq-1-mathematical-routines) - [Magnetic flux and induced emf in parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aa6ace97825fc4c8a3bf4&type=frq-1-mathematical-routines) - [Magnetic forces between parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aa7a2e97825fc4c8a3bff&type=frq-1-mathematical-routines) - [Magnetic flux and induced current in rectangular loop near current-carrying wire](/ap-physics-2-revised/frq-practice?id=698aa976e97825fc4c8a3c0e&type=frq-1-mathematical-routines) - [Magnetic forces between parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aaa9ce97825fc4c8a3c19&type=frq-1-mathematical-routines) - [Magnetic force on moving charge near current-carrying wire](/ap-physics-2-revised/frq-practice?id=698aac8de97825fc4c8a3c2c&type=frq-1-mathematical-routines) - [Magnetic flux direction from parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698aae4ee97825fc4c8a3c39&type=frq-1-mathematical-routines) - [Magnetic forces between parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698ab078e97825fc4c8a3c49&type=frq-1-mathematical-routines) - [Magnetic force on moving charged particle in wire field](/ap-physics-2-revised/frq-practice?id=698ab166e97825fc4c8a3c51&type=frq-1-mathematical-routines) - [Magnetic field from parallel current-carrying wires](/ap-physics-2-revised/frq-practice?id=698ab2c9e97825fc4c8a3c5f&type=frq-1-mathematical-routines) - [Magnetic field from current-carrying wire in material](/ap-physics-2-revised/frq-practice?id=698ab3d2e97825fc4c8a3c66&type=frq-1-mathematical-routines) ### Unit 13: Geometric Optics (10) - [Light reflection, refraction, and converging lens imaging](/ap-physics-2-revised/frq-practice?id=698a9f80e97825fc4c8a3bc0&type=frq-1-mathematical-routines) - [Concave mirror image formation and location](/ap-physics-2-revised/frq-practice?id=698aa508e97825fc4c8a3be9&type=frq-1-mathematical-routines) - [Concave mirror image formation and refraction](/ap-physics-2-revised/frq-practice?id=698aa5fae97825fc4c8a3bef&type=frq-1-mathematical-routines) - [Light reflection and refraction through mirror and block](/ap-physics-2-revised/frq-practice?id=698aa8dce97825fc4c8a3c0a&type=frq-1-mathematical-routines) - [Lens refraction, mirror reflection, image formation](/ap-physics-2-revised/frq-practice?id=698aaa3ce97825fc4c8a3c15&type=frq-1-mathematical-routines) - [Light refraction across air-material boundary](/ap-physics-2-revised/frq-practice?id=698aab5ee97825fc4c8a3c1e&type=frq-1-mathematical-routines) - [Light reflection and refraction through glass slab](/ap-physics-2-revised/frq-practice?id=698aac43e97825fc4c8a3c28&type=frq-1-mathematical-routines) - [Spherical mirror and converging lens image formation](/ap-physics-2-revised/frq-practice?id=698aad1de97825fc4c8a3c2f&type=frq-1-mathematical-routines) - [Light reflection and image formation by mirrors](/ap-physics-2-revised/frq-practice?id=698ab232e97825fc4c8a3c58&type=frq-1-mathematical-routines) - [Refraction at air-glass interface with incident angle](/ap-physics-2-revised/frq-practice?id=698ab339e97825fc4c8a3c63&type=frq-1-mathematical-routines) ### Unit 14: Waves, Sound, and Physical Optics (6) - [Sound wave frequency detection via Doppler effect](/ap-physics-2-revised/frq-practice?id=698aa18de97825fc4c8a3bd0&type=frq-1-mathematical-routines) - [Doppler effect with moving sound source and observer](/ap-physics-2-revised/frq-practice?id=698aa7a9e97825fc4c8a3c00&type=frq-1-mathematical-routines) - [Doppler effect and wave interference patterns](/ap-physics-2-revised/frq-practice?id=698aad92e97825fc4c8a3c32&type=frq-1-mathematical-routines) - [Wave behavior at boundaries and medium interfaces](/ap-physics-2-revised/frq-practice?id=6a8f95e9545c4b69e56c375c&type=frq-1-mathematical-routines) - [Standing waves, pulses, and thin film interference](/ap-physics-2-revised/frq-practice?id=6a8f9747545c4b69e56c379f&type=frq-1-mathematical-routines) - [Double-slit interference pattern central fringe location](/ap-physics-2-revised/frq-practice?id=6a8fa2d6d169687f5e6955bc&type=frq-1-mathematical-routines) ### Unit 15: Modern Physics (8) - [Photon wavelength changes in Compton scattering](/ap-physics-2-revised/frq-practice?id=6977fb4242b14de2424c5a5a&type=frq-1-mathematical-routines) - [Blackbody radiation, photoelectric effect, Compton scattering, radioactive decay](/ap-physics-2-revised/frq-practice?id=698aa6b8e97825fc4c8a3bf5&type=frq-1-mathematical-routines) - [Photoelectric effect and stopping potential in sodium](/ap-physics-2-revised/frq-practice?id=698aa7dfe97825fc4c8a3c02&type=frq-1-mathematical-routines) - [Compton scattering wavelength shift and energy transfer](/ap-physics-2-revised/frq-practice?id=698aaa11e97825fc4c8a3c14&type=frq-1-mathematical-routines) - [Wave-particle duality and photoelectric effect](/ap-physics-2-revised/frq-practice?id=698aabe3e97825fc4c8a3c24&type=frq-1-mathematical-routines) - [Photoelectric effect and thermal radiation wavelength](/ap-physics-2-revised/frq-practice?id=698aae0ce97825fc4c8a3c35&type=frq-1-mathematical-routines) - [Electron diffraction, photoelectric effect, radiation phenomena](/ap-physics-2-revised/frq-practice?id=698aafdee97825fc4c8a3c44&type=frq-1-mathematical-routines) - [Electron diffraction, photoelectric effect, photon scattering](/ap-physics-2-revised/frq-practice?id=698ab0b3e97825fc4c8a3c4b&type=frq-1-mathematical-routines) ## Sample question **Monatomic ideal gas isothermal expansion entropy change** A sealed cylinder contains n = 0.500 mol of a monatomic ideal gas and is fitted with a frictionless, movable piston of cross-sectional area A = 4.00 × 10⁻³ m². The cylinder is initially in thermal equilibrium with a large reservoir at temperature T₀ = 300 K. The gas is initially at pressure P₀ = 1.00 × 10⁵ Pa and volume V₀. A schematic of the apparatus is shown in Figure 1. Figures: Type: setup_diagram Caption: Figure 1. Sealed cylinder with movable frictionless piston in thermal contact with an adjustable-temperature reservoir (initial state at T₀ and P₀). URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig1.jpg Placement: after-background Figure 2: Type: setup_diagram Caption: Figure 2. Microscopic model of gas particles at state 0 (T₀) and state 1 (T₁) for comparing average speed and momentum transfer rate at the same volume. URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fix1_fig2.png Placement: before-part Part Letter: A Figure 3: Type: graph Caption: Figure 3. Blank P–V diagram with states 0 and 1 at the same pressure P₀; students sketch the constant-pressure path from 0 to 1. URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig3.jpg Placement: before-part Part Letter: A Part A (7 points): i. The reservoir temperature is increased so that the gas reaches a new equilibrium temperature T₁ = 450 K while the piston is free to move. Complete the following tasks in Figure 2. - **Indicate** in Panel (b) whether the average speed of the particles is greater than, less than, or equal to that in Panel (a). - **Indicate** in Panel (b) whether the average rate of momentum transfer to the cylinder wall (and thus the pressure) is greater than, less than, or equal to that in Panel (a) *at the same volume*. (3 pt) Verb: indicate ii. The gas is heated from T₀ to T₁ in a quasistatic process at constant pressure P₀. Using Figure 3, sketch the process path from state 0 to state 1. **Derive** an expression for the final volume V₁ in terms of n, T₁, and P₀. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. (4 pt) Verb: derive Figure 4: Type: setup_diagram Caption: Figure 4. Heat transfer through a metal plate of thickness L between a hot reservoir and the gas; plate area equals the cylinder cross-sectional area. URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig4.jpg Placement: before-part Part Letter: B Part B (3 points): Prompt: **Indicate** whether the change in entropy of the gas is positive, negative, or zero. ______ Positive ______ Negative ______ Zero **Justify** your answer. Context: The gas undergoes the constant-pressure process from state 0 to state 1, where the final temperature is T₁ = 450 K and the pressure remains P₀ = 1.00 × 10⁵ Pa. The amount of gas is n = 0.500 mol, and the gas is monatomic and ideal. Assume the process is quasistatic and that no phase changes occur. The heat transfer mechanism is shown in Figure 4. Verb: indicate and justify Image URLs (4): - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig1.jpg - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fix1_fig2.png - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig3.jpg - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-2-revised/d7d31141-615d-4d24-98ce-aeda9c8b7c31_fig4.jpg ## Questions **How many responses can I get scored?** You can read every question without a plan. Your first scored response is included. A plan unlocks unlimited scoring. **How do I complete the diagram in part A?** Open the drawing tool in the editor and draw it there. You can also switch to handwrite and add a photo of your page. **How do I type equations and symbols?** Use the subscript, superscript, and special-character buttons in the editor toolbar. Or write your derivations on paper, switch to handwrite, and add a photo. **How is the Mathematical Routines question different from the Qualitative/Quantitative Translation (QQT) question?** Both ask for a derivation. The Mathematical Routines question also asks for a diagram and is worth 10 points. The QQT starts with reasoning in words and is worth 8 points. **Are these real College Board questions?** No. We wrote them in the AP format, with the same parts and points, and wrote scoring guidelines for each one. ## Related - [Mathematical Routines FRQ exam guide](/ap-physics-2-revised/ap-physics-2-exam/ap-physics-2-ap-phys-2-frq-mathematical-routines/study-guide/ap-physics-2-ap-phys-2-frq-mathematical-routines) - [AP Physics 2 Representations FRQ practice](/ap-physics-2-revised/frqs/translation-between-representations) - [AP Physics 2 Experimental Design FRQ practice](/ap-physics-2-revised/frqs/experimental-design) - [AP Physics 2 Qualitative/Quantitative FRQ practice](/ap-physics-2-revised/frqs/qualitative-quantitative-translation) - [AP Physics 2 FRQs](/ap-physics-2-revised/frqs) - Any question by id: the `get_frq` tool on the [Fiveable MCP server](/mcp/docs) ## About This Document Canonical Fiveable pages are available as Markdown at the same path plus `.md`. - [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns - [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing - [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`) - [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`)