---
title: "Ultimate AP Physics C: Mechanics Equation Sheets"
description: "Every AP Physics C: Mechanics equation you need, straight from the official reference sheet, to help you get a 5 on the AP Physics C: Mechanics Exam."
url: "https://nerd-notes.com/ap-physics-c-mechanics-formula-sheets/"
date_modified: "2026-10-02T05:47:48+00:00"
---

# Ultimate AP Physics C: Mechanics Equation Sheets

Every equation and constant on the official AP Physics C: Mechanics exam reference sheet, in one interactive digital sheet with each formula tagged by unit.

- [View Digital Equation Sheet](https://nerd-notes.com/ap-physics-c-mechanics-formula-sheets/#eq-official) (jumps to the digital sheet at the bottom of this page)

## And even more free resources to help you get a 5

- For AP-style questions, [check out UBQ.](https://nerd-notes.com/ubq/?subject=physics-c-mechanics)
- For AP-style Practice Exams, [check out UBQ Quiz Labs.](https://nerd-notes.com/quiz/?subject=physics-c-mechanics)
- For AP Physics C: Mechanics practice questions sorted by unit, [click here.](https://nerd-notes.com/subjects/ap-physics-c-mechanics/)
- Use Phy to solve and explain hard questions. [Try here.](https://nerd-notes.com/ai/)

## Official 2026 Digital Equation Sheet

Also built into every AP Physics C: Mechanics question on [UBQ](https://nerd-notes.com/ubq/?subject=physics-c-mechanics), or [download the official PDF](https://apcentral.collegeboard.org/media/pdf/ap-physics-c-mechanics-equations-sheet.pdf).

On the page this is an interactive sheet: a sticky tab bar jumps between its sections, and hovering (or focusing) an equation shows its name and the AP Physics C: Mechanics unit it belongs to. Its full content follows: College Board's "AP Physics C: Mechanics 2026 Exam Reference Information", transcribed, with each equation's name and CED unit as the page shows them on hover.

AP Physics C: Mechanics units: 1 Kinematics · 2 Force and Translational Dynamics · 3 Work, Energy, and Power · 4 Linear Momentum · 5 Torque and Rotational Dynamics · 6 Energy and Momentum of Rotating Systems · 7 Oscillations.

### Mechanics Equations

- Velocity (constant acceleration) (Unit 1: Kinematics): $v_x = v_{x0} + a_x t$
- Position (constant acceleration) (Unit 1: Kinematics): $x = x_0 + v_{x0}t + \frac{1}{2}a_x t^2$
- Velocity and displacement (constant acceleration) (Unit 1: Kinematics): $v_x^2 = v_{x0}^2 + 2a_x\left(x - x_0\right)$
- Displacement from velocity (Unit 1: Kinematics): $\Delta x = \int v_x\left(t\right)dt$
- Change in velocity from acceleration (Unit 1: Kinematics): $\Delta v_x = \int a_x\left(t\right)dt$
- Center of mass position (Unit 2: Force and Translational Dynamics): $\vec{x}_{\text{cm}} = \frac{\sum m_i \vec{x}_i}{\sum m_i}$
- Center of mass (continuous) (Unit 2: Force and Translational Dynamics): $\vec{r}_{\text{cm}} = \frac{\int \vec{r}\,dm}{\int dm}$
- Linear mass density (Unit 2: Force and Translational Dynamics): $\lambda = \frac{d}{d\ell}m\left(\ell\right)$
- Newton's second law (system) (Unit 2: Force and Translational Dynamics): $\vec{a}_{\text{sys}} = \frac{\sum \vec{F}}{m_{\text{sys}}} = \frac{\vec{F}_{\text{net}}}{m_{\text{sys}}}$
- Newton's law of universal gravitation (Unit 2: Force and Translational Dynamics): $\left|\vec{F}_g\right| = G\frac{m_1 m_2}{r^2}$
- Friction force (Unit 2: Force and Translational Dynamics): $\left|\vec{F}_f\right| \le \left|\mu \vec{F}_N\right|$
- Hooke's law (spring force) (Unit 2: Force and Translational Dynamics): $\vec{F}_s = -k\Delta\vec{x}$
- Centripetal acceleration (Unit 2: Force and Translational Dynamics): $a_c = \frac{v^2}{r} = r\omega^2$
- Period and frequency (Unit 2: Force and Translational Dynamics): $T = \frac{1}{f}$
- Translational kinetic energy (Unit 3: Work, Energy, and Power): $K = \frac{1}{2}mv^2$
- Work done by a force (Unit 3: Work, Energy, and Power): $W = \int\limits_a^b \vec{F}\cdot d\vec{r}$
- Work-energy theorem (Unit 3: Work, Energy, and Power): $\Delta K = \sum W_i = \sum F_{\parallel,i}\,d_i$
- Potential energy from a conservative force (Unit 3: Work, Energy, and Power): $\Delta U = -\int\limits_a^b \vec{F}_{\text{cf}}(r)\cdot d\vec{r}$
- Force from potential energy (Unit 3: Work, Energy, and Power): $F_x = -\frac{dU(x)}{dx}$
- Spring potential energy (Unit 3: Work, Energy, and Power): $U_s = \frac{1}{2}k\left(\Delta x\right)^2$
- Gravitational potential energy (general) (Unit 3: Work, Energy, and Power): $U_G = -G\frac{m_1 m_2}{r}$
- Gravitational potential energy near Earth (Unit 3: Work, Energy, and Power): $\Delta U_g = mg\Delta y$
- Average power (Unit 3: Work, Energy, and Power): $P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
- Instantaneous power (Unit 3: Work, Energy, and Power): $P_{\text{inst}} = \frac{dW}{dt}$
- Momentum (Unit 4: Linear Momentum): $\vec{p} = m\vec{v}$
- Newton's second law (momentum form) (Unit 4: Linear Momentum): $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$
- Impulse-momentum theorem (Unit 4: Linear Momentum): $\vec{J} = \int\limits_{t_1}^{t_2} \vec{F}_{\text{net}}\left(t\right)dt = \Delta\vec{p}$
- Center of mass velocity (Unit 4: Linear Momentum): $\vec{v}_{\text{cm}} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$

Variables: $a$ acceleration; $d$ distance; $E$ energy; $f$ frequency; $F$ force; $J$ impulse; $k$ spring constant; $K$ kinetic energy; $\ell$ length; $m$ mass; $M$ mass; $p$ momentum; $P$ power; $r$ radius, distance, or position; $t$ time; $T$ period; $U$ potential energy; $v$ velocity or speed; $W$ work; $x$ position or distance; $y$ vertical position; $\lambda$ linear mass density; $\mu$ coefficient of friction

- Angular velocity (Unit 5: Torque and Rotational Dynamics): $\omega = \frac{d\theta}{dt}$
- Angular acceleration (Unit 5: Torque and Rotational Dynamics): $\alpha = \frac{d\omega}{dt}$
- Angular velocity (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\omega = \omega_0 + \alpha t$
- Angular position (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
- Angular velocity and displacement (constant angular acceleration) (Unit 5: Torque and Rotational Dynamics): $\omega^2 = \omega_0^2 + 2\alpha\left(\theta - \theta_0\right)$
- Linear and angular speed (Unit 5: Torque and Rotational Dynamics): $v = r\omega$
- Tangential acceleration (Unit 5: Torque and Rotational Dynamics): $a_T = r\alpha$
- Torque (Unit 5: Torque and Rotational Dynamics): $\vec{\tau} = \vec{r}\times\vec{F}$
- Rotational inertia of point masses (Unit 5: Torque and Rotational Dynamics): $I_{\text{tot}} = \sum I_i = \sum m_i r_i^2$
- Rotational inertia (continuous) (Unit 5: Torque and Rotational Dynamics): $I = \int r^2\,dm$
- Parallel axis theorem (Unit 5: Torque and Rotational Dynamics): $I' = I_{\text{cm}} + Md^2$
- Newton's second law for rotation (Unit 5: Torque and Rotational Dynamics): $\alpha_{\text{sys}} = \frac{\Sigma\tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}$
- Rotational kinetic energy (Unit 6: Energy and Momentum of Rotating Systems): $K_{\text{rot}} = \frac{1}{2}I\omega^2$
- Work done by a torque (Unit 6: Energy and Momentum of Rotating Systems): $W = \int \tau\cdot d\theta$
- Angular momentum (Unit 6: Energy and Momentum of Rotating Systems): $\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}$
- Angular impulse-momentum theorem (Unit 6: Energy and Momentum of Rotating Systems): $\Delta L = \int \tau\,dt$
- Rolling without slipping (Unit 6: Energy and Momentum of Rotating Systems): $\Delta x_{\text{cm}} = r\Delta\theta$
- Period, angular frequency, and frequency (Unit 7: Oscillations): $T = \frac{2\pi}{\omega} = \frac{1}{f}$
- Period of a mass-spring system (Unit 7: Oscillations): $T_s = 2\pi\sqrt{\frac{m}{k}}$
- Period of a simple pendulum (Unit 7: Oscillations): $T_p = 2\pi\sqrt{\frac{\ell}{g}}$
- Period of a physical pendulum (Unit 7: Oscillations): $T_{\text{phys}} = 2\pi\sqrt{\frac{I}{mgd}}$
- Simple harmonic motion position (Unit 7: Oscillations): $x = x_{\text{max}}\cos\left(\omega t + \phi\right)$

Variables: $a$ acceleration; $d$ distance; $f$ frequency; $F$ force; $I$ rotational inertia; $k$ spring constant; $K$ kinetic energy; $\ell$ length; $L$ angular momentum; $m$ mass; $M$ mass; $p$ momentum; $r$ radius, distance, or position; $t$ time; $T$ period; $v$ velocity or speed; $W$ work; $x$ position or distance; $\alpha$ angular acceleration; $\theta$ angular position; $\tau$ torque; $\phi$ phase angle; $\omega$ angular frequency or angular speed

### Constants and Conversion Factors

- Universal gravitational constant: $\begin{aligned} G &= 6.67\times10^{-11}\ \mathrm{m^3/(kg\cdot s^2)} \\ &= 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2} \end{aligned}$
- Magnitude of the acceleration due to gravity at Earth's surface: $g = 9.8\ \mathrm{m/s^2}$
- Magnitude of the gravitational field strength at Earth's surface: $g = 9.8\ \mathrm{N/kg}$

### Prefixes

| Factor | Prefix | Symbol |
|---|---|---|
| $10^{12}$ | tera | $\mathrm{T}$ |
| $10^{9}$ | giga | $\mathrm{G}$ |
| $10^{6}$ | mega | $\mathrm{M}$ |
| $10^{3}$ | kilo | $\mathrm{k}$ |
| $10^{-2}$ | centi | $\mathrm{c}$ |
| $10^{-3}$ | milli | $\mathrm{m}$ |
| $10^{-6}$ | micro | $\mu$ |
| $10^{-9}$ | nano | $\mathrm{n}$ |
| $10^{-12}$ | pico | $\mathrm{p}$ |

### Unit Symbols

| Unit | Symbol |
|---|---|
| hertz | $\mathrm{Hz}$ |
| joule | $\mathrm{J}$ |
| kilogram | $\mathrm{kg}$ |
| meter | $\mathrm{m}$ |
| newton | $\mathrm{N}$ |
| second | $\mathrm{s}$ |
| watt | $\mathrm{W}$ |

### Values of Trigonometric Functions for Common Angles

| $\theta$ | $0^\circ$ | $30^\circ$ | $37^\circ$ | $45^\circ$ | $53^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $1/2$ | $3/5$ | $\sqrt{2}/2$ | $4/5$ | $\sqrt{3}/2$ | $1$ |
| $\cos\theta$ | $1$ | $\sqrt{3}/2$ | $4/5$ | $\sqrt{2}/2$ | $3/5$ | $1/2$ | $0$ |
| $\tan\theta$ | $0$ | $\sqrt{3}/3$ | $3/4$ | $1$ | $4/3$ | $\sqrt{3}$ | $\infty$ |

### Exam Conventions

- The frame of reference of any problem is assumed to be inertial unless otherwise stated.
- Air resistance is assumed to be negligible unless otherwise stated.
- Springs and strings are assumed to be ideal unless otherwise stated.

### Geometry and Trigonometry

**Rectangle**

- Area of a rectangle: $A = bh$

**Triangle**

- Area of a triangle: $A = \frac{1}{2}bh$

**Circle**

- Area of a circle: $A = \pi r^2$
- Circumference of a circle: $C = 2\pi r$
- Arc length: $s = r\theta$

**Rectangular Solid**

- Volume of a rectangular solid: $V = \ell wh$

**Cylinder**

- Volume of a cylinder: $V = \pi r^2\ell$
- Surface area of a cylinder: $S = 2\pi r\ell + 2\pi r^2$

**Sphere**

- Volume of a sphere: $V = \frac{4}{3}\pi r^3$
- Surface area of a sphere: $S = 4\pi r^2$

**Right Triangle**

- Pythagorean theorem: $a^2 + b^2 = c^2$
- Sine: $\sin\theta = \frac{a}{c}$
- Cosine: $\cos\theta = \frac{b}{c}$
- Tangent: $\tan\theta = \frac{a}{b}$

Variables: $A$ area; $b$ base; $C$ circumference; $h$ height; $\ell$ length; $r$ radius; $s$ arc length; $S$ surface area; $V$ volume; $w$ width; $\theta$ angle

### Vectors

- Dot product: $\vec{A}\cdot\vec{B} = AB\cos\theta$
- Magnitude of the cross product: $\left|\vec{A}\times\vec{B}\right| = AB\sin\theta$
- Vector in unit-vector notation: $\vec{r} = \left(A\hat{\imath} + B\hat{\jmath} + C\hat{k}\right)$
- Vector addition: $\vec{C} = \vec{A} + \vec{B}$
- Vector addition by components: $\vec{C} = \left(A_x + B_x\right)\hat{\imath} + \left(A_y + B_y\right)\hat{\jmath}$

### Calculus

- Chain rule: $\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}$
- Derivative of a power: $\frac{d}{dx}\left(x^n\right) = nx^{n-1}$
- Derivative of an exponential: $\frac{d}{dx}\left(e^{ax}\right) = ae^{ax}$
- Derivative of a natural log: $\frac{d}{dx}\left(\ln ax\right) = \frac{1}{x}$
- Derivative of sine: $\frac{d}{dx}\left[\sin(ax)\right] = a\cos(ax)$
- Derivative of cosine: $\frac{d}{dx}\left[\cos(ax)\right] = -a\sin(ax)$
- Integral of a power: $\int x^n\,dx = \frac{1}{n+1}x^{n+1},\ n \ne -1$
- Integral of an exponential: $\int e^{ax}\,dx = \frac{1}{a}e^{ax}$
- Integral giving a natural log: $\int \frac{dx}{x+a} = \ln\left|x+a\right|$
- Integral of cosine: $\int \cos(ax)\,dx = \frac{1}{a}\sin(ax)$
- Integral of sine: $\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax)$

### Identities

- Logarithm of a product and power: $\log\left(a\cdot b^x\right) = \log a + x\log b$
- Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
- Double-angle identity: $\sin\left(2\theta\right) = 2\sin\theta\cos\theta$
- Tangent as a ratio: $\frac{\sin\theta}{\cos\theta} = \tan\theta$

---

The resources above are plain links; the digital sheet is an interactive JavaScript component whose data loads from the Nerd Notes API — its full content is reproduced above for automated readers.