Constants and conversion factors
$G = 6.67 \times 10^{-11}\ \mathrm{m^{3}/(kg \cdot s^{2})} = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^{2}/kg^{2}}$universal gravitational constant
$g = 9.8\ \mathrm{m/s^{2}}$magnitude of the acceleration due to gravity at the surface of the Earth
$g = 9.8\ \mathrm{N/kg}$magnitude of the gravitational field strength at the surface of the Earth
Prefixes
| Factor | Prefix | Symbol |
|---|---|---|
| $10^{12}$ | tera | $T$ |
| $10^{9}$ | giga | $G$ |
| $10^{6}$ | mega | $M$ |
| $10^{3}$ | kilo | $k$ |
| $10^{-2}$ | centi | $c$ |
| $10^{-3}$ | milli | $m$ |
| $10^{-6}$ | micro | $\mu$ |
| $10^{-9}$ | nano | $n$ |
| $10^{-12}$ | pico | $p$ |
Unit symbols
| hertz | $\mathrm{Hz}$ |
| newton | $\mathrm{N}$ |
| joule | $\mathrm{J}$ |
| second | $\mathrm{s}$ |
| kilogram | $\mathrm{kg}$ |
| watt | $\mathrm{W}$ |
| meter | $\mathrm{m}$ |
Trigonometric functions at common angles
| $\theta$ | $0^{\circ}$ | $30^{\circ}$ | $37^{\circ}$ | $45^{\circ}$ | $53^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $1/2$ | $3/5$ | $\sqrt{2}/2$ | $4/5$ | $\sqrt{3}/2$ | $1$ |
| $\cos\theta$ | $1$ | $\sqrt{3}/2$ | $4/5$ | $\sqrt{2}/2$ | $3/5$ | $1/2$ | $0$ |
| $\tan\theta$ | $0$ | $\sqrt{3}/3$ | $3/4$ | $1$ | $4/3$ | $\sqrt{3}$ | $\infty$ |
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
$A = bh$
Triangle
$A = \tfrac{1}{2}bh$
Circle
$A = \pi r^{2}$
$C = 2\pi r$
$s = r\theta$
Rectangular solid
$V = \ell w h$
Cylinder
$V = \pi r^{2}\ell$
$S = 2\pi r\ell + 2\pi r^{2}$
Sphere
$V = \tfrac{4}{3}\pi r^{3}$
$S = 4\pi r^{2}$
Right triangle
$a^{2} + b^{2} = c^{2}$
$\sin\theta = a/c$
$\cos\theta = b/c$
$\tan\theta = a/b$
Kinematics
$v_{x} = v_{x0} + a_{x}t$
$x = x_{0} + v_{x0}t + \tfrac{1}{2}a_{x}t^{2}$
$v_{x}^{2} = v_{x0}^{2} + 2a_{x}\left(x – x_{0}\right)$
$\Delta x = \int v_{x}(t)\,dt$
$\Delta v_{x} = \int a_{x}(t)\,dt$
Center of mass
$\vec{x}_{\mathrm{cm}} = \frac{\sum m_{i}\vec{x}_{i}}{\sum m_{i}}$
$\vec{r}_{\mathrm{cm}} = \frac{\int \vec{r}\,dm}{\int dm}$
$\lambda = \frac{d}{d\ell}m(\ell)$
$\vec{v}_{\mathrm{cm}} = \frac{\sum \vec{p}_{i}}{\sum m_{i}} = \frac{\sum m_{i}\vec{v}_{i}}{\sum m_{i}}$
Force and translational dynamics
$\vec{a}_{\mathrm{sys}} = \frac{\sum \vec{F}}{m_{\mathrm{sys}}} = \frac{\vec{F}_{\mathrm{net}}}{m_{\mathrm{sys}}}$
$\vec{F}_{\mathrm{net}} = \frac{d\vec{p}}{dt}$
$\left|\vec{F}_{g}\right| = G\frac{m_{1}m_{2}}{r^{2}}$
$\left|\vec{F}_{f}\right| \le \left|\mu \vec{F}_{N}\right|$
$\vec{F}_{s} = -k\Delta\vec{x}$
$a_{c} = \frac{v^{2}}{r} = r\omega^{2}$
$T = \frac{1}{f}$
Work, energy and power
$K = \tfrac{1}{2}mv^{2}$
$W = \int_{a}^{b} \vec{F}\cdot d\vec{r}$
$\Delta K = \sum W_{i} = \sum F_{\parallel,i}\,d_{i}$
$\Delta U = -\int_{a}^{b} \vec{F}_{\mathrm{cf}}(r)\cdot d\vec{r}$
$F_{x} = -\frac{dU(x)}{dx}$
$U_{s} = \tfrac{1}{2}k\left(\Delta x\right)^{2}$
$U_{G} = -G\frac{m_{1}m_{2}}{r}$
$\Delta U_{g} = mg\Delta y$
$P_{\mathrm{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
$P_{\mathrm{inst}} = \frac{dW}{dt}$
Linear momentum
$\vec{p} = m\vec{v}$
$\vec{J} = \int_{t_{1}}^{t_{2}} \vec{F}_{\mathrm{net}}(t)\,dt = \Delta\vec{p}$
Torque and rotational dynamics
$\omega = \frac{d\theta}{dt}$
$\alpha = \frac{d\omega}{dt}$
$\omega = \omega_{0} + \alpha t$
$\theta = \theta_{0} + \omega_{0}t + \tfrac{1}{2}\alpha t^{2}$
$\omega^{2} = \omega_{0}^{2} + 2\alpha\left(\theta – \theta_{0}\right)$
$v = r\omega$
$a_{T} = r\alpha$
$\vec{\tau} = \vec{r} \times \vec{F}$
$I_{\mathrm{tot}} = \sum I_{i} = \sum m_{i}r_{i}^{2}$
$I = \int r^{2}\,dm$
$I’ = I_{\mathrm{cm}} + Md^{2}$
$\alpha_{\mathrm{sys}} = \frac{\Sigma\tau}{I_{\mathrm{sys}}} = \frac{\tau_{\mathrm{net}}}{I_{\mathrm{sys}}}$
Energy and momentum of rotating systems
$K_{\mathrm{rot}} = \tfrac{1}{2}I\omega^{2}$
$W = \int \tau \cdot d\theta$
$\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$
$\Delta L = \int \tau\,dt$
$\Delta x_{\mathrm{cm}} = r\Delta\theta$
Oscillations
$T = \frac{2\pi}{\omega} = \frac{1}{f}$
$T_{s} = 2\pi\sqrt{\frac{m}{k}}$
$T_{p} = 2\pi\sqrt{\frac{\ell}{g}}$
$T_{\mathrm{phys}} = 2\pi\sqrt{\frac{I}{mgd}}$
$x = x_{\max}\cos\left(\omega t + \phi\right)$
Vectors
$\vec{A}\cdot\vec{B} = AB\cos\theta$
$\left|\vec{A}\times\vec{B}\right| = AB\sin\theta$
$\vec{r} = \left(A\hat{\imath} + B\hat{\jmath} + C\hat{k}\right)$
$\vec{C} = \vec{A} + \vec{B}$
$\vec{C} = \left(A_{x} + B_{x}\right)\hat{\imath} + \left(A_{y} + B_{y}\right)\hat{\jmath}$
Calculus
$\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}$
$\frac{d}{dx}\left(x^{n}\right) = nx^{n-1}$
$\frac{d}{dx}\left(e^{ax}\right) = ae^{ax}$
$\frac{d}{dx}\left(\ln ax\right) = \frac{1}{x}$
$\frac{d}{dx}\left[\sin(ax)\right] = a\cos(ax)$
$\frac{d}{dx}\left[\cos(ax)\right] = -a\sin(ax)$
$\int x^{n}\,dx = \frac{1}{n+1}x^{n+1},\ n \neq -1$
$\int e^{ax}\,dx = \frac{1}{a}e^{ax}$
$\int \frac{dx}{x+a} = \ln\left|x+a\right|$
$\int \cos(ax)\,dx = \frac{1}{a}\sin(ax)$
$\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax)$
Identities
$\log\left(a \cdot b^{x}\right) = \log a + x\log b$
$\sin^{2}\theta + \cos^{2}\theta = 1$
$\sin\left(2\theta\right) = 2\sin\theta\cos\theta$
$\frac{\sin\theta}{\cos\theta} = \tan\theta$
What the letters mean
$a$ acceleration
$d$ distance
$E$ energy
$f$ frequency
$F$ force
$I$ rotational inertia
$J$ impulse
$k$ spring constant
$K$ kinetic energy
$\ell$ length
$L$ angular momentum
$m$ 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
$\alpha$ angular acceleration
$\theta$ angular position
$\lambda$ linear mass density
$\mu$ coefficient of friction
$\tau$ torque
$\phi$ phase angle
$\omega$ angular frequency or angular speed
The constants, conventions and equations on this sheet are those the College Board provides on the AP Physics exam reference information.