Equation sheet · Atlas of Physics

AP Physics 1

Algebra-based · effective fall 2024

Constants and conversion factors

Universal gravitational constant	$G = 6.67\times10^{-11}\ \mathrm{m^3/(kg\cdot s^2)} = 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$	3.7 Gravitation 3.8 Orbits
1 atmosphere of pressure	$1\ \mathrm{atm} = 1.0\times10^{5}\ \mathrm{N/m^2} = 1.0\times10^{5}\ \mathrm{Pa}$	8.2 Pressure and depth
Magnitude of the acceleration due to gravity at Earth’s surface	$g = 9.8\ \mathrm{m/s^2}$	1.3 Acceleration 3.7 Gravitation
Magnitude of the gravitational field strength at Earth’s surface	$g = 9.8\ \mathrm{N/kg}$	3.7 Gravitation

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	μ
$10^{-9}$	nano	n
$10^{-12}$	pico	p

Unit symbols

hertz,	Hz
joule,	J
kilogram,	kg
meter,	m
newton,	N
pascal,	Pa
second,	s
watt,	W

Values of trigonometric functions for common angles

$\theta$	$0^\circ$	$30^\circ$	$37^\circ$	$45^\circ$	$53^\circ$	$60^\circ$	$90^\circ$
$\sin\theta$	$0$	$\tfrac12$	$\tfrac35$	$\tfrac{\sqrt2}{2}$	$\tfrac45$	$\tfrac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac45$	$\tfrac{\sqrt2}{2}$	$\tfrac35$	$\tfrac12$	$0$
$\tan\theta$	$0$	$\tfrac{\sqrt3}{3}$	$\tfrac34$	$1$	$\tfrac43$	$\sqrt3$	$\infty$

Conventions

The following conventions are used in this exam:

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.
Fluids are assumed to be ideal, and pipes are assumed to be completely filled by fluid, unless otherwise stated.

Geometry and trigonometry

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

$A = bh$
Rectangle
$A = \tfrac12bh$
Triangle4.5 Spring potential energy 5.10 Plates
$A = \pi r^2 \qquad C = 2\pi r \qquad s = r\theta$
CircleCh. 3 Circular motion 8.6 Continuity 3.6 Angular velocity 7.1 Two riders
$V = \ell wh$
Rectangular solidCh. 8 Fluids
$V = \pi r^2\ell \qquad S = 2\pi r\ell + 2\pi r^2$
Cylinder10.2 A line of charge 10.6 Charged rods
$V = \tfrac43\pi r^3 \qquad S = 4\pi r^2$
Sphere10.1 Gauss’s law 10.5 Insulating sphere
$\displaystyle a^2 + b^2 = c^2 \qquad \sin\theta = \frac{a}{c} \qquad \cos\theta = \frac{b}{c} \qquad \tan\theta = \frac{a}{b}$
Right triangle1.7 Reference frames 1.8 Crossing a river 2.10 The inclined plane 5.7 Two dimensions

Mechanics and fluids · left column

$a$ = acceleration · $d$ = distance · $E$ = energy · $F$ = force · $J$ = impulse · $k$ = spring constant · $K$ = kinetic energy · $m$ = mass · $p$ = momentum · $P$ = power · $r$ = radius or distance · $t$ = time · $U$ = potential energy · $v$ = velocity or speed · $W$ = work · $x$ = position · $y$ = vertical position · $\theta$ = angle · $\mu$ = coefficient of friction

$v_x = v_{x0} + a_xt$
Velocity at constant acceleration1.5 The kinematic equations 1.3 Acceleration
$x = x_0 + v_{x0}t + \tfrac12a_xt^2$
Position at constant acceleration1.5 The kinematic equations 1.6 Projectile motion 9.8 A uniform field
$v_x^2 = v_{x0}^2 + 2a_x(x - x_0)$
Velocity without time1.5 The kinematic equations
$\displaystyle \vec{x}_\text{cm} = \frac{\sum m_i\vec{x}_i}{\sum m_i}$
Centre of mass2.8 Choosing a system 5.8 Centre of mass 5.11 Parts and holes
$\displaystyle \vec{a}_\text{sys} = \frac{\sum\vec{F}}{m_\text{sys}} = \frac{\vec{F}_\text{net}}{m_\text{sys}}$
Newton’s second law2.4 The second law 2.9 The centre of mass accelerates 3.1 Centripetal force
$\displaystyle |\vec{F}_g| = \frac{Gm_1m_2}{r^2}$
Universal gravitation3.7 Gravitation 3.8 Orbits
$|\vec{F}_f| \le |\mu\vec{F}_N|$
Friction2.6 Friction 3.2 A car on a curve 4.10 When it isn’t conserved 7.4 The ladder 7.10 Sliding, then rolling
$\vec{F}_s = -k\Delta\vec{x}$
Hooke’s law: spring force4.5 Spring potential energy 6.1 Hooke’s law 6.4 A hanging spring
$\displaystyle a_c = \frac{v^2}{r}$
Centripetal accelerationCh. 3 Circular motion 3.1 Centripetal force 3.3 Swung in a cone 3.4 Vertical circles 3.5 Banked curves 3.8 Orbits 7.1 Two riders
$K = \tfrac12mv^2$
Kinetic energy4.3 Kinetic energy 4.1 Positive and negative work Ch. 5 Momentum
$W = F_\parallel d = Fd\cos\theta$
Work by a constant forceCh. 4 Work 4.2 Forces that do no work 7.8 Work and power
$\Delta K = \sum W_i = \sum F_{\parallel,i}d_i$
Work–energy theorem4.3 Kinetic energy 4.1 Positive and negative work
$U_s = \tfrac12k(\Delta x)^2$
Spring potential energy4.5 Spring potential energy 6.3 Energy 6.11 Everyday oscillators
$\displaystyle U_G = -\frac{Gm_1m_2}{r}$
Gravitational potential energy, any distance3.10 Energy on an orbit 3.11 Potential energy 3.12 Escape
$\Delta U_g = mg\Delta y$
Gravitational potential energy near the surface4.4 Gravitational potential energy 4.7 Conservation of energy 3.11 Potential energy 5.6 Ballistic pendulum
$\displaystyle P_\text{avg} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
Average power4.11 Power
$P_\text{inst} = F_\parallel v = Fv\cos\theta$
Instantaneous power4.11 Power 15.6 A bar on rails
$\vec{p} = m\vec{v}$
MomentumCh. 5 Momentum 5.3 Conservation
$\displaystyle \vec{F}_\text{net} = \frac{\Delta\vec{p}}{\Delta t} = \frac{m\Delta\vec{v}}{\Delta t} = m\vec{a}$
Net force and change of momentum5.1 Impulse 5.2 Less force
$\vec{J} = \vec{F}_\text{avg}\Delta t = \Delta\vec{p}$
Impulse5.1 Impulse 5.2 Less force 5.6 Ballistic pendulum
$\displaystyle \vec{v}_\text{cm} = \frac{\sum\vec{p}_i}{\sum m_i} = \frac{\sum m_i\vec{v}_i}{\sum m_i}$
Velocity of the centre of mass5.3 Conservation 5.8 Centre of mass 2.8 Choosing a system

Mechanics and fluids · right column

$a$ = acceleration · $A$ = amplitude or area · $d$ = distance · $f$ = frequency · $F$ = force · $h$ = height · $I$ = rotational inertia · $k$ = spring constant · $K$ = kinetic energy · $\ell$ = length · $L$ = angular momentum · $m$ = mass · $M$ = mass · $P$ = pressure · $r$ = radius or distance · $t$ = time · $T$ = period · $v$ = velocity or speed · $V$ = volume · $W$ = work · $x$ = position · $y$ = vertical position · $\alpha$ = angular acceleration · $\theta$ = angle or angular position · $\rho$ = density · $\tau$ = torque · $\omega$ = angular speed

$\omega = \omega_0 + \alpha t$
Angular velocity at constant αCh. 7 Rotation
$\theta = \theta_0 + \omega_0t + \tfrac12\alpha t^2$
Angle at constant αCh. 7 Rotation
$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
Angular velocity without timeCh. 7 Rotation
$v = r\omega$
Linear and angular speed3.6 Angular velocity 7.1 Two riders 7.9 Rolling
$a_T = r\alpha$
Tangential acceleration7.1 Two riders 7.11 The well
$\tau = r_\perp F = rF\sin\theta$
Torque7.2 Torque 7.3 Equilibrium 7.4 The ladder 6.9 Physical pendulums
$I = \sum m_ir_i^2$
Rotational inertia7.5 Moment of inertia 7.6 Adding up I
$I' = I_\text{cm} + Md^2$
Parallel-axis theorem7.7 Parallel axes 6.9 Physical pendulums
$\displaystyle \alpha_\text{sys} = \frac{\sum\tau}{I_\text{sys}} = \frac{\tau_\text{net}}{I_\text{sys}}$
Newton’s second law for rotation7.5 Moment of inertia 7.7 Parallel axes 7.10 Sliding, then rolling 7.11 The well
$K = \tfrac12I\omega^2$
Rotational kinetic energy7.9 Rolling 7.13 The spin 7.8 Work and power
$W = \tau\Delta\theta$
Work done by a torque7.8 Work and power
$L = I\omega$
Angular momentum of a rigid body7.12 Angular impulse 7.13 The spin
$L = rmv\sin\theta$
Angular momentum of a particle7.14 L of a particle 3.9 Elliptical orbits
$\Delta L = \tau\Delta t$
Angular impulse7.12 Angular impulse
$\Delta x_\text{cm} = r\Delta\theta$
Rolling without slipping7.9 Rolling 7.10 Sliding, then rolling
$\displaystyle T = \frac{1}{f}$
Period and frequencyCh. 6 Oscillations
$\displaystyle T_s = 2\pi\sqrt{\frac{m}{k}}$
Period of a mass on a spring6.6 Double the period 6.7 What it ignores 6.11 Everyday oscillators
$\displaystyle T_p = 2\pi\sqrt{\frac{\ell}{g}}$
Period of a simple pendulum6.5 The pendulum 6.6 Double the period 6.11 Everyday oscillators
$x = A\cos(2\pi ft)$
Position in simple harmonic motion, from x = A6.2 x, v and a
$x = A\sin(2\pi ft)$
Position in simple harmonic motion, from x = 0not shown yet
$\displaystyle \rho = \frac{m}{V}$
DensityCh. 8 Fluids 8.5 Float or sink
$\displaystyle P = \frac{F_\perp}{A}$
Pressure8.1 Pressure 8.3 Pascal’s principle
$P = P_0 + \rho gh$
Absolute pressure at depth8.2 Pressure and depth 8.4 Buoyancy
$P_\text{gauge} = \rho gh$
Gauge pressure8.2 Pressure and depth
$F_b = \rho Vg$
Buoyant force8.4 Buoyancy 8.5 Float or sink 2.12 Real fluids
$A_1v_1 = A_2v_2$
Continuity8.6 Continuity
$P_1 + \rho gy_1 + \tfrac12\rho v_1^2 = P_2 + \rho gy_2 + \tfrac12\rho v_2^2$
Bernoulli’s equation8.7 Bernoulli 8.8 Torricelli’s law

AP Physics C: Mechanics

Calculus-based · effective fall 2024

Constants and conversion factors

Universal gravitational constant	$G = 6.67\times10^{-11}\ \mathrm{m^3/(kg\cdot s^2)} = 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$	3.7 Gravitation 3.8 Orbits
Magnitude of the acceleration due to gravity at Earth’s surface	$g = 9.8\ \mathrm{m/s^2}$	1.3 Acceleration 3.7 Gravitation
Magnitude of the gravitational field strength at Earth’s surface	$g = 9.8\ \mathrm{N/kg}$	3.7 Gravitation

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	μ
$10^{-9}$	nano	n
$10^{-12}$	pico	p

Unit symbols

hertz,	Hz
joule,	J
kilogram,	kg
meter,	m
newton,	N
second,	s
watt,	W

Values of trigonometric functions for common angles

$\theta$	$0^\circ$	$30^\circ$	$37^\circ$	$45^\circ$	$53^\circ$	$60^\circ$	$90^\circ$
$\sin\theta$	$0$	$\tfrac12$	$\tfrac35$	$\tfrac{\sqrt2}{2}$	$\tfrac45$	$\tfrac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac45$	$\tfrac{\sqrt2}{2}$	$\tfrac35$	$\tfrac12$	$0$
$\tan\theta$	$0$	$\tfrac{\sqrt3}{3}$	$\tfrac34$	$1$	$\tfrac43$	$\sqrt3$	$\infty$

Conventions

The following conventions are used in this exam:

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.

Mechanics · left column

$a$ = 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

$v_x = v_{x0} + a_xt$
Velocity at constant acceleration1.5 The kinematic equations
$x = x_0 + v_{x0}t + \tfrac12a_xt^2$
Position at constant acceleration1.5 The kinematic equations 1.6 Projectile motion
$v_x^2 = v_{x0}^2 + 2a_x(x - x_0)$
Velocity without time1.5 The kinematic equations
$\displaystyle \Delta x = \int v_x(t)\,\mathrm{d}t$
Displacement as an integral1.9 Calculus kinematics
$\displaystyle \Delta v_x = \int a_x(t)\,\mathrm{d}t$
Change of velocity as an integral1.9 Calculus kinematics
$\displaystyle \vec{x}_\text{cm} = \frac{\sum m_i\vec{x}_i}{\sum m_i}$
Centre of mass2.8 Choosing a system 5.8 Centre of mass 5.11 Parts and holes
$\displaystyle \vec{r}_\text{cm} = \frac{\int\vec{r}\,\mathrm{d}m}{\int\mathrm{d}m}$
Centre of mass of a solid body5.9 Solid bodies 5.10 Plates
$\displaystyle \lambda = \frac{\mathrm{d}}{\mathrm{d}\ell}m(\ell)$
Linear mass density5.9 Solid bodies 7.6 Adding up I
$\displaystyle \vec{a}_\text{sys} = \frac{\sum\vec{F}}{m_\text{sys}} = \frac{\vec{F}_\text{net}}{m_\text{sys}}$
Newton’s second law2.4 The second law 2.9 The centre of mass accelerates 2.11 Drag proportional to v
$\displaystyle |\vec{F}_g| = \frac{Gm_1m_2}{r^2}$
Universal gravitation3.7 Gravitation 3.8 Orbits
$|\vec{F}_f| \le |\mu\vec{F}_N|$
Friction2.6 Friction 3.2 A car on a curve 4.10 When it isn’t conserved 7.10 Sliding, then rolling
$\vec{F}_s = -k\Delta\vec{x}$
Hooke’s law: spring force4.5 Spring potential energy 6.1 Hooke’s law
$\displaystyle a_c = \frac{v^2}{r} = r\omega^2$
Centripetal accelerationCh. 3 Circular motion 3.1 Centripetal force 3.4 Vertical circles 3.6 Angular velocity 7.1 Two riders
$\displaystyle T = \frac{1}{f}$
Period and frequencyCh. 6 Oscillations
$K = \tfrac12mv^2$
Kinetic energy4.3 Kinetic energy
$\displaystyle W = \int_a^b\vec{F}\cdot\mathrm{d}\vec{r}$
Work4.6 Work by a changing force Ch. 4 Work
$\Delta K = \sum W_i = \sum F_{\parallel,i}d_i$
Work–energy theorem4.3 Kinetic energy 4.6 Work by a changing force
$\displaystyle \Delta U = -\int_a^b\vec{F}_\text{cf}(r)\cdot\mathrm{d}\vec{r}$
Potential energy from a conservative force3.11 Potential energy 4.8 Force from potential energy
$\displaystyle F_x = -\frac{\mathrm{d}U(x)}{\mathrm{d}x}$
Force from potential energy4.8 Force from potential energy 4.9 Energy diagrams 3.11 Potential energy
$U_s = \tfrac12k(\Delta x)^2$
Spring potential energy4.5 Spring potential energy 6.3 Energy
$\displaystyle U_G = -\frac{Gm_1m_2}{r}$
Gravitational potential energy3.10 Energy on an orbit 3.11 Potential energy 3.12 Escape
$\Delta U_g = mg\Delta y$
Gravitational potential energy near the surface4.4 Gravitational potential energy 3.11 Potential energy
$\displaystyle P_\text{avg} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
Average power4.11 Power
$\displaystyle P_\text{inst} = \frac{\mathrm{d}W}{\mathrm{d}t}$
Instantaneous power4.11 Power 15.6 A bar on rails
$\vec{p} = m\vec{v}$
MomentumCh. 5 Momentum
$\displaystyle \vec{F}_\text{net} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$
Net force is the rate of change of momentum5.1 Impulse
$\displaystyle \vec{J} = \int_{t_1}^{t_2}\vec{F}_\text{net}(t)\,\mathrm{d}t = \Delta\vec{p}$
Impulse5.1 Impulse 5.2 Less force
$\displaystyle \vec{v}_\text{cm} = \frac{\sum\vec{p}_i}{\sum m_i} = \frac{\sum m_i\vec{v}_i}{\sum m_i}$
Velocity of the centre of mass5.3 Conservation 5.8 Centre of mass

Mechanics · right column

$a$ = 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 · $\varphi$ = phase angle · $\omega$ = angular frequency or angular speed

$\displaystyle \omega = \frac{\mathrm{d}\theta}{\mathrm{d}t}$
Angular velocityCh. 7 Rotation 3.6 Angular velocity
$\displaystyle \alpha = \frac{\mathrm{d}\omega}{\mathrm{d}t}$
Angular accelerationCh. 7 Rotation 7.10 Sliding, then rolling
$\omega = \omega_0 + \alpha t$
Angular velocity at constant αCh. 7 Rotation
$\theta = \theta_0 + \omega_0t + \tfrac12\alpha t^2$
Angle at constant αCh. 7 Rotation
$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
Angular velocity without timeCh. 7 Rotation
$v = r\omega$
Linear and angular speed3.6 Angular velocity 7.1 Two riders
$a_T = r\alpha$
Tangential acceleration7.1 Two riders 7.11 The well
$\vec{\tau} = \vec{r}\times\vec{F}$
Torque7.2 Torque 7.3 Equilibrium 7.4 The ladder 6.9 Physical pendulums
$I_\text{tot} = \sum I_i = \sum m_ir_i^2$
Rotational inertia7.5 Moment of inertia 7.6 Adding up I
$\displaystyle I = \int r^2\,\mathrm{d}m$
Rotational inertia of a solid body7.6 Adding up I
$I' = I_\text{cm} + Md^2$
Parallel-axis theorem7.7 Parallel axes 6.9 Physical pendulums
$\displaystyle \alpha_\text{sys} = \frac{\sum\tau}{I_\text{sys}} = \frac{\tau_\text{net}}{I_\text{sys}}$
Newton’s second law for rotation7.5 Moment of inertia 7.10 Sliding, then rolling 7.11 The well 6.10 The torsion pendulum
$K_\text{rot} = \tfrac12I\omega^2$
Rotational kinetic energy7.9 Rolling 7.13 The spin
$\displaystyle W = \int\tau\cdot\mathrm{d}\theta$
Work done by a torque7.8 Work and power
$\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}$
Angular momentum7.14 L of a particle 3.9 Elliptical orbits 7.12 Angular impulse 7.13 The spin
$\displaystyle \Delta L = \int\tau\,\mathrm{d}t$
Angular impulse7.12 Angular impulse
$\Delta x_\text{cm} = r\Delta\theta$
Rolling without slipping7.9 Rolling 7.10 Sliding, then rolling
$\displaystyle T = \frac{2\pi}{\omega} = \frac{1}{f}$
Period and frequencyCh. 6 Oscillations 6.8 The SHM equation
$\displaystyle T_s = 2\pi\sqrt{\frac{m}{k}}$
Period of a mass on a spring6.6 Double the period 6.4 A hanging spring
$\displaystyle T_p = 2\pi\sqrt{\frac{\ell}{g}}$
Period of a simple pendulum6.5 The pendulum 6.6 Double the period
$\displaystyle T_\text{phys} = 2\pi\sqrt{\frac{I}{mgd}}$
Period of a physical pendulum6.9 Physical pendulums
$x = x_\text{max}\cos(\omega t + \varphi)$
Position in simple harmonic motion6.8 The SHM equation 6.2 x, v and a

Geometry and trigonometry

Also on this sheet, identical to the Physics 1 sheet.

Vectors

The sheet’s math reference.

$\vec{A}\cdot\vec{B} = AB\cos\theta$
Dot productCh. 4 Work Ch. 10 Electric flux
$|\vec{A}\times\vec{B}| = AB\sin\theta$
Cross product7.2 Torque 14.1 The magnetic force
$\vec{r} = (A\hat{\imath} + B\hat{\jmath} + C\hat{k})$
Vector from components
$\vec{C} = \vec{A} + \vec{B}$
Vector sum1.7 Reference frames 1.8 Crossing a river
$\vec{C} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath}$
Vector sum by components

Calculus

The sheet’s math reference.

$\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x} = \frac{\mathrm{d}f}{\mathrm{d}u}\,\frac{\mathrm{d}u}{\mathrm{d}x}$
Chain rule
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$
Derivative of a power1.10 Motion from x(t)
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}(e^{ax}) = ae^{ax}$
Derivative of an exponential
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}(\ln ax) = \frac{1}{x}$
Derivative of a logarithm
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}[\sin(ax)] = a\cos(ax)$
Derivative of sine6.2 x, v and a 6.8 The SHM equation
$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}[\cos(ax)] = -a\sin(ax)$
Derivative of cosine6.2 x, v and a 6.8 The SHM equation
$\displaystyle \int x^n\,\mathrm{d}x = \frac{1}{n + 1}x^{n+1},\quad n \ne -1$
Integral of a power1.9 Calculus kinematics 5.9 Solid bodies 7.6 Adding up I
$\displaystyle \int e^{ax}\,\mathrm{d}x = \frac{1}{a}e^{ax}$
Integral of an exponential2.12 Real fluids
$\displaystyle \int\frac{\mathrm{d}x}{x + a} = \ln|x + a|$
Integral giving a logarithm2.11 Drag proportional to v 12.3 Two cylinders 13.8 RC circuits
$\displaystyle \int\cos(ax)\,\mathrm{d}x = \frac{1}{a}\sin(ax)$
Integral of cosine
$\displaystyle \int\sin(ax)\,\mathrm{d}x = -\frac{1}{a}\cos(ax)$
Integral of sine5.1 Impulse

Identities

The sheet’s math reference.

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

AP Physics C: Electricity and Magnetism

Calculus-based · effective fall 2024

Constants and conversion factors

Coulomb constant	$k = \dfrac{1}{4\pi\varepsilon_0} = 9.0\times10^{9}\ \mathrm{(N\cdot m^2)/C^2}$	9.2 Coulomb’s law
Vacuum permittivity	$\varepsilon_0 = 8.85\times10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$	10.1 Gauss’s law
Vacuum permeability	$\mu_0 = 4\pi\times10^{-7}\ \mathrm{(T\cdot m)/A}$	14.5 Ampère’s law
Proton mass	$m_p = 1.67\times10^{-27}\ \mathrm{kg}$	Ch. 9 Electric charge 14.2 Circling in a field
Neutron mass	$m_n = 1.67\times10^{-27}\ \mathrm{kg}$	Ch. 9 Electric charge
Electron mass	$m_e = 9.11\times10^{-31}\ \mathrm{kg}$	Ch. 9 Electric charge 9.8 A uniform field
Elementary charge	$e = 1.60\times10^{-19}\ \mathrm{C}$	Ch. 9 Electric charge
1 electron volt	$1\ \mathrm{eV} = 1.60\times10^{-19}\ \mathrm{J}$	11.8 The electron volt
Speed of light	$c = 3.00\times10^{8}\ \mathrm{m/s}$	not shown yet
1 unified atomic mass unit	$1\ \mathrm{u} = 1.66\times10^{-27}\ \mathrm{kg} = 931\ \mathrm{MeV}/c^2$	not shown yet
Universal gravitational constant	$G = 6.67\times10^{-11}\ \mathrm{m^3/(kg\cdot s^2)} = 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$	3.7 Gravitation 3.8 Orbits
Magnitude of the acceleration due to gravity at Earth’s surface	$g = 9.8\ \mathrm{m/s^2}$	1.3 Acceleration 3.7 Gravitation
Magnitude of the gravitational field strength at Earth’s surface	$g = 9.8\ \mathrm{N/kg}$	3.7 Gravitation

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	μ
$10^{-9}$	nano	n
$10^{-12}$	pico	p

Unit symbols

ampere,	A
coulomb,	C
electron volt,	eV
farad,	F
henry,	H
hertz,	Hz
joule,	J
kilogram,	kg
meter,	m
newton,	N
ohm,	Ω
second,	s
tesla,	T
volt,	V
watt,	W

Values of trigonometric functions for common angles

$\theta$	$0^\circ$	$30^\circ$	$37^\circ$	$45^\circ$	$53^\circ$	$60^\circ$	$90^\circ$
$\sin\theta$	$0$	$\tfrac12$	$\tfrac35$	$\tfrac{\sqrt2}{2}$	$\tfrac45$	$\tfrac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac45$	$\tfrac{\sqrt2}{2}$	$\tfrac35$	$\tfrac12$	$0$
$\tan\theta$	$0$	$\tfrac{\sqrt3}{3}$	$\tfrac34$	$1$	$\tfrac43$	$\sqrt3$	$\infty$

Conventions

The following conventions are used in this exam unless otherwise stated:

The frame of reference for any problem is inertial.
Frictional forces and air resistance are negligible.
Strings, springs, batteries, wires, and meters are ideal.
The electric potential is zero at an infinite distance from an isolated point charge.
The direction of current is the direction in which positive charges would drift.
Resistors and lightbulbs are ohmic.
Capacitors are air-filled ( $\kappa = 1.0$ ).

Electricity and magnetism · left column

$A$ = area · $C$ = capacitance · $d$ = distance · $E$ = electric field · $F$ = force · $I$ = current · $J$ = current density · $\ell$ = length · $P$ = power · $q$ = charge · $Q$ = charge · $r$ = radius, distance, or position · $R$ = resistance · $t$ = time · $U$ = potential energy · $V$ = electric potential or volume · $\varepsilon$ = electric permittivity · $\rho$ = resistivity or charge density · $\kappa$ = dielectric constant · $\Phi$ = flux

$\displaystyle |\vec{F}_E| = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2} = \frac{k|q_1q_2|}{r^2}$
Coulomb’s law9.2 Coulomb’s law 9.3 Net force
$\displaystyle \vec{E} = \frac{\vec{F}_E}{q}$
Electric field9.4 The electric field 9.8 A uniform field
$\displaystyle \vec{E} = \frac{1}{4\pi\varepsilon_0}\int\frac{\mathrm{d}q}{r^2}\hat{r}$
Field of a continuous charge9.6 A charged rod 9.7 A ring of charge
$\displaystyle \Phi_E = \int\vec{E}\cdot\mathrm{d}\vec{A}$
Electric fluxCh. 10 Electric flux
$\displaystyle \oint\vec{E}\cdot\mathrm{d}\vec{A} = \frac{q_\text{enc}}{\varepsilon_0}$
Gauss’s law10.1 Gauss’s law 10.2 A line of charge 10.3 Sheets and plates 10.4 A metal sphere 10.5 Insulating sphere 10.6 Charged rods 10.7 Uneven charge 12.1 Parallel plates
$\displaystyle Q_\text{total} = \int\rho(r)\,\mathrm{d}V$
Total charge from a charge density10.7 Uneven charge
$\displaystyle U_E = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}$
Electric potential energy of two chargesCh. 11 Electric potential energy 11.1 Assembling charges
$\displaystyle V = \frac{1}{4\pi\varepsilon_0}\int\frac{\mathrm{d}q}{r}$
Potential of a continuous charge11.5 A charged rod 11.6 Ring, arc and disc
$\displaystyle \Delta V = -\int_a^b\vec{E}\cdot\mathrm{d}\vec{r}$
Potential difference from the field11.2 Electric potential 11.3 The potential hill 11.7 Charged spheres 12.1 Parallel plates 12.2 Two spheres 12.3 Two cylinders
$\displaystyle E_x = -\frac{\mathrm{d}V}{\mathrm{d}x}$
Field from potential11.3 The potential hill
$\Delta U_E = q\Delta V$
Potential energy from potential11.2 Electric potential 11.8 The electron volt 14.2 Circling in a field
$\displaystyle C = \frac{Q}{\Delta V}$
CapacitanceCh. 12 Capacitors 12.1 Parallel plates 12.2 Two spheres 12.3 Two cylinders
$\displaystyle C = \frac{\kappa\varepsilon_0A}{d}$
Parallel-plate capacitor12.1 Parallel plates 12.6 Dielectrics 12.8 Conductors
$U_C = \tfrac12Q\Delta V$
Energy stored in a capacitor12.4 Stored energy 12.7 Connected or not
$\displaystyle \kappa = \frac{\varepsilon}{\varepsilon_0}$
Dielectric constant12.6 Dielectrics
$\displaystyle I = \frac{\mathrm{d}q}{\mathrm{d}t}$
CurrentCh. 13 Current 13.8 RC circuits
$\displaystyle I = \int\vec{J}\cdot\mathrm{d}\vec{A}$
Current from current densityCh. 13 Current
$\vec{E} = \rho\vec{J}$
Ohm’s law in a materialCh. 13 Current 13.2 Ohm’s law
$\displaystyle R = \frac{\rho\ell}{A}$
Resistance of a wire13.1 Resistance
$\displaystyle I = \frac{\Delta V}{R}$
Ohm’s law13.2 Ohm’s law 13.5 Series and parallel 13.6 Combinations
$P = I\Delta V$
Electric power13.3 Power 13.4 Emf

Electricity and magnetism · right column

$A$ = area · $B$ = magnetic field · $C$ = capacitance · $E$ = electric field · $F$ = force · $I$ = current · $\ell$ = length · $L$ = inductance · $n$ = number of loops per unit length · $N$ = number of loops · $q$ = charge · $r$ = radius, distance, or position · $R$ = resistance · $t$ = time · $U$ = potential energy · $v$ = velocity or speed · $\varepsilon$ = emf · $\mu$ = magnetic permeability · $\tau$ = time constant · $\Phi$ = flux · $\omega$ = angular frequency

$R_{\text{eq},s} = \sum_i R_i$
Resistors in series13.5 Series and parallel 13.6 Combinations
$\displaystyle \frac{1}{R_{\text{eq},p}} = \sum_i\frac{1}{R_i}$
Resistors in parallel13.5 Series and parallel 13.6 Combinations
$\displaystyle \frac{1}{C_{\text{eq},s}} = \sum_i\frac{1}{C_i}$
Capacitors in series12.5 Series and parallel 12.8 Conductors
$C_{\text{eq},p} = \sum_i C_i$
Capacitors in parallel12.5 Series and parallel 12.7 Connected or not
$\tau = R_\text{eq}C_\text{eq}$
RC time constant13.8 RC circuits
$\displaystyle \oint\vec{B}\cdot\mathrm{d}\vec{A} = 0$
No magnetic monopolesCh. 14 Magnetic fields
$\vec{F}_B = q(\vec{v}\times\vec{B})$
Magnetic force on a moving charge14.1 The magnetic force 14.2 Circling in a field 14.3 Crossed E and B 15.4 Motional emf
$\displaystyle \mathrm{d}\vec{B} = \frac{\mu_0}{4\pi}\frac{I(\mathrm{d}\vec{\ell}\times\hat{r})}{r^2}$
Biot–Savart law14.6 Biot–Savart
$\displaystyle \vec{F}_B = \int I(\mathrm{d}\vec{\ell}\times\vec{B})$
Magnetic force on a current14.4 Force on a current 14.7 Parallel wires 15.6 A bar on rails
$\displaystyle \oint\vec{B}\cdot\mathrm{d}\vec{\ell} = \mu_0I_\text{enc}$
Ampère’s law14.5 Ampère’s law 14.8 Solenoids
$B_\text{sol} = \mu_0nI$
Field inside a solenoid14.8 Solenoids 15.7 Inductance
$\displaystyle \Phi_B = \int\vec{B}\cdot\mathrm{d}\vec{A}$
Magnetic fluxCh. 15 Induction 15.3 Leaving a wire 15.6 A bar on rails Ch. 14 Magnetic fields
$\displaystyle \varepsilon = \oint\vec{E}\cdot\mathrm{d}\vec{\ell} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}$
Faraday’s lawCh. 15 Induction 15.1 Lenz’s law 15.2 Mutual induction 15.3 Leaving a wire 15.5 Through a field
$\displaystyle |\varepsilon_\text{sol}| = N\left|\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|$
Emf of a coil of N loopsCh. 15 Induction
$\displaystyle L_\text{sol} = \frac{\mu_\text{core}N^2A}{\ell}$
Inductance of a solenoid15.7 Inductance
$U_L = \tfrac12LI^2$
Energy stored in an inductor15.7 Inductance 15.9 LC oscillations
$\displaystyle \varepsilon = -L\frac{\mathrm{d}I}{\mathrm{d}t}$
Self-induced emf15.7 Inductance 15.8 RL circuits
$\displaystyle \tau = \frac{L}{R_\text{eq}}$
RL time constant15.8 RL circuits
$\displaystyle \omega_{LC} = \frac{1}{\sqrt{LC}}$
LC angular frequency15.9 LC oscillations

Mechanics

Also on this sheet, identical to the C: Mechanics sheet (both columns).

Geometry and trigonometry

Also on this sheet, identical to the Physics 1 sheet.

Vectors, calculus and identities

Also on this sheet, identical to the C: Mechanics sheet.

No equation matches.