2027 exam · Equation sheet

AP® Physics C: Mechanics equation sheet (2027), explained

After its cover, the official Physics C: Mechanics sheet has three pages. A table of information gives you G and g, a table of prefixes, unit symbols, sine, cosine and tangent for common angles, and the conventions the exam uses. One big Mechanics block lists the equations, with a symbol key, and a last page covers geometry, trigonometry, vectors, calculus and identities. Below, every equation is grouped by unit, with what each symbol means, when to use it, the mistake students make most often, and the topic that teaches it.

The official sheet: AP Physics C: Mechanics Exam Reference Information (PDF, College Board) (opens in a new tab). Keep it open next to this page. We link to it instead of copying it, so you always see College Board's current version.

Checked against the 2027 version on October 5, 2026. The explanations are ours, not College Board's.

Using the sheet on exam day

The sheet is there for the whole exam, on paper and in the testing app. Look it over before exam day so you know where things are: the left column of the Mechanics block is straight-line motion, forces, energy and momentum, and the right column is rotation and oscillations.
The sheet sets three conventions: the frame of reference is inertial, air resistance is negligible, and springs and strings are ideal (massless, with ideal springs obeying Hooke's law), unless the problem says otherwise. So only add drag when a problem gives you a resistive force.
The sheet never tells you when an equation applies. The constant-acceleration lines need constant acceleration, $T_p = 2\pi\sqrt{\ell/g}$ needs small swings, and $v = r\omega$ needs ω in rad/s. When the acceleration changes, use the integral versions.
The trig table includes 37° and 53°, the angles of a 3-4-5 triangle, so you can get exact values like $\sin 37^\circ = 3/5$ without a calculator. When you do use a calculator, check that it's in degree mode for degrees and radian mode for anything like $\cos(\omega t)$ .
Letters get reused: T is a period on this sheet, but you'll also write T for tension, and d is the distance to the center of mass in $T_{\text{phys}}$ but the parallel-axis distance in $I' = I_{\text{cm}} + Md^2$ . Define your symbols when you write an answer.

Constants and conversions

Universal gravitational constant: $\displaystyle G = 6.67 \times 10^{-11}\ \text{m}^3/(\text{kg}\cdot\text{s}^2) = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$; Use it in Newton's law of gravitation and in $U_G = -Gm_1m_2/r$ , for orbits, escape speed and the pull between any two masses.
Acceleration due to gravity at Earth's surface: $\displaystyle g = 9.8\ \text{m/s}^2$; Use it for free fall, projectiles, weight $mg$ and $\Delta U_g = mg\Delta y$ near Earth's surface. It's a magnitude: you pick the sign from your axes.
Gravitational field strength at Earth's surface: $\displaystyle g = 9.8\ \text{N/kg}$; The same number as above, read as newtons of gravitational force per kilogram of mass. It's handy when you think of weight as $mg$ or compare gravity at different places.

Kinematics

Motion along one axis. The first three equations need constant acceleration. The two integrals work for any motion.

Velocity with constant acceleration

\displaystyle v_x = v_{x0} + a_xt

What the symbols mean

$v_x$: velocity along x at time t; Unit: m/s
$v_{x0}$: velocity along x at t = 0; Unit: m/s
$a_x$: constant acceleration along x; Unit: m/s²
$t$: time since the start; Unit: s

Use it when: Use it when the acceleration is constant and the problem links speed and time, like how long a car takes to stop or how fast a dropped ball is moving after 2 s.

Watch out: Using it when the acceleration changes, like when a force depends on time or speed. If $a_x$ isn't constant, integrate $a_x(t)$ instead.

Learn it: 1.3 Representing Motion

Position with constant acceleration

\displaystyle x = x_0 + v_{x0}t + \frac{1}{2}a_xt^2

What the symbols mean

$x$: position at time t; Unit: m
$x_0$: position at t = 0; Unit: m
$v_{x0}$: velocity along x at t = 0; Unit: m/s
$a_x$: constant acceleration along x; Unit: m/s²
$t$: time since the start; Unit: s

Use it when: Use it to find where an object is after a given time, or how long it takes to get somewhere, when the acceleration is constant. In projectile problems, use it once for x and once for y.

Watch out: Getting the sign of the acceleration wrong. If up is positive, a thrown ball has $a_y = -9.8$ m/s² on the way up and on the way down.

Learn it: 1.3 Representing Motion · 1.5 Motion in Two or Three Dimensions

Velocity and position without time

\displaystyle v_x^2 = v_{x0}^2 + 2a_x(x - x_0)

What the symbols mean

$v_x$: velocity along x at the final position; Unit: m/s
$v_{x0}$: starting velocity along x; Unit: m/s
$a_x$: constant acceleration along x; Unit: m/s²
$x - x_0$: displacement; Unit: m

Use it when: Use it when a constant-acceleration problem doesn't give or ask for time, like the stopping distance of a car braking from a known speed.

Watch out: Forgetting that taking the square root gives two signs. The equation tells you the speed; pick the direction of $v_x$ from the situation.

Learn it: 1.3 Representing Motion

Displacement from velocity

\displaystyle \Delta x = \int v_x(t)\,dt

What the symbols mean

$\Delta x$: displacement over the time interval; Unit: m
$v_x(t)$: velocity along x as a function of time; Unit: m/s
$dt$: a tiny slice of time; Unit: s

Use it when: Use it whenever you're given velocity as a function of time, constant acceleration or not. Integrate between the start and end times. On a graph, it's the area between the v–t curve and the time axis.

Watch out: Treating area below the time axis as positive. That part is negative displacement. If the question asks for total distance traveled, split the integral where $v_x$ changes sign and add the sizes.

Try it: An object moves with $v_x(t) = 3t^2$ , in m/s with t in seconds. How far does it move from t = 0 to t = 2.0 s?

Answer: $\Delta x = \int_0^{2.0} 3t^2\,dt = \left[t^3\right]_0^{2.0} = 8.0$ m. The velocity never changes sign, so the distance is also 8.0 m.

Learn it: 1.3 Representing Motion

Velocity change from acceleration

\displaystyle \Delta v_x = \int a_x(t)\,dt

What the symbols mean

$\Delta v_x$: change in velocity along x; Unit: m/s
$a_x(t)$: acceleration along x as a function of time; Unit: m/s²
$dt$: a tiny slice of time; Unit: s

Use it when: Use it when acceleration changes with time. It's the area under the a–t graph. Add the starting velocity to get $v_x(t)$ , then integrate again for position.

Watch out: Forgetting the constant of integration. An indefinite integral gives $v_x(t)$ only up to a constant; set it with the starting velocity, or use limits from the start time to t.

Learn it: 1.3 Representing Motion

Forces and translational dynamics

Center of mass, Newton's second law and the individual forces of Unit 2, plus circular motion.

Center of mass of separate particles

\displaystyle \vec{x}_{\text{cm}} = \frac{\sum m_i\vec{x}_i}{\sum m_i}

What the symbols mean

$\vec{x}_{\text{cm}}$: position of the center of mass; Unit: m
$m_i$: mass of the i-th particle; Unit: kg
$\vec{x}_i$: position of the i-th particle; Unit: m

Use it when: Use it to find the balance point of a few separate objects, like masses on a light rod or the parts of a system. Do x and y separately.

Watch out: Measuring positions from different origins. Pick one origin and measure every $x_i$ from it, with signs.

Learn it: 2.1 Systems and Center of Mass

Center of mass of a solid object

\displaystyle \vec{r}_{\text{cm}} = \frac{\int \vec{r}\,dm}{\int dm}

What the symbols mean

$\vec{r}_{\text{cm}}$: position of the center of mass; Unit: m
$\vec{r}$: position of a tiny piece of the object; Unit: m
$dm$: mass of that tiny piece; Unit: kg

Use it when: Use it for a continuous object whose mass isn't spread evenly, like a rod that gets denser toward one end. For a rod along x, write $dm = \lambda\,dx$ and integrate over the rod. The bottom integral is just the total mass.

Watch out: Integrating over dm without changing to dx. You can't integrate $x\,dm$ until you replace dm with $\lambda(x)\,dx$ and put in the rod's end positions as limits.

Try it: A 1.5 m rod lies from x = 0 to x = 1.5 m, with linear mass density $\lambda = cx$ for some constant c. Where is its center of mass?

Answer: $x_{\text{cm}} = \frac{\int_0^{1.5} x\,(cx)\,dx}{\int_0^{1.5} cx\,dx} = \frac{(1.5)^3/3}{(1.5)^2/2} = \frac{2}{3}(1.5) = 1.0$ m, toward the heavy end. The constant c cancels.

Learn it: 2.1 Systems and Center of Mass

Linear mass density

\displaystyle \lambda = \frac{d}{d\ell}m(\ell)

What the symbols mean

$\lambda$: linear mass density, mass per length; Unit: kg/m
$m(\ell)$: mass of the part of the object from one end out to length ℓ; Unit: kg
$\ell$: length measured along the object; Unit: m

Use it when: Use it to turn a rod or string problem into an integral: $dm = \lambda\,d\ell$ . For a uniform rod, λ is just M/L. If a problem gives you λ as a function of position, integrate it to get the total mass.

Watch out: Treating a varying λ as constant. If $\lambda = cx$ , the mass is $\int \lambda\,dx$ , not λ times the length.

Learn it: 2.1 Systems and Center of Mass · 5.4 Rotational Inertia

Newton's second law for a system

\displaystyle \vec{a}_{\text{sys}} = \frac{\sum\vec{F}}{m_{\text{sys}}} = \frac{\vec{F}_{\text{net}}}{m_{\text{sys}}}

What the symbols mean

$\vec{a}_{\text{sys}}$: acceleration of the system's center of mass; Unit: m/s²
$\sum\vec{F}$: vector sum of the external forces on the system; Unit: N
$\vec{F}_{\text{net}}$: net external force, the same sum; Unit: N
$m_{\text{sys}}$: total mass of the system; Unit: kg

Use it when: Use it after drawing a free-body diagram: add the forces along each axis and divide by mass. Choose a system of several objects to make internal forces like tension drop out.

Watch out: Using one force instead of the net force, or including internal forces. Only forces from outside the system you chose belong in the sum.

Learn it: 2.5 Newton’s Second Law · 2.2 Forces and Free-Body Diagrams

Newton's law of gravitation

\displaystyle \lvert\vec{F}_g\rvert = G\frac{m_1m_2}{r^2}

What the symbols mean

$\lvert\vec{F}_g\rvert$: size of the gravitational force each mass exerts on the other; Unit: N
$G$: universal gravitational constant; Unit: N·m²/kg²
$m_1, m_2$: the two masses; Unit: kg
$r$: distance between their centers; Unit: m

Use it when: Use it for gravity between planets, moons and satellites, or anywhere g isn't constant. Set it equal to $mv^2/r$ for a circular orbit.

Watch out: Using the height above the surface for r. For a satellite, r is the planet's radius plus the altitude, and it's squared.

Learn it: 2.6 Gravitational Force · 2.10 Circular Motion

Friction force

\displaystyle \lvert\vec{F}_f\rvert \le \lvert\mu\vec{F}_N\rvert

What the symbols mean

$\lvert\vec{F}_f\rvert$: size of the friction force; Unit: N
$\mu$: coefficient of friction, static or kinetic; Unit: none
$\vec{F}_N$: normal force between the surfaces; Unit: N

Use it when: For sliding surfaces, kinetic friction equals $\mu_kF_N$ . For surfaces that aren't slipping, static friction is whatever is needed to stop slipping, up to a maximum of $\mu_sF_N$ . Use the maximum only when the object is just about to slip.

Watch out: Setting static friction equal to $\mu_sF_N$ every time. That's its largest possible value. Also, $F_N$ isn't always mg: on a slope it's $mg\cos\theta$ , and pushes or pulls can change it.

Learn it: 2.7 Kinetic and Static Friction

Hooke's law

\displaystyle \vec{F}_s = -k\Delta\vec{x}

What the symbols mean

$\vec{F}_s$: force the spring exerts; Unit: N
$k$: spring constant, the spring's stiffness; Unit: N/m
$\Delta\vec{x}$: stretch or compression from the spring's relaxed length; Unit: m

Use it when: Use it for the force from an ideal spring. The minus sign says the force points back toward the relaxed length. It's also the restoring force behind simple harmonic motion.

Watch out: Measuring Δx from the wrong place. It's from the relaxed length, not from the equilibrium point of a hanging mass and not from the wall.

Learn it: 2.8 Spring Forces · 7.1 Defining Simple Harmonic Motion (SHM)

Centripetal acceleration

\displaystyle a_c = \frac{v^2}{r} = r\omega^2

What the symbols mean

$a_c$: acceleration toward the center of the circle; Unit: m/s²
$v$: speed; Unit: m/s
$r$: radius of the circle; Unit: m
$\omega$: angular speed; Unit: rad/s

Use it when: Use it whenever something moves in a circle. Find the net force toward the center from your free-body diagram and set it equal to $ma_c$ .

Watch out: Drawing a separate "centripetal force" on the free-body diagram. It isn't a new force; it's the net of real forces like tension, gravity, friction or the normal force.

Learn it: 2.10 Circular Motion · 5.2 Connecting Linear and Rotational Motion

Period and frequency

\displaystyle T = \frac{1}{f}

What the symbols mean

$T$: period, the time for one full cycle or revolution; Unit: s
$f$: frequency, cycles per second; Unit: Hz

Use it when: Use it to switch between period and frequency for anything that repeats, like an object going around a circle. For uniform circular motion, the speed is $v = 2\pi r/T$ .

Watch out: Mixing up frequency f and angular speed ω. They differ by 2π: $\omega = 2\pi f$ .

Learn it: 2.10 Circular Motion · 7.2 Frequency and Period of SHM

Work, energy, and power

Kinetic energy, work done by forces, potential energy and how fast energy moves.

Translational kinetic energy

\displaystyle K = \frac{1}{2}mv^2

What the symbols mean

$K$: kinetic energy; Unit: J
$m$: mass; Unit: kg
$v$: speed; Unit: m/s

Use it when: Use it for the energy of motion of a moving object, in any energy or work problem. It's a scalar, so direction doesn't matter.

Watch out: Forgetting to square the speed. Doubling the speed makes K four times bigger.

Learn it: 3.1 Translational Kinetic Energy

Work done by a force

\displaystyle W = \int_a^b \vec{F}\cdot d\vec{r}

What the symbols mean

$W$: work done by the force; Unit: J
$\vec{F}$: the force, which can change along the path; Unit: N
$d\vec{r}$: a tiny step along the path; Unit: m
$a, b$: the start and end points of the path; Unit: m

Use it when: Use it when a force changes with position, like a spring or a force given as $F(x)$ . In one dimension it's $\int F_x\,dx$ , the area under the F–x graph. For a constant force it becomes $Fd\cos\theta$ .

Watch out: Dropping the dot product. Only the part of the force along the motion does work, so a force perpendicular to the path, like the normal force on a flat floor, does zero work.

Try it: A force $F_x = 6x^2$ , in N with x in meters, pushes an object from x = 0 to x = 2.0 m. How much work does it do?

Answer: $W = \int_0^{2.0} 6x^2\,dx = \left[2x^3\right]_0^{2.0} = 16$ J.

Learn it: 3.2 Work

Work-energy theorem

\displaystyle \Delta K = \sum W_i = \sum F_{\parallel,i}d_i

What the symbols mean

$\Delta K$: change in kinetic energy; Unit: J
$W_i$: work done by the i-th force; Unit: J
$F_{\parallel,i}$: part of the i-th force along the displacement; Unit: N
$d_i$: distance moved while that force acts; Unit: m

Use it when: Use it to get a speed from the forces on an object without finding the acceleration. Add the work by every force, then set it equal to the change in K.

Watch out: Leaving out the sign of a force's work. Friction and other forces against the motion do negative work and take kinetic energy away.

Learn it: 3.2 Work · 3.4 Conservation of Energy

Potential energy from a conservative force

\displaystyle \Delta U = -\int_a^b \vec{F}_{\text{cf}}(r)\cdot d\vec{r}

What the symbols mean

$\Delta U$: change in potential energy from point a to point b; Unit: J
$\vec{F}_{\text{cf}}(r)$: a conservative force, like gravity or a spring force, as a function of position; Unit: N
$d\vec{r}$: a tiny step along the path; Unit: m

Use it when: Use it to build a potential energy function from a conservative force, like getting $U_G = -Gm_1m_2/r$ from gravity or $\tfrac{1}{2}kx^2$ from a spring. The subscript cf means conservative force: only those have a potential energy.

Watch out: Dropping the minus sign. When a conservative force does positive work, like gravity on a falling ball, the potential energy goes down.

Learn it: 3.3 Potential Energy

Force from potential energy

\displaystyle F_x = -\frac{dU(x)}{dx}

What the symbols mean

$F_x$: the conservative force along x; Unit: N
$U(x)$: potential energy as a function of position; Unit: J
$x$: position; Unit: m

Use it when: Use it when you're given U(x) as an equation or a graph and asked for the force. The force is the negative slope, so it points downhill on the U–x graph. Where the slope is zero, the object is in equilibrium.

Watch out: Reading the force off the height of the U graph instead of its slope. A point with large U can have zero force if the curve is flat there.

Learn it: 3.3 Potential Energy

Spring potential energy

\displaystyle U_s = \frac{1}{2}k(\Delta x)^2

What the symbols mean

$U_s$: energy stored in the spring; Unit: J
$k$: spring constant; Unit: N/m
$\Delta x$: stretch or compression from the relaxed length; Unit: m

Use it when: Use it in energy problems with springs, like a block launched by a compressed spring or a mass oscillating on one.

Watch out: Using the change in stretch instead of the stretch itself. If a spring goes from 0.10 m to 0.20 m of stretch, $\Delta U_s = \tfrac{1}{2}k(0.20^2 - 0.10^2)$ , not $\tfrac{1}{2}k(0.10)^2$ .

Learn it: 3.3 Potential Energy · 7.4 Energy of Simple Harmonic Oscillators

Gravitational potential energy between two masses

\displaystyle U_G = -G\frac{m_1m_2}{r}

What the symbols mean

$U_G$: gravitational potential energy of the pair, zero when they're infinitely far apart; Unit: J
$G$: universal gravitational constant; Unit: N·m²/kg²
$m_1, m_2$: the two masses; Unit: kg
$r$: distance between their centers; Unit: m

Use it when: Use it for satellites, escape speed and anything that moves far from a planet's surface, where g isn't constant.

Watch out: Dropping the minus sign, or dividing by r². This U is always negative and gets less negative as the masses move apart, and it has r to the first power.

Learn it: 3.3 Potential Energy · 6.6 Motion of Orbiting Satellites

Change in gravitational potential energy near Earth

\displaystyle \Delta U_g = mg\Delta y

What the symbols mean

$\Delta U_g$: change in gravitational potential energy; Unit: J
$m$: mass; Unit: kg
$g$: gravitational field strength, 9.8 N/kg near Earth; Unit: N/kg
$\Delta y$: change in height, positive going up; Unit: m

Use it when: Use it for objects that move up or down near a planet's surface, where g barely changes. Only the change in height matters, not where you put zero.

Watch out: Using the length of a ramp or the path instead of the vertical change in height.

Learn it: 3.3 Potential Energy · 3.4 Conservation of Energy

Average power

\displaystyle P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}

What the symbols mean

$P_{\text{avg}}$: average power; Unit: W
$W$: work done; Unit: J
$\Delta E$: energy transferred or changed; Unit: J
$\Delta t$: time it took; Unit: s

Use it when: Use it when you know a total amount of work or energy and the time, like a motor lifting a load to a height in a given time.

Watch out: Mixing up the italic W for work with the unit W for watts. In $P = W/\Delta t$ , the W on top is work in joules.

Learn it: 3.5 Power

Instantaneous power

\displaystyle P_{\text{inst}} = \frac{dW}{dt}

What the symbols mean

$P_{\text{inst}}$: power at one instant; Unit: W
$dW$: a tiny amount of work; Unit: J
$dt$: a tiny time interval; Unit: s

Use it when: Use it when the rate of work changes with time. For a force on a moving object it works out to $P = \vec{F}\cdot\vec{v}$ , so you can get the power at any moment from the force and the velocity.

Watch out: Using the average power when a question asks for the power at a particular moment, while the speed or force is changing.

Learn it: 3.5 Power

Linear momentum

Momentum, impulse and the motion of a system's center of mass.

Linear momentum

\displaystyle \vec{p} = m\vec{v}

What the symbols mean

$\vec{p}$: momentum; Unit: kg·m/s
$m$: mass; Unit: kg
$\vec{v}$: velocity; Unit: m/s

Use it when: Use it in collisions, explosions and any problem where objects push on each other. Add the momenta of every object to get the system's total.

Watch out: Treating momentum as a scalar. Objects moving in opposite directions have momenta with opposite signs, and in two dimensions you add components.

Learn it: 4.1 Linear Momentum · 4.3 Conservation of Linear Momentum

Net force as rate of change of momentum

\displaystyle \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}

What the symbols mean

$\vec{F}_{\text{net}}$: net external force; Unit: N
$d\vec{p}/dt$: rate at which momentum changes; Unit: kg·m/s²

Use it when: Use it when you have momentum as a function of time and want the force, or as the general form of Newton's second law. If the net external force on a system is zero, its momentum stays constant.

Watch out: Thinking a force is needed to keep momentum constant. Zero net force means constant momentum, not zero momentum.

Learn it: 4.2 Change in Momentum and Impulse · 2.5 Newton’s Second Law

Impulse

\displaystyle \vec{J} = \int_{t_1}^{t_2}\vec{F}_{\text{net}}(t)\,dt = \Delta\vec{p}

What the symbols mean

$\vec{J}$: impulse; Unit: N·s
$\vec{F}_{\text{net}}(t)$: net force as a function of time; Unit: N
$t_1, t_2$: start and end of the time interval; Unit: s
$\Delta\vec{p}$: change in momentum; Unit: kg·m/s

Use it when: Use it when a force changes with time, like a bat hitting a ball or a force given as F(t). It's the area under the F–t graph, and it equals the change in momentum.

Watch out: Ignoring direction when an object bounces. A ball hitting a wall at 5 m/s and bouncing back at 5 m/s has $\Delta v = 10$ m/s, not zero.

Try it: A net force $F(t) = 4t$ , in N with t in seconds, acts on a 2.0 kg cart at rest from t = 0 to t = 3.0 s. What is the cart's final speed?

Answer: $J = \int_0^{3.0} 4t\,dt = \left[2t^2\right]_0^{3.0} = 18$ N·s. So $\Delta p = 18$ kg·m/s and $v = 18/2.0 = 9.0$ m/s.

Learn it: 4.2 Change in Momentum and Impulse

Velocity of the center of mass

\displaystyle \vec{v}_{\text{cm}} = \frac{\sum\vec{p}_i}{\sum m_i} = \frac{\sum m_i\vec{v}_i}{\sum m_i}

What the symbols mean

$\vec{v}_{\text{cm}}$: velocity of the system's center of mass; Unit: m/s
$\vec{p}_i$: momentum of the i-th object; Unit: kg·m/s
$m_i$: mass of the i-th object; Unit: kg
$\vec{v}_i$: velocity of the i-th object; Unit: m/s

Use it when: Use it to track a whole system through a collision or explosion. If no net external force acts, $\vec{v}_{\text{cm}}$ stays the same before and after, even when the pieces fly apart.

Watch out: Adding speeds without signs. Velocities in opposite directions partly cancel.

Learn it: 4.3 Conservation of Linear Momentum · 2.1 Systems and Center of Mass

Rotation

The right column of the Mechanics block: rotational kinematics, torque, rotational inertia, rotational energy, angular momentum and rolling. Angles are in radians throughout.

Angular velocity

\displaystyle \omega = \frac{d\theta}{dt}

What the symbols mean

$\omega$: angular velocity; Unit: rad/s
$\theta$: angular position; Unit: rad
$t$: time; Unit: s

Use it when: Use it to get angular velocity from an angle given as a function of time. It works just like $v = dx/dt$ for straight-line motion.

Watch out: Leaving the answer in revolutions or degrees. Convert to radians: one revolution is 2π rad.

Learn it: 5.1 Rotational Kinematics

Angular acceleration

\displaystyle \alpha = \frac{d\omega}{dt}

What the symbols mean

$\alpha$: angular acceleration; Unit: rad/s²
$\omega$: angular velocity; Unit: rad/s
$t$: time; Unit: s

Use it when: Use it to get angular acceleration from ω(t), or integrate α to get the change in ω.

Watch out: Getting the sign wrong for something slowing down. If ω is positive and decreasing, α is negative.

Learn it: 5.1 Rotational Kinematics

Angular velocity with constant angular acceleration

\displaystyle \omega = \omega_0 + \alpha t

What the symbols mean

$\omega$: angular velocity at time t; Unit: rad/s
$\omega_0$: angular velocity at t = 0; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²
$t$: time; Unit: s

Use it when: Use it when a spinning object speeds up or slows down at a steady rate and the problem links angular speed and time.

Watch out: Using it when the torque, and so α, changes. Then you need $\alpha = d\omega/dt$ and an integral.

Learn it: 5.1 Rotational Kinematics

Angular position with constant angular acceleration

\displaystyle \theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2

What the symbols mean

$\theta$: angular position at time t; Unit: rad
$\theta_0$: angular position at t = 0; Unit: rad
$\omega_0$: angular velocity at t = 0; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²
$t$: time; Unit: s

Use it when: Use it to find how far something turns in a given time at constant angular acceleration. Divide by 2π to get the number of revolutions.

Watch out: Answering in radians when the question asks for revolutions, or the other way around.

Learn it: 5.1 Rotational Kinematics

Angular velocity and angle without time

\displaystyle \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)

What the symbols mean

$\omega$: final angular velocity; Unit: rad/s
$\omega_0$: starting angular velocity; Unit: rad/s
$\alpha$: constant angular acceleration; Unit: rad/s²
$\theta - \theta_0$: angle turned through; Unit: rad

Use it when: Use it at constant angular acceleration when time isn't given or asked for, like how many turns a wheel makes while braking to a stop.

Watch out: Putting in the angle in revolutions or degrees. It must be in radians for ω to come out in rad/s.

Learn it: 5.1 Rotational Kinematics

Linear speed from angular speed

\displaystyle v = r\omega

What the symbols mean

$v$: speed of a point on the rotating object; Unit: m/s
$r$: that point's distance from the axis; Unit: m
$\omega$: angular speed; Unit: rad/s

Use it when: Use it to connect a spinning object to things moving in a line, like a string unwinding from a pulley or a point on a wheel's rim.

Watch out: Using ω in revolutions per minute. Convert to rad/s first, or the answer is off by a factor like 2π/60.

Learn it: 5.2 Connecting Linear and Rotational Motion

Tangential acceleration

\displaystyle a_T = r\alpha

What the symbols mean

$a_T$: tangential acceleration, along the direction of motion; Unit: m/s²
$r$: distance from the axis; Unit: m
$\alpha$: angular acceleration; Unit: rad/s²

Use it when: Use it to link the angular acceleration of a pulley or wheel to the linear acceleration of a hanging mass or rope, when the rope doesn't slip.

Watch out: Mixing it up with centripetal acceleration. $a_T$ changes the speed; $a_c = r\omega^2$ points toward the center and changes the direction. A point on a speeding-up wheel has both.

Learn it: 5.2 Connecting Linear and Rotational Motion

Torque

\displaystyle \vec{\tau} = \vec{r}\times\vec{F}

What the symbols mean

$\vec{\tau}$: torque about the chosen axis or point; Unit: N·m
$\vec{r}$: position from the axis to where the force acts; Unit: m
$\vec{F}$: the force; Unit: N

Use it when: Use it to find how strongly a force twists an object. The size is $rF\sin\theta$ , where θ is the angle between $\vec{r}$ and $\vec{F}$ , and the direction comes from the right-hand rule.

Watch out: Using the full distance when the force isn't perpendicular, or swapping sine for cosine. A force pointing straight at the axis has zero torque.

Learn it: 5.3 Torque

Rotational inertia of particles

\displaystyle I_{\text{tot}} = \sum I_i = \sum m_ir_i^2

What the symbols mean

$I_{\text{tot}}$: total rotational inertia about the axis; Unit: kg·m²
$I_i$: rotational inertia of the i-th part; Unit: kg·m²
$m_i$: mass of the i-th particle; Unit: kg
$r_i$: its perpendicular distance from the axis; Unit: m

Use it when: Use it for a few small masses, or to add the rotational inertias of parts, like a rod plus a ball on its end. Every part must be about the same axis.

Watch out: Forgetting to square r, or measuring r from the center of the object instead of from the axis of rotation.

Learn it: 5.4 Rotational Inertia

Rotational inertia of a solid object

\displaystyle I = \int r^2\,dm

What the symbols mean

$I$: rotational inertia about the axis; Unit: kg·m²
$r$: perpendicular distance of a tiny piece from the axis; Unit: m
$dm$: mass of that tiny piece; Unit: kg

Use it when: Use it to derive I for a rod, hoop or disk, especially when the density isn't uniform. For a rod, write $dm = \lambda\,dx$ and integrate from one end to the other, measuring x from the axis.

Watch out: Putting the limits in the wrong place. For a rod pivoted at its end, integrate from 0 to L; about its center, from −L/2 to L/2.

Learn it: 5.4 Rotational Inertia

Parallel-axis theorem

\displaystyle I' = I_{\text{cm}} + Md^2

What the symbols mean

$I'$: rotational inertia about the new axis; Unit: kg·m²
$I_{\text{cm}}$: rotational inertia about a parallel axis through the center of mass; Unit: kg·m²
$M$: total mass; Unit: kg
$d$: distance between the two axes; Unit: m

Use it when: Use it when an object rotates about an axis that doesn't pass through its center of mass, and you know $I_{\text{cm}}$ . It's common in physical pendulum problems.

Watch out: Using it between two axes when neither goes through the center of mass. You must start from $I_{\text{cm}}$ , and the axes must be parallel.

Try it: A uniform rod has M = 3.0 kg and L = 2.0 m, so $I_{\text{cm}} = \tfrac{1}{12}ML^2 = 1.0$ kg·m². What is its rotational inertia about one end?

Answer: The end is d = 1.0 m from the center, so $I' = 1.0 + (3.0)(1.0)^2 = 4.0$ kg·m². That matches $\tfrac{1}{3}ML^2$ .

Learn it: 5.4 Rotational Inertia

Newton's second law for rotation

\displaystyle \alpha_{\text{sys}} = \frac{\sum\tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}

What the symbols mean

$\alpha_{\text{sys}}$: angular acceleration of the system; Unit: rad/s²
$\sum\tau$: sum of the external torques about the axis; Unit: N·m
$\tau_{\text{net}}$: net external torque, the same sum; Unit: N·m
$I_{\text{sys}}$: rotational inertia about the same axis; Unit: kg·m²

Use it when: Use it for anything spinning up or down because of torques, like a pulley with a hanging mass. Pair it with $F_{\text{net}} = ma$ for the hanging mass and $a = r\alpha$ to link them. With zero net torque, the object is in rotational equilibrium.

Watch out: Ignoring the signs of torques. Pick a positive direction, like counterclockwise, and give each torque a sign before adding.

Learn it: 5.6 Newton’s Second Law in Rotational Form · 5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form

Rotational kinetic energy

\displaystyle K_{\text{rot}} = \frac{1}{2}I\omega^2

What the symbols mean

$K_{\text{rot}}$: kinetic energy of rotation; Unit: J
$I$: rotational inertia about the axis; Unit: kg·m²
$\omega$: angular speed; Unit: rad/s

Use it when: Use it in energy problems with spinning or rolling objects. A rolling object has both kinds: $K = \tfrac{1}{2}Mv_{\text{cm}}^2 + \tfrac{1}{2}I_{\text{cm}}\omega^2$ .

Watch out: Leaving out the rotational part for something that rolls. A ball that rolls down a ramp ends up with less translational kinetic energy than a block that slides down a frictionless ramp from the same height, because some energy goes into spinning.

Learn it: 6.1 Rotational Kinetic Energy · 6.5 Rolling

Work done by a torque

\displaystyle W = \int \tau\cdot d\theta

What the symbols mean

$W$: work done by the torque; Unit: J
$\tau$: torque about the axis, which can change with angle; Unit: N·m
$d\theta$: a tiny angle turned through; Unit: rad

Use it when: Use it to find how much energy a torque adds to a spinning object. For a constant torque, it's $\tau\Delta\theta$ . It's the rotational partner of $W = \int F\,dx$ .

Watch out: Using the angle in degrees or revolutions. With θ in radians, N·m times rad gives joules.

Learn it: 6.2 Torque and Work

Angular momentum

\displaystyle \vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}

What the symbols mean

$\vec{L}$: angular momentum about the chosen point or axis; Unit: kg·m²/s
$\vec{r}$: position from the chosen point to the object; Unit: m
$\vec{p}$: linear momentum; Unit: kg·m/s
$I$: rotational inertia about the axis; Unit: kg·m²
$\vec{\omega}$: angular velocity; Unit: rad/s

Use it when: Use $\vec{r}\times\vec{p}$ for an object moving in a line or orbit, with size $rp\sin\theta$ , and $I\vec{\omega}$ for a spinning rigid object. Use them together for a ball hitting and sticking to a rod.

Watch out: Thinking an object moving in a straight line has no angular momentum. About any point not on its path, it has $L = mvr_\perp$ .

Learn it: 6.3 Angular Momentum and Angular Impulse

Angular impulse

\displaystyle \Delta L = \int \tau\,dt

What the symbols mean

$\Delta L$: change in angular momentum; Unit: kg·m²/s
$\tau$: net external torque, which can change with time; Unit: N·m
$dt$: a tiny time interval; Unit: s

Use it when: Use it when a torque acts for a time, like a motor spinning up a wheel. It's the area under the τ–t graph. With no net external torque, L stays constant, which is conservation of angular momentum.

Watch out: Expecting angular momentum to stay constant when an external torque acts, or forgetting that internal torques, like a skater pulling in her arms, don't change L.

Learn it: 6.3 Angular Momentum and Angular Impulse · 6.4 Conservation of Angular Momentum

Rolling without slipping

\displaystyle \Delta x_{\text{cm}} = r\Delta\theta

What the symbols mean

$\Delta x_{\text{cm}}$: distance the center of mass moves; Unit: m
$r$: radius of the rolling object; Unit: m
$\Delta\theta$: angle the object turns through; Unit: rad

Use it when: Use it for a wheel or ball that rolls without slipping: each turn moves it forward one circumference. Take the time derivative to get $v_{\text{cm}} = r\omega$ and $a_{\text{cm}} = r\alpha$ .

Watch out: Using it when the object slips or skids. Then the center moves a different distance than $r\Delta\theta$ , and kinetic friction does work.

Learn it: 6.5 Rolling

Oscillations

Simple harmonic motion: period, frequency and position over time for springs and pendulums.

Period and angular frequency

\displaystyle T = \frac{2\pi}{\omega} = \frac{1}{f}

What the symbols mean

$T$: period, the time for one full cycle; Unit: s
$\omega$: angular frequency; Unit: rad/s
$f$: frequency; Unit: Hz

Use it when: Use it to switch between period, frequency and angular frequency. In $x = x_{\text{max}}\cos(\omega t + \phi)$ , the ω is this angular frequency.

Watch out: Mixing up f and ω. They differ by a factor of 2π, so check which one a question asks for.

Learn it: 7.2 Frequency and Period of SHM

Period of a mass on a spring

\displaystyle T_s = 2\pi\sqrt{\frac{m}{k}}

What the symbols mean

$T_s$: period of the oscillating mass; Unit: s
$m$: mass attached to the spring; Unit: kg
$k$: spring constant; Unit: N/m

Use it when: Use it for any mass on an ideal spring, horizontal or vertical. The period doesn't depend on the amplitude or on g.

Watch out: Flipping the fraction. A heavier mass oscillates more slowly, so m goes on top.

Try it: A 0.50 kg block oscillates on a spring with k = 200 N/m. What is its period?

Answer: $T_s = 2\pi\sqrt{\frac{0.50}{200}} = 2\pi(0.050) \approx 0.31$ s.

Learn it: 7.2 Frequency and Period of SHM

Period of a simple pendulum

\displaystyle T_p = 2\pi\sqrt{\frac{\ell}{g}}

What the symbols mean

$T_p$: period of the pendulum; Unit: s
$\ell$: length of the string, from the pivot to the center of the bob; Unit: m
$g$: acceleration due to gravity; Unit: m/s²

Use it when: Use it for a small mass on a light string swinging through small angles. The period doesn't depend on the mass of the bob.

Watch out: Using it for large swings, or for a swinging object whose mass is spread out, like a rod. For a rod or other solid object, use the physical pendulum equation.

Learn it: 7.2 Frequency and Period of SHM · 7.5 Simple and Physical Pendulums

Period of a physical pendulum

\displaystyle T_{\text{phys}} = 2\pi\sqrt{\frac{I}{mgd}}

What the symbols mean

$T_{\text{phys}}$: period of the physical pendulum; Unit: s
$I$: rotational inertia about the pivot; Unit: kg·m²
$m$: mass of the object; Unit: kg
$g$: acceleration due to gravity; Unit: m/s²
$d$: distance from the pivot to the center of mass; Unit: m

Use it when: Use it for any rigid object swinging about a pivot through small angles, like a rod, a ruler or a disk hung from its edge. A simple pendulum is the special case $I = m\ell^2$ , $d = \ell$ .

Watch out: Using $I_{\text{cm}}$ instead of the rotational inertia about the pivot. Use the parallel-axis theorem to move I to the pivot first.

Try it: A uniform 1.0 m rod swings from one end. About that end, $I = \tfrac{1}{3}mL^2$ , and its center of mass is d = 0.50 m from the pivot. What is the period for small swings?

Answer: $T = 2\pi\sqrt{\frac{mL^2/3}{mg(L/2)}} = 2\pi\sqrt{\frac{2L}{3g}} = 2\pi\sqrt{\frac{2(1.0)}{3(9.8)}} \approx 1.6$ s. The mass cancels.

Learn it: 7.5 Simple and Physical Pendulums

Position in simple harmonic motion

\displaystyle x = x_{\text{max}}\cos(\omega t + \phi)

What the symbols mean

$x$: displacement from equilibrium at time t; Unit: m
$x_{\text{max}}$: amplitude, the largest displacement; Unit: m
$\omega$: angular frequency; Unit: rad/s
$t$: time; Unit: s
$\phi$: phase angle, which sets where the cycle starts; Unit: rad

Use it when: Use it to write the position of an oscillator as a function of time, then take derivatives for velocity and acceleration. If the object starts at $+x_{\text{max}}$ at t = 0, φ is 0.

Watch out: Leaving the calculator in degree mode. The angle $\omega t + \phi$ is in radians.

Learn it: 7.3 Representing and Analyzing SHM · 7.1 Defining Simple Harmonic Motion (SHM)

Geometry and trigonometry

Areas, volumes and right-triangle rules. In mechanics you use them for areas under graphs, components of vectors and arc lengths.

Area of a rectangle

\displaystyle A = bh

What the symbols mean

$A$: area; Unit: m²
$b$: base; Unit: m
$h$: height; Unit: m

Use it when: Use it for the area under a flat section of a graph, like a constant velocity on a v–t graph, which gives a displacement.

Watch out: Forgetting the units of a graph's axes. Area under a v–t graph has units of m/s times s, which is meters, not square meters.

Learn it: 1.3 Representing Motion

Area of a triangle

\displaystyle A = \frac{1}{2}bh

What the symbols mean

$A$: area; Unit: m²
$b$: base; Unit: m
$h$: height; Unit: m

Use it when: Use it for the area under a sloped straight-line section of a graph, like a v–t graph during constant acceleration or an F–x graph for a spring.

Watch out: Leaving off the one half.

Learn it: 1.3 Representing Motion · 3.2 Work

Area of a circle

\displaystyle A = \pi r^2

What the symbols mean

$A$: area; Unit: m²
$r$: radius; Unit: m

Use it when: Use it when you need the mass of a disk from its mass per area, as when you set up $I = \int r^2\,dm$ for a disk with thin rings.

Watch out: Using the diameter in place of the radius.

Learn it: 5.4 Rotational Inertia

Circumference of a circle

\displaystyle C = 2\pi r

What the symbols mean

$C$: circumference, the distance around; Unit: m
$r$: radius; Unit: m

Use it when: Use it for the distance covered in one trip around a circle, as in $v = 2\pi r/T$ for uniform circular motion, or how far a rolling wheel moves in one turn.

Watch out: Using the diameter in place of the radius, which doubles the answer.

Learn it: 2.10 Circular Motion · 6.5 Rolling

Arc length

\displaystyle s = r\theta

What the symbols mean

$s$: arc length, the distance along the circle; Unit: m
$r$: radius; Unit: m
$\theta$: angle the arc covers; Unit: rad

Use it when: Use it to turn an angle into a distance along a circle, like the length of rope unwound from a pulley. Taking derivatives gives $v = r\omega$ and $a_T = r\alpha$ .

Watch out: Putting θ in degrees. It only works with θ in radians.

Try it: A pulley of radius 0.30 m turns through 2.0 rad. How much rope unwinds?

Answer: $s = r\theta = (0.30)(2.0) = 0.60$ m.

Learn it: 5.1 Rotational Kinematics · 5.2 Connecting Linear and Rotational Motion

Volume of a rectangular solid

\displaystyle V = \ell wh

What the symbols mean

$V$: volume; Unit: m³
$\ell$: length; Unit: m
$w$: width; Unit: m
$h$: height; Unit: m

Use it when: Use it to get the mass of a block from its density, like when you find the center of mass of an object built from blocks of different materials.

Watch out: Mixing units, like centimeters for one side and meters for another. Convert everything to meters first.

Learn it: 2.1 Systems and Center of Mass

Volume of a cylinder

\displaystyle V = \pi r^2\ell

What the symbols mean

$V$: volume; Unit: m³
$r$: radius; Unit: m
$\ell$: length; Unit: m

Use it when: Use it to get the mass of a solid cylinder or disk from its density, before finding its rotational inertia.

Watch out: Using the diameter in place of the radius, which makes the volume four times too big.

Learn it: 5.4 Rotational Inertia

Surface area of a cylinder

\displaystyle S = 2\pi r\ell + 2\pi r^2

What the symbols mean

$S$: total surface area, side plus both ends; Unit: m²
$r$: radius; Unit: m
$\ell$: length; Unit: m

Use it when: Use the side part, $2\pi r\ell$ , when you need the area of a thin cylindrical shell, for example to get its mass from a mass per area.

Watch out: Using the whole formula for an open tube. A shell with no ends has only the $2\pi r\ell$ part.

Learn it: 5.4 Rotational Inertia

Volume of a sphere

\displaystyle V = \frac{4}{3}\pi r^3

What the symbols mean

$V$: volume; Unit: m³
$r$: radius; Unit: m

Use it when: Use it to get the mass of a planet or ball from its density, or the share of a uniform planet's mass that lies inside a smaller radius, which grows as $r^3$ .

Watch out: Squaring the radius instead of cubing it.

Learn it: 2.6 Gravitational Force

Surface area of a sphere

\displaystyle S = 4\pi r^2

What the symbols mean

$S$: surface area; Unit: m²
$r$: radius; Unit: m

Use it when: Use it when you need the area of a thin spherical shell, for example to get its mass from a mass per area.

Watch out: Mixing it up with the volume formula. Area goes with r², volume with r³.

Learn it: 2.6 Gravitational Force

Pythagorean theorem

\displaystyle a^2 + b^2 = c^2

What the symbols mean

$a, b$: the two shorter sides of a right triangle; Unit: m, or the unit of the vector
$c$: the hypotenuse, the side across from the right angle; Unit: m, or the unit of the vector

Use it when: Use it to find the size of a vector from its two components, like a projectile's speed from $v_x$ and $v_y$ .

Watch out: Adding the components directly instead of their squares. A 3 m/s and a 4 m/s component give 5 m/s, not 7 m/s.

Learn it: 1.1 Scalars and Vectors · 1.5 Motion in Two or Three Dimensions

Sine in a right triangle

\displaystyle \sin\theta = \frac{a}{c}

What the symbols mean

$\theta$: the angle; Unit: degrees or rad
$a$: side opposite the angle; Unit: m, or the unit of the vector
$c$: hypotenuse; Unit: m, or the unit of the vector

Use it when: Use it to find the component of a vector on the side opposite the angle, like the part of gravity along a ramp, $mg\sin\theta$ .

Watch out: Using sine when the component you want is next to the angle. Check which side is opposite θ in your sketch.

Learn it: 1.1 Scalars and Vectors · 2.2 Forces and Free-Body Diagrams

Cosine in a right triangle

\displaystyle \cos\theta = \frac{b}{c}

What the symbols mean

$\theta$: the angle; Unit: degrees or rad
$b$: side next to the angle; Unit: m, or the unit of the vector
$c$: hypotenuse; Unit: m, or the unit of the vector

Use it when: Use it to find the component of a vector next to the angle, like the horizontal part of a launch velocity, $v_0\cos\theta$ .

Watch out: Assuming the horizontal part always uses cosine. It does only when θ is measured from the horizontal.

Learn it: 1.1 Scalars and Vectors · 1.5 Motion in Two or Three Dimensions

Tangent in a right triangle

\displaystyle \tan\theta = \frac{a}{b}

What the symbols mean

$\theta$: the angle; Unit: degrees or rad
$a$: side opposite the angle; Unit: m, or the unit of the vector
$b$: side next to the angle; Unit: m, or the unit of the vector

Use it when: Use it to find a vector's direction from its components: $\theta = \tan^{-1}(a/b)$ .

Watch out: Trusting the calculator's inverse tangent blindly. It can't tell a vector from one pointing the opposite way, so check the quadrant from the signs of the components.

Learn it: 1.1 Scalars and Vectors

Vectors

Products and sums of vectors. The dot product shows up in work and power, the cross product in torque and angular momentum.

Dot product

\displaystyle \vec{A}\cdot\vec{B} = AB\cos\theta

What the symbols mean

$\vec{A}, \vec{B}$: any two vectors; Unit: the units of each vector
$A, B$: their sizes; Unit: the units of each vector
$\theta$: angle between them, tail to tail; Unit: degrees or rad

Use it when: Use it for work, $W = \vec{F}\cdot\Delta\vec{r}$ , and power, $P = \vec{F}\cdot\vec{v}$ . With components, it's also $A_xB_x + A_yB_y + A_zB_z$ . The answer is a scalar.

Watch out: Measuring θ from the wrong line, like from a ramp instead of from the other vector. θ is the angle between the two vectors.

Try it: A constant force $\vec{F} = (3\hat{\imath} + 4\hat{\jmath})$ N moves an object through $\Delta\vec{r} = (2\hat{\imath} + 1\hat{\jmath})$ m. How much work does it do?

Answer: $W = \vec{F}\cdot\Delta\vec{r} = (3)(2) + (4)(1) = 10$ J.

Learn it: 3.2 Work · 3.5 Power · 1.1 Scalars and Vectors

Size of the cross product

\displaystyle \lvert\vec{A}\times\vec{B}\rvert = AB\sin\theta

What the symbols mean

$\vec{A}, \vec{B}$: any two vectors; Unit: the units of each vector
$A, B$: their sizes; Unit: the units of each vector
$\theta$: angle between them, tail to tail; Unit: degrees or rad

Use it when: Use it for the size of a torque, $\lvert\vec{r}\times\vec{F}\rvert = rF\sin\theta$ , or an angular momentum, $rp\sin\theta$ . The direction is perpendicular to both vectors, from the right-hand rule.

Watch out: Forgetting that order matters. $\vec{B}\times\vec{A}$ points the opposite way from $\vec{A}\times\vec{B}$ , which flips the sign of a torque.

Learn it: 5.3 Torque · 6.3 Angular Momentum and Angular Impulse

A vector in unit-vector form

\displaystyle \vec{r} = (A\hat{\imath} + B\hat{\jmath} + C\hat{k})

What the symbols mean

$\vec{r}$: a vector, here a position; Unit: m
$A, B, C$: its x, y and z components; Unit: m
$\hat{\imath}, \hat{\jmath}, \hat{k}$: unit vectors along x, y and z; Unit: none

Use it when: Use it to write a vector by its components. Many Physics C problems give position, velocity or force this way, and you differentiate or integrate each component separately.

Watch out: Treating the components as the size. The size is $\sqrt{A^2 + B^2 + C^2}$ , not A + B + C.

Learn it: 1.1 Scalars and Vectors · 1.5 Motion in Two or Three Dimensions

Adding vectors

\displaystyle \vec{C} = \vec{A} + \vec{B}

What the symbols mean

$\vec{C}$: the resultant, the vector sum; Unit: the shared unit
$\vec{A}, \vec{B}$: the vectors being added; Unit: the shared unit

Use it when: Use it to add forces, velocities or displacements, like a boat's velocity relative to the water plus the water's velocity relative to the ground.

Watch out: Adding the sizes. Vectors that point in different directions don't add like plain numbers; use components.

Learn it: 1.1 Scalars and Vectors · 1.4 Reference Frames and Relative Motion

Adding vectors by components

\displaystyle \vec{C} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath}

What the symbols mean

$\vec{C}$: the sum of the two vectors; Unit: the shared unit
$A_x, A_y$: the x and y components of the first vector; Unit: the shared unit
$B_x, B_y$: the x and y components of the second vector; Unit: the shared unit
$\hat{\imath}, \hat{\jmath}$: unit vectors along x and y; Unit: none

Use it when: Use it to add any vectors: split each into components, add the x parts and the y parts, then rebuild the size and direction if needed.

Watch out: Dropping the signs of components. A force pointing left has a negative x component.

Learn it: 1.1 Scalars and Vectors · 1.5 Motion in Two or Three Dimensions · 2.2 Forces and Free-Body Diagrams

Calculus

The derivatives and integrals you need for Physics C. Position, velocity and acceleration are linked by derivatives; work, impulse and center of mass by integrals.

Chain rule

\displaystyle \frac{df}{dx} = \frac{df}{du}\frac{du}{dx}

What the symbols mean

$f$: a function of u; Unit: depends on the quantity
$u$: an inner function of x; Unit: depends on the quantity
$x$: the variable you differentiate with respect to; Unit: depends on the quantity

Use it when: Use it whenever there's a function inside a function, like $\cos(\omega t)$ in SHM or $e^{-kt/m}$ for drag. It also gives the trick $a = \frac{dv}{dt} = v\frac{dv}{dx}$ .

Watch out: Forgetting to multiply by the inner derivative. The derivative of $\sin(5t)$ is $5\cos(5t)$ , not $\cos(5t)$ .

Learn it: 7.3 Representing and Analyzing SHM · 3.2 Work

Derivative of a power

\displaystyle \frac{d}{dx}(x^n) = nx^{n-1}

What the symbols mean

$x$: the variable; Unit: depends on the quantity
$n$: a constant exponent; Unit: none

Use it when: Use it to get velocity from a position like $x = 2t^3$ , acceleration from velocity, or force from a potential energy like $U = cx^4$ .

Watch out: Forgetting to lower the exponent by one. The derivative of $t^3$ is $3t^2$ .

Learn it: 1.2 Displacement, Velocity, and Acceleration · 3.3 Potential Energy

Derivative of an exponential

\displaystyle \frac{d}{dx}(e^{ax}) = ae^{ax}

What the symbols mean

$a$: a constant; Unit: 1 over the unit of x
$x$: the variable; Unit: depends on the quantity

Use it when: Use it for drag problems, where speed approaches terminal velocity like $1 - e^{-kt/m}$ , to get the acceleration or check a solution.

Watch out: Dropping the constant a that comes down, including its minus sign when the exponent is negative.

Learn it: 2.9 Resistive Forces

Derivative of a natural log

\displaystyle \frac{d}{dx}(\ln ax) = \frac{1}{x}

What the symbols mean

$a$: a positive constant; Unit: 1 over the unit of x
$x$: the variable; Unit: depends on the quantity

Use it when: Use it to check a solution that involves a natural log, like the time it takes an object with drag to reach a given speed.

Watch out: Expecting the a to show up in the answer. It doesn't, because $\ln ax = \ln a + \ln x$ and $\ln a$ is a constant.

Learn it: 2.9 Resistive Forces

Derivative of a sine

\displaystyle \frac{d}{dx}[\sin(ax)] = a\cos(ax)

What the symbols mean

$a$: a constant, like an angular frequency; Unit: 1 over the unit of x
$x$: the variable, often time; Unit: depends on the quantity

Use it when: Use it to get velocity or acceleration in simple harmonic motion when the position is written with sine.

Watch out: Forgetting the factor a out front. In SHM it's ω, so the top speed is $\omega x_{\text{max}}$ , not $x_{\text{max}}$ .

Learn it: 7.3 Representing and Analyzing SHM

Derivative of a cosine

\displaystyle \frac{d}{dx}[\cos(ax)] = -a\sin(ax)

What the symbols mean

$a$: a constant, like an angular frequency; Unit: 1 over the unit of x
$x$: the variable, often time; Unit: depends on the quantity

Use it when: Use it to get velocity from $x = x_{\text{max}}\cos(\omega t + \phi)$ , and again to get acceleration, which comes out as $-\omega^2x$ .

Watch out: Dropping the minus sign. The derivative of cosine is negative sine.

Try it: An oscillator's position is $x = 0.20\cos(5.0t)$ , in meters with t in seconds. What is its top speed?

Answer: $v = \frac{dx}{dt} = -(0.20)(5.0)\sin(5.0t) = -1.0\sin(5.0t)$ , so the top speed is 1.0 m/s.

Learn it: 7.3 Representing and Analyzing SHM · 7.1 Defining Simple Harmonic Motion (SHM)

Integral of a power

\displaystyle \int x^n\,dx = \frac{1}{n+1}x^{n+1}, n \ne -1

What the symbols mean

$x$: the variable; Unit: depends on the quantity
$n$: a constant exponent, anything except −1; Unit: none

Use it when: Use it for work with a force like $F = cx^2$ , impulse from $F = ct$ , the center of mass of a rod with $\lambda = cx$ , and $I = \int r^2\,dm$ .

Watch out: Using it for $n = -1$ , like $\int dx/x$ . That one is a natural log.

Learn it: 1.3 Representing Motion · 3.2 Work · 5.4 Rotational Inertia

Integral of an exponential

\displaystyle \int e^{ax}\,dx = \frac{1}{a}e^{ax}

What the symbols mean

$a$: a nonzero constant; Unit: 1 over the unit of x
$x$: the variable; Unit: depends on the quantity

Use it when: Use it to get position from a velocity that changes exponentially, like a boat coasting to a stop against drag.

Watch out: Multiplying by a instead of dividing. Differentiate your answer to check it.

Learn it: 2.9 Resistive Forces

Integral of one over x plus a

\displaystyle \int \frac{dx}{x+a} = \ln\lvert x + a\rvert

What the symbols mean

$x$: the variable; Unit: depends on the quantity
$a$: a constant; Unit: same unit as x

Use it when: Use it when you separate variables in a drag problem, like $\int \frac{dv}{v - v_T}$ , which leads to the exponential approach to terminal velocity.

Watch out: Using log base 10 instead of ln. On a calculator, ln is the natural log.

Learn it: 2.9 Resistive Forces

Integral of a cosine

\displaystyle \int \cos(ax)\,dx = \frac{1}{a}\sin(ax)

What the symbols mean

$a$: a nonzero constant; Unit: 1 over the unit of x
$x$: the variable, often time; Unit: depends on the quantity

Use it when: Use it to get position from a velocity that varies like a cosine, as in simple harmonic motion, or impulse from a force that oscillates.

Watch out: Forgetting to divide by a.

Learn it: 7.3 Representing and Analyzing SHM

Integral of a sine

\displaystyle \int \sin(ax)\,dx = -\frac{1}{a}\cos(ax)

What the symbols mean

$a$: a nonzero constant; Unit: 1 over the unit of x
$x$: the variable, often time; Unit: depends on the quantity

Use it when: Use it to get position from a velocity that varies like a sine, as in simple harmonic motion.

Watch out: Dropping the minus sign, or the 1/a.

Learn it: 7.3 Representing and Analyzing SHM

Identities

Rules for logs and trig that help you simplify answers.

Log of a product and a power

\displaystyle \log(a\cdot b^x) = \log a + x\log b

What the symbols mean

$a, b$: positive numbers; Unit: none
$x$: an exponent; Unit: none

Use it when: Use it to bring a variable down from an exponent, like solving $e^{-kt/m} = 0.5$ for t, or to turn an exponential relationship into a straight-line graph. It works the same for ln.

Watch out: Writing $\log(a + b)$ as $\log a + \log b$ . The rule is for a product, not a sum.

Learn it: 2.9 Resistive Forces

Pythagorean identity

\displaystyle \sin^2\theta + \cos^2\theta = 1

What the symbols mean

$\theta$: any angle; Unit: degrees or rad

Use it when: Use it to rebuild a vector's size from its components, or to show that the total energy of an oscillator, $\tfrac{1}{2}kx_{\text{max}}^2\cos^2(\omega t) + \tfrac{1}{2}kx_{\text{max}}^2\sin^2(\omega t) = \tfrac{1}{2}kx_{\text{max}}^2$ , stays constant.

Watch out: Reading $\sin^2\theta$ as $\sin(\theta^2)$ . It means $(\sin\theta)^2$ .

Learn it: 1.1 Scalars and Vectors · 7.4 Energy of Simple Harmonic Oscillators

Double-angle identity

\displaystyle \sin(2\theta) = 2\sin\theta\cos\theta

What the symbols mean

$\theta$: any angle, like a launch angle; Unit: degrees or rad

Use it when: Use it to simplify projectile results, like the range on level ground, $R = \frac{v_0^2\sin(2\theta)}{g}$ , which is largest at 45°.

Watch out: Writing $\sin(2\theta)$ as $2\sin\theta$ . Doubling the angle doesn't double the sine.

Learn it: 1.5 Motion in Two or Three Dimensions

Tangent as sine over cosine

\displaystyle \frac{\sin\theta}{\cos\theta} = \tan\theta

What the symbols mean

$\theta$: any angle; Unit: degrees or rad

Use it when: Use it when a ratio of sine and cosine shows up, like a block on the verge of sliding down a ramp, where $mg\sin\theta = \mu_smg\cos\theta$ gives $\mu_s = \tan\theta$ .

Watch out: Flipping the ratio. Tangent is sine over cosine, opposite over adjacent.

Learn it: 1.1 Scalars and Vectors · 2.7 Kinetic and Static Friction

Not on the sheet: know these

The exam expects you to know these without being given them.

Resistive force and terminal velocity
$\displaystyle \vec{F}_r = -k\vec{v}, \quad v_T = \frac{mg}{k}, \quad v(t) = \frac{mg}{k}\left(1 - e^{-kt/m}\right)$
The course models drag as proportional to velocity. Set the net force to zero for terminal velocity, and separate variables in Newton's second law for v(t) when an object falls from rest. Some books write the constant as b instead of k.
Learn it: 2.9 Resistive Forces
Orbital speed and escape speed
$\displaystyle v_{\text{orbit}} = \sqrt{\frac{GM}{r}}, \quad v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$
Neither is printed. Get orbital speed by setting gravity equal to $mv^2/r$ , and escape speed by setting total energy $\tfrac{1}{2}mv^2 - GMm/R$ to zero.
Learn it: 2.10 Circular Motion · 6.6 Motion of Orbiting Satellites
Rolling without slipping in rate form
$\displaystyle v_{\text{cm}} = R\omega, \quad a_{\text{cm}} = R\alpha$
The sheet only gives $\Delta x_{\text{cm}} = r\Delta\theta$ . Its time derivatives link the speed and acceleration of a rolling object's center to its spin, which you need for rolling down ramps.
Learn it: 6.5 Rolling
Energy with nonconservative forces
$\displaystyle \Delta K + \Delta U + \Delta E_{\text{th}} = W_{\text{ext}}$
Mechanical energy is constant only with no friction and no outside work. Kinetic friction sliding over a distance d turns $F_fd$ of mechanical energy into thermal energy.
Learn it: 3.4 Conservation of Energy
Rotational inertia of common shapes
$\displaystyle I_{\text{hoop}} = MR^2, \quad I_{\text{disk}} = \tfrac{1}{2}MR^2, \quad I_{\text{rod, center}} = \tfrac{1}{12}ML^2, \quad I_{\text{rod, end}} = \tfrac{1}{3}ML^2$
The sheet gives $I = \int r^2\,dm$ but no results. You're expected to derive these for rods, hoops, cylindrical shells and disks, and knowing them saves time and lets you check your integral.
Learn it: 5.4 Rotational Inertia
The condition for simple harmonic motion
$\displaystyle a_x = -\omega^2x, \quad \omega = \sqrt{\frac{k}{m}}$
SHM happens when the acceleration is proportional to the displacement and opposite to it. Writing Newton's second law in this form lets you read off ω, and then the period, for any oscillator.
Learn it: 7.1 Defining Simple Harmonic Motion (SHM) · 7.2 Frequency and Period of SHM · 7.3 Representing and Analyzing SHM
Springs in parallel and in series
$\displaystyle k_{\text{eq}} = k_1 + k_2, \quad \frac{1}{k_{\text{eq}}} = \frac{1}{k_1} + \frac{1}{k_2}$
Side-by-side springs share the same stretch and act stiffer. End-to-end springs share the same force and act softer.
Learn it: 2.8 Spring Forces
Conservation of linear and angular momentum
$\displaystyle \sum\vec{p}_i = \sum\vec{p}_f, \quad L_i = L_f$
The sheet gives $\vec{F}_{\text{net}} = d\vec{p}/dt$ and $\Delta L = \int \tau\,dt$ , but not the rule that follows: with no net external force, total momentum is constant, and with no net external torque, angular momentum is constant.
Learn it: 4.3 Conservation of Linear Momentum · 6.4 Conservation of Angular Momentum
Elastic and inelastic collisions
Momentum is conserved in every collision when no outside force acts, but kinetic energy is conserved only in an elastic one. In a perfectly inelastic collision the objects stick together and move with one final velocity.
Learn it: 4.4 Elastic and Inelastic Collisions
Relative velocity
$\displaystyle \vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$
The velocity of A relative to C is A's velocity relative to B plus B's velocity relative to C. You use it for boats in rivers, planes in wind and people walking on moving trains.
Learn it: 1.4 Reference Frames and Relative Motion