AP Physics 1
Algebra-based · effective fall 2024Constants and conversion factors
| Universal gravitational constant | G = 6.67 × 10−11 m3/(kg·s2) = 6.67 × 10−11 N·m2/kg2 | 3.7 Gravitation3.8 Orbits |
|---|---|---|
| 1 atmosphere of pressure | 1 atm = 1.0 × 105 N/m2 = 1.0 × 105 Pa | 8.2 Pressure and depth |
| Magnitude of the acceleration due to gravity at Earth’s surface | g = 9.8 m/s2 | 1.3 Acceleration3.7 Gravitation |
| Magnitude of the gravitational field strength at Earth’s surface | g = 9.8 N/kg | 3.7 Gravitation |
Prefixes
| Factor | Prefix | Symbol |
|---|---|---|
| 1012 | tera | T |
| 109 | giga | G |
| 106 | mega | M |
| 103 | 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
| θ | 0° | 30° | 37° | 45° | 53° | 60° | 90° |
|---|---|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 3/5 | √2/2 | 4/5 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 4/5 | √2/2 | 3/5 | 1/2 | 0 |
| tan θ | 0 | √3/3 | 3/4 | 1 | 4/3 | √3 | ∞ |
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 · ℓ = length · r = radius · s = arc length · S = surface area · V = volume · w = width · θ = angle
- A = bhRectangle
- A = ½bh
- A = πr2 C = 2πr s = rθ
- V = ℓwhRectangular solidCh. 8 Fluids
- V = πr2ℓ S = 2πrℓ + 2πr2
- V = 4⁄3πr3 S = 4πr2
- a2 + b2 = c2 sin θ = a/c cos θ = b/c tan θ = a/b
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 · θ = angle · μ = coefficient of friction
- vx = vx0 + axtVelocity at constant acceleration1.5 The kinematic equations1.3 Acceleration
- x = x0 + vx0t + ½axt2Position at constant acceleration1.5 The kinematic equations1.6 Projectile motion9.8 A uniform field
- vx2 = vx02 + 2ax(x − x0)Velocity without time1.5 The kinematic equations
- xcm = Σmixi / Σmi
- asys = ΣF / msys = Fnet / msysNewton’s second law2.4 The second law2.9 The centre of mass accelerates3.1 Centripetal force
- |Fg| = Gm1m2 / r2Universal gravitation3.7 Gravitation3.8 Orbits
- |Ff| ≤ |μFN|
- Fs = −kΔxHooke’s law: spring force4.5 Spring potential energy6.1 Hooke’s law6.4 A hanging spring
- ac = v2 / r
- K = ½mv2
- W = F∥d = Fd cos θWork by a constant forceCh. 4 Work4.2 Forces that do no work7.8 Work and power
- ΔK = ΣWi = ΣF∥,idiWork–energy theorem4.3 Kinetic energy4.1 Positive and negative work
- Us = ½k(Δx)2Spring potential energy4.5 Spring potential energy6.3 Energy6.11 Everyday oscillators
- UG = −Gm1m2 / rGravitational potential energy, any distance3.10 Energy on an orbit3.11 Potential energy3.12 Escape
- ΔUg = mgΔyGravitational potential energy near the surface4.4 Gravitational potential energy4.7 Conservation of energy3.11 Potential energy5.6 Ballistic pendulum
- Pavg = W / Δt = ΔE / ΔtAverage power4.11 Power
- Pinst = F∥v = Fv cos θInstantaneous power4.11 Power15.6 A bar on rails
- p = mvMomentumCh. 5 Momentum5.3 Conservation
- Fnet = Δp / Δt = mΔv / Δt = maNet force and change of momentum5.1 Impulse5.2 Less force
- J = FavgΔt = Δp
- vcm = Σpi / Σmi = Σmivi / ΣmiVelocity of the centre of mass5.3 Conservation5.8 Centre of mass2.8 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 · ℓ = 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 · α = angular acceleration · θ = angle or angular position · ρ = density · τ = torque · ω = angular speed
- ω = ω0 + αtAngular velocity at constant αCh. 7 Rotation
- θ = θ0 + ω0t + ½αt2Angle at constant αCh. 7 Rotation
- ω2 = ω02 + 2α(θ − θ0)Angular velocity without timeCh. 7 Rotation
- v = rωLinear and angular speed3.6 Angular velocity7.1 Two riders7.9 Rolling
- aT = rαTangential acceleration7.1 Two riders7.11 The well
- τ = r⊥F = rF sin θ
- I = Σmiri2Rotational inertia7.5 Moment of inertia7.6 Adding up I
- I′ = Icm + Md2Parallel-axis theorem7.7 Parallel axes6.9 Physical pendulums
- αsys = Στ / Isys = τnet / IsysNewton’s second law for rotation7.5 Moment of inertia7.7 Parallel axes7.10 Sliding, then rolling7.11 The well
- K = ½Iω2Rotational kinetic energy7.9 Rolling7.13 The spin7.8 Work and power
- W = τΔθWork done by a torque7.8 Work and power
- L = IωAngular momentum of a rigid body7.12 Angular impulse7.13 The spin
- L = rmv sin θAngular momentum of a particle7.14 L of a particle3.9 Elliptical orbits
- ΔL = τΔtAngular impulse7.12 Angular impulse
- Δxcm = rΔθRolling without slipping7.9 Rolling7.10 Sliding, then rolling
- T = 1 / fPeriod and frequencyCh. 6 Oscillations
- Ts = 2π√(m / k)Period of a mass on a spring6.6 Double the period6.7 What it ignores6.11 Everyday oscillators
- Tp = 2π√(ℓ / g)Period of a simple pendulum6.5 The pendulum6.6 Double the period6.11 Everyday oscillators
- x = A cos(2πft)Position in simple harmonic motion, from x = A6.2 x, v and a
- x = A sin(2πft)Position in simple harmonic motion, from x = 0not shown yet
- ρ = m / V
- P = F⊥ / A
- P = P0 + ρghAbsolute pressure at depth8.2 Pressure and depth8.4 Buoyancy
- Pgauge = ρghGauge pressure8.2 Pressure and depth
- Fb = ρVgBuoyant force8.4 Buoyancy8.5 Float or sink2.12 Real fluids
- A1v1 = A2v2Continuity8.6 Continuity
- P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv22Bernoulli’s equation8.7 Bernoulli8.8 Torricelli’s law
AP Physics C: Mechanics
Calculus-based · effective fall 2024Constants and conversion factors
| Universal gravitational constant | G = 6.67 × 10−11 m3/(kg·s2) = 6.67 × 10−11 N·m2/kg2 | 3.7 Gravitation3.8 Orbits |
|---|---|---|
| Magnitude of the acceleration due to gravity at Earth’s surface | g = 9.8 m/s2 | 1.3 Acceleration3.7 Gravitation |
| Magnitude of the gravitational field strength at Earth’s surface | g = 9.8 N/kg | 3.7 Gravitation |
Prefixes
| Factor | Prefix | Symbol |
|---|---|---|
| 1012 | tera | T |
| 109 | giga | G |
| 106 | mega | M |
| 103 | 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
| θ | 0° | 30° | 37° | 45° | 53° | 60° | 90° |
|---|---|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 3/5 | √2/2 | 4/5 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 4/5 | √2/2 | 3/5 | 1/2 | 0 |
| tan θ | 0 | √3/3 | 3/4 | 1 | 4/3 | √3 | ∞ |
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 · ℓ = 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 · λ = linear mass density · μ = coefficient of friction
- vx = vx0 + axtVelocity at constant acceleration1.5 The kinematic equations
- x = x0 + vx0t + ½axt2Position at constant acceleration1.5 The kinematic equations1.6 Projectile motion
- vx2 = vx02 + 2ax(x − x0)Velocity without time1.5 The kinematic equations
- Δx = ∫vx(t) dtDisplacement as an integral1.9 Calculus kinematics
- Δvx = ∫ax(t) dtChange of velocity as an integral1.9 Calculus kinematics
- xcm = Σmixi / Σmi
- rcm = ∫r dm / ∫dmCentre of mass of a solid body5.9 Solid bodies5.10 Plates
- λ = (d/dℓ) m(ℓ)Linear mass density5.9 Solid bodies7.6 Adding up I
- asys = ΣF / msys = Fnet / msys
- |Fg| = Gm1m2 / r2Universal gravitation3.7 Gravitation3.8 Orbits
- |Ff| ≤ |μFN|
- Fs = −kΔxHooke’s law: spring force4.5 Spring potential energy6.1 Hooke’s law
- ac = v2 / r = rω2Centripetal accelerationCh. 3 Circular motion3.1 Centripetal force3.4 Vertical circles3.6 Angular velocity7.1 Two riders
- T = 1 / fPeriod and frequencyCh. 6 Oscillations
- K = ½mv2Kinetic energy4.3 Kinetic energy
- W = ∫ab F · dr
- ΔK = ΣWi = ΣF∥,idiWork–energy theorem4.3 Kinetic energy4.6 Work by a changing force
- ΔU = −∫ab Fcf(r) · drPotential energy from a conservative force3.11 Potential energy4.8 Force from potential energy
- Fx = −dU(x) / dxForce from potential energy4.8 Force from potential energy4.9 Energy diagrams3.11 Potential energy
- Us = ½k(Δx)2Spring potential energy4.5 Spring potential energy6.3 Energy
- UG = −Gm1m2 / rGravitational potential energy3.10 Energy on an orbit3.11 Potential energy3.12 Escape
- ΔUg = mgΔyGravitational potential energy near the surface4.4 Gravitational potential energy3.11 Potential energy
- Pavg = W / Δt = ΔE / ΔtAverage power4.11 Power
- Pinst = dW / dtInstantaneous power4.11 Power15.6 A bar on rails
- p = mvMomentumCh. 5 Momentum
- Fnet = dp / dtNet force is the rate of change of momentum5.1 Impulse
- J = ∫t1t2 Fnet(t) dt = ΔpImpulse5.1 Impulse5.2 Less force
- vcm = Σpi / Σmi = Σmivi / ΣmiVelocity of the centre of mass5.3 Conservation5.8 Centre of mass
Mechanics · right column
a = acceleration · d = distance · f = frequency · F = force · I = rotational inertia · k = spring constant · K = kinetic energy · ℓ = 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 · α = angular acceleration · θ = angular position · τ = torque · φ = phase angle · ω = angular frequency or angular speed
- ω = dθ / dtAngular velocityCh. 7 Rotation3.6 Angular velocity
- α = dω / dtAngular accelerationCh. 7 Rotation7.10 Sliding, then rolling
- ω = ω0 + αtAngular velocity at constant αCh. 7 Rotation
- θ = θ0 + ω0t + ½αt2Angle at constant αCh. 7 Rotation
- ω2 = ω02 + 2α(θ − θ0)Angular velocity without timeCh. 7 Rotation
- v = rωLinear and angular speed3.6 Angular velocity7.1 Two riders
- aT = rαTangential acceleration7.1 Two riders7.11 The well
- τ = r × F
- Itot = ΣIi = Σmiri2Rotational inertia7.5 Moment of inertia7.6 Adding up I
- I = ∫r2 dmRotational inertia of a solid body7.6 Adding up I
- I′ = Icm + Md2Parallel-axis theorem7.7 Parallel axes6.9 Physical pendulums
- αsys = Στ / Isys = τnet / IsysNewton’s second law for rotation7.5 Moment of inertia7.10 Sliding, then rolling7.11 The well6.10 The torsion pendulum
- Krot = ½Iω2Rotational kinetic energy7.9 Rolling7.13 The spin
- W = ∫τ · dθWork done by a torque7.8 Work and power
- L = r × p = Iω
- ΔL = ∫τ dtAngular impulse7.12 Angular impulse
- Δxcm = rΔθRolling without slipping7.9 Rolling7.10 Sliding, then rolling
- T = 2π / ω = 1 / fPeriod and frequencyCh. 6 Oscillations6.8 The SHM equation
- Ts = 2π√(m / k)Period of a mass on a spring6.6 Double the period6.4 A hanging spring
- Tp = 2π√(ℓ / g)Period of a simple pendulum6.5 The pendulum6.6 Double the period
- Tphys = 2π√(I / mgd)Period of a physical pendulum6.9 Physical pendulums
- x = xmax cos(ωt + φ)Position in simple harmonic motion6.8 The SHM equation6.2 x, v and a
Geometry and trigonometry
Also on this sheet, identical to the Physics 1 sheet.
Vectors
The sheet’s math reference.
- A · B = AB cos θDot productCh. 4 WorkCh. 10 Electric flux
- |A × B| = AB sin θCross product7.2 Torque14.1 The magnetic force
- r = (Aî + Bĵ + Ck̂)Vector from components
- C = A + BVector sum1.7 Reference frames1.8 Crossing a river
- C = (Ax + Bx)î + (Ay + By)ĵVector sum by components
Calculus
The sheet’s math reference.
- df/dx = (df/du)(du/dx)Chain rule
- d/dx (xn) = nxn−1Derivative of a power1.10 Motion from x(t)
- d/dx (eax) = aeaxDerivative of an exponential
- d/dx (ln ax) = 1/xDerivative of a logarithm
- d/dx [sin(ax)] = a cos(ax)Derivative of sine6.2 x, v and a6.8 The SHM equation
- d/dx [cos(ax)] = −a sin(ax)Derivative of cosine6.2 x, v and a6.8 The SHM equation
- ∫xn dx = (1/(n + 1)) xn+1, n ≠ −1Integral of a power1.9 Calculus kinematics5.9 Solid bodies7.6 Adding up I
- ∫eax dx = (1/a) eaxIntegral of an exponential2.12 Real fluids
- ∫dx / (x + a) = ln|x + a|Integral giving a logarithm2.11 Drag proportional to v12.3 Two cylinders13.8 RC circuits
- ∫cos(ax) dx = (1/a) sin(ax)Integral of cosine
- ∫sin(ax) dx = −(1/a) cos(ax)Integral of sine5.1 Impulse
Identities
The sheet’s math reference.
- log(a · bx) = log a + x log bLogarithm of a product and a power
- sin2 θ + cos2 θ = 1Pythagorean identity
- sin(2θ) = 2 sin θ cos θDouble angle
- sin θ / cos θ = tan θTangent
AP Physics C: Electricity and Magnetism
Calculus-based · effective fall 2024Constants and conversion factors
| Coulomb constant | k = 1/4πε0 = 9.0 × 109 (N·m2)/C2 | 9.2 Coulomb’s law |
|---|---|---|
| Vacuum permittivity | ε0 = 8.85 × 10−12 C2/(N·m2) | 10.1 Gauss’s law |
| Vacuum permeability | μ0 = 4π × 10−7 (T·m)/A | 14.5 Ampère’s law |
| Proton mass | mp = 1.67 × 10−27 kg | Ch. 9 Electric charge14.2 Circling in a field |
| Neutron mass | mn = 1.67 × 10−27 kg | Ch. 9 Electric charge |
| Electron mass | me = 9.11 × 10−31 kg | Ch. 9 Electric charge9.8 A uniform field |
| Elementary charge | e = 1.60 × 10−19 C | Ch. 9 Electric charge |
| 1 electron volt | 1 eV = 1.60 × 10−19 J | 11.8 The electron volt |
| Speed of light | c = 3.00 × 108 m/s | not shown yet |
| 1 unified atomic mass unit | 1 u = 1.66 × 10−27 kg = 931 MeV/c2 | not shown yet |
| Universal gravitational constant | G = 6.67 × 10−11 m3/(kg·s2) = 6.67 × 10−11 N·m2/kg2 | 3.7 Gravitation3.8 Orbits |
| Magnitude of the acceleration due to gravity at Earth’s surface | g = 9.8 m/s2 | 1.3 Acceleration3.7 Gravitation |
| Magnitude of the gravitational field strength at Earth’s surface | g = 9.8 N/kg | 3.7 Gravitation |
Prefixes
| Factor | Prefix | Symbol |
|---|---|---|
| 1012 | tera | T |
| 109 | giga | G |
| 106 | mega | M |
| 103 | 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
| θ | 0° | 30° | 37° | 45° | 53° | 60° | 90° |
|---|---|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 3/5 | √2/2 | 4/5 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 4/5 | √2/2 | 3/5 | 1/2 | 0 |
| tan θ | 0 | √3/3 | 3/4 | 1 | 4/3 | √3 | ∞ |
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 (κ = 1.0).
Electricity and magnetism · left column
A = area · C = capacitance · d = distance · E = electric field · F = force · I = current · J = current density · ℓ = length · P = power · q = charge · Q = charge · r = radius, distance, or position · R = resistance · t = time · U = potential energy · V = electric potential or volume · ε = electric permittivity · ρ = resistivity or charge density · κ = dielectric constant · Φ = flux
- |FE| = (1/4πε0) |q1q2| / r2 = k|q1q2| / r2Coulomb’s law9.2 Coulomb’s law9.3 Net force
- E = FE / qElectric field9.4 The electric field9.8 A uniform field
- E = (1/4πε0) ∫ (dq / r2) r̂Field of a continuous charge9.6 A charged rod9.7 A ring of charge
- ΦE = ∫E · dAElectric fluxCh. 10 Electric flux
- ∮E · dA = qenc / ε0
- Qtotal = ∫ρ(r) dVTotal charge from a charge density10.7 Uneven charge
- UE = (1/4πε0) q1q2 / rElectric potential energy of two chargesCh. 11 Electric potential energy11.1 Assembling charges
- V = (1/4πε0) ∫ dq / rPotential of a continuous charge11.5 A charged rod11.6 Ring, arc and disc
- ΔV = −∫ab E · drPotential difference from the field11.2 Electric potential11.3 The potential hill11.7 Charged spheres12.1 Parallel plates12.2 Two spheres12.3 Two cylinders
- Ex = −dV / dxField from potential11.3 The potential hill
- ΔUE = qΔVPotential energy from potential11.2 Electric potential11.8 The electron volt14.2 Circling in a field
- C = Q / ΔV
- C = κε0A / dParallel-plate capacitor12.1 Parallel plates12.6 Dielectrics12.8 Conductors
- UC = ½QΔVEnergy stored in a capacitor12.4 Stored energy12.7 Connected or not
- κ = ε / ε0Dielectric constant12.6 Dielectrics
- I = dq / dt
- I = ∫J · dACurrent from current densityCh. 13 Current
- E = ρJOhm’s law in a materialCh. 13 Current13.2 Ohm’s law
- R = ρℓ / AResistance of a wire13.1 Resistance
- I = ΔV / R
- P = IΔVElectric power13.3 Power13.4 Emf
Electricity and magnetism · right column
A = area · B = magnetic field · C = capacitance · E = electric field · F = force · I = current · ℓ = 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 · ε = emf · μ = magnetic permeability · τ = time constant · Φ = flux · ω = angular frequency
- Req,s = Σi RiResistors in series13.5 Series and parallel13.6 Combinations
- 1 / Req,p = Σi 1 / RiResistors in parallel13.5 Series and parallel13.6 Combinations
- 1 / Ceq,s = Σi 1 / CiCapacitors in series12.5 Series and parallel12.8 Conductors
- Ceq,p = Σi CiCapacitors in parallel12.5 Series and parallel12.7 Connected or not
- τ = ReqCeqRC time constant13.8 RC circuits
- ∮B · dA = 0No magnetic monopolesCh. 14 Magnetic fields
- FB = q(v × B)Magnetic force on a moving charge14.1 The magnetic force14.2 Circling in a field14.3 Crossed E and B15.4 Motional emf
- dB = (μ0/4π) I(dℓ × r̂) / r2Biot–Savart law14.6 Biot–Savart
- FB = ∫I(dℓ × B)Magnetic force on a current14.4 Force on a current14.7 Parallel wires15.6 A bar on rails
- ∮B · dℓ = μ0IencAmpère’s law14.5 Ampère’s law14.8 Solenoids
- Bsol = μ0nIField inside a solenoid14.8 Solenoids15.7 Inductance
- ΦB = ∫B · dA
- ε = ∮E · dℓ = −dΦB / dt
- |εsol| = N|dΦB / dt|Emf of a coil of N loopsCh. 15 Induction
- Lsol = μcoreN2A / ℓInductance of a solenoid15.7 Inductance
- UL = ½LI2Energy stored in an inductor15.7 Inductance15.9 LC oscillations
- ε = −L dI / dtSelf-induced emf15.7 Inductance15.8 RL circuits
- τ = L / ReqRL time constant15.8 RL circuits
- ωLC = 1 / √(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.