1. Kinematics, Dynamics
1.1. Units
● SI units: meters (m), kilograms (kg), seconds (s), etc.
1.2. Vectors, Scalars
● Vector addition: add tip to tail
● Scalar vector multiplication
● Dot product: a • b = ||a|| ||b|| cos θ = axbx + ayby + …
● Cross product: a × b = ||a|| ||b|| sin θ n̂ = det [[î, ĵ, k̂], [ax, ay, az], [bx, by, bz]] for 3D
○ Right-hand rule
1.3. Displacement, Velocity
● Distance (x, scalar), displacement (x, vector)
● Speed (v, scalar), velocity (v, vector)
○ vavg = Δx/Δt
1.4. Forces, Acceleration
● Force (F): push/pull, in newtons (N = kg·m/s2)
○ Gravity: attractive force felt, exerted by all objects
■ F = GmM/r2, where G = universal gravitational constant
○ Friction: opposes motion
■ Static friction (fs): stationary object, depends on force being applied
● 0 ≤ fs ≤ μsN, where μ = coefficient of friction, N = normal force
● Object moves when force exceeds max static friction: fs, max = μsN
● Rolling w/o slipping: rolling object experiences instantaneous
static, not kinetic, friction
■ Kinetic friction (fk): sliding object
● fk = μkN
● Usually μs ≥ μk
● Mass (m, object’s inertia, scalar), weight (w, gravity on object, vector)
○ w = mg, where g = acceleration due to gravity
■ gEarth = GMEarth/rEarth2
○ Weight is applied to center of gravity: xCM = Σ mixi / Σ mi, etc.
● Acceleration (a): rate of change of v
○ aavg = Δv/Δt
1.5. Newton’s Laws of Motion
● 1st Law: inertia, Σ F = 0 if a = 0
● 2nd Law: Σ F = ma
● 3rd Law: action and reaction, FA→B = –FB→A
1.6. Motion w/ Constant Acceleration
● Linear motion
○ v = v0 + at
○ x = x0 + vt + ½ at2
○ v2 = v02 + 2a Δx
○ Δx = ½ (v + v0)t
● Free fall: object falls w/ constant acceleration (g), no air resistance
○ Terminal velocity: when gravitational force = drag force
● Projectile motion: free fall w/ ax = 0, constant ay = g
, ● Inclined plane: split forces into components perpendicular, parallel to the plane
○ Fg, ⊥ = mg cos θ = –N, where θ = angle of incline
○ Fg, || = mg sin θ
● Circular motion
○ Uniform circular motion: v is always tangent to the path, centripetal force always
points inward
○ Centripetal acceleration: ac = v2/r
■ Centripetal force: Fc = mac = mv2/r
1.7. Mechanical Equilibrium
● Equilibrium: Σ F = 0, Σ τ = 0
● Torque (τ): in N·m (= kg·m2/s2)
○ τ = r × F, where r = length from fulcrum to point of force
○ CCW τ > 0, CW τ < 0
2. Work, Energy
2.1. Energy
● Kinetic energy (K): energy of motion, in joules (J = N·m = kg·m2/s2)
○ K = ½ mv2
● Potential energy (U): energy of position/etc.
○ Gravitational: Ug = mgh, where h = height
■ Generally, Ug = –GmM/r
○ Elastic: Uel = ½ kx2, where k = spring constant, x = extension of spring
● Mechanical energy (E): total energy, conserved (1st Law of Thermodynamics)
○ ΔE = ΔK + ΔU
○ Energy can still be “lost” as heat, etc.
● Conservation
○ Conservative forces: path-independent, conserves E (e.g., gravitational,
electrostatic)
■ ΔE = 0
○ Nonconservative forces: path-dependent, dissipates E as thermal/chemical
energy (e.g., friction, etc.)
■ ΔE = WNC
2.2. Work
● Work (W): transfer of mechanical energy, in J
○ W = F • d = Fdcosθ
■ Only forces anti/parallel to displacement do work
○ Wnet = Kf – Ki
■ Work-energy theorem
○ Gas compression: (+) work done on gas, (–) work done by gas
○ Gas expansion: (–) work done on gas, (+) work done by gas
● Pressure (P), volume (V)
○ P–V graphs: pressure vs. volume
■ Work = integral of curve
○ W = P ΔV when process is isobaric (ΔP = 0)
● Power (Ƥ): rate of work, in watts (W = J/s)
○ Ƥavg = ΔW/Δt = ΔE/Δt
1.1. Units
● SI units: meters (m), kilograms (kg), seconds (s), etc.
1.2. Vectors, Scalars
● Vector addition: add tip to tail
● Scalar vector multiplication
● Dot product: a • b = ||a|| ||b|| cos θ = axbx + ayby + …
● Cross product: a × b = ||a|| ||b|| sin θ n̂ = det [[î, ĵ, k̂], [ax, ay, az], [bx, by, bz]] for 3D
○ Right-hand rule
1.3. Displacement, Velocity
● Distance (x, scalar), displacement (x, vector)
● Speed (v, scalar), velocity (v, vector)
○ vavg = Δx/Δt
1.4. Forces, Acceleration
● Force (F): push/pull, in newtons (N = kg·m/s2)
○ Gravity: attractive force felt, exerted by all objects
■ F = GmM/r2, where G = universal gravitational constant
○ Friction: opposes motion
■ Static friction (fs): stationary object, depends on force being applied
● 0 ≤ fs ≤ μsN, where μ = coefficient of friction, N = normal force
● Object moves when force exceeds max static friction: fs, max = μsN
● Rolling w/o slipping: rolling object experiences instantaneous
static, not kinetic, friction
■ Kinetic friction (fk): sliding object
● fk = μkN
● Usually μs ≥ μk
● Mass (m, object’s inertia, scalar), weight (w, gravity on object, vector)
○ w = mg, where g = acceleration due to gravity
■ gEarth = GMEarth/rEarth2
○ Weight is applied to center of gravity: xCM = Σ mixi / Σ mi, etc.
● Acceleration (a): rate of change of v
○ aavg = Δv/Δt
1.5. Newton’s Laws of Motion
● 1st Law: inertia, Σ F = 0 if a = 0
● 2nd Law: Σ F = ma
● 3rd Law: action and reaction, FA→B = –FB→A
1.6. Motion w/ Constant Acceleration
● Linear motion
○ v = v0 + at
○ x = x0 + vt + ½ at2
○ v2 = v02 + 2a Δx
○ Δx = ½ (v + v0)t
● Free fall: object falls w/ constant acceleration (g), no air resistance
○ Terminal velocity: when gravitational force = drag force
● Projectile motion: free fall w/ ax = 0, constant ay = g
, ● Inclined plane: split forces into components perpendicular, parallel to the plane
○ Fg, ⊥ = mg cos θ = –N, where θ = angle of incline
○ Fg, || = mg sin θ
● Circular motion
○ Uniform circular motion: v is always tangent to the path, centripetal force always
points inward
○ Centripetal acceleration: ac = v2/r
■ Centripetal force: Fc = mac = mv2/r
1.7. Mechanical Equilibrium
● Equilibrium: Σ F = 0, Σ τ = 0
● Torque (τ): in N·m (= kg·m2/s2)
○ τ = r × F, where r = length from fulcrum to point of force
○ CCW τ > 0, CW τ < 0
2. Work, Energy
2.1. Energy
● Kinetic energy (K): energy of motion, in joules (J = N·m = kg·m2/s2)
○ K = ½ mv2
● Potential energy (U): energy of position/etc.
○ Gravitational: Ug = mgh, where h = height
■ Generally, Ug = –GmM/r
○ Elastic: Uel = ½ kx2, where k = spring constant, x = extension of spring
● Mechanical energy (E): total energy, conserved (1st Law of Thermodynamics)
○ ΔE = ΔK + ΔU
○ Energy can still be “lost” as heat, etc.
● Conservation
○ Conservative forces: path-independent, conserves E (e.g., gravitational,
electrostatic)
■ ΔE = 0
○ Nonconservative forces: path-dependent, dissipates E as thermal/chemical
energy (e.g., friction, etc.)
■ ΔE = WNC
2.2. Work
● Work (W): transfer of mechanical energy, in J
○ W = F • d = Fdcosθ
■ Only forces anti/parallel to displacement do work
○ Wnet = Kf – Ki
■ Work-energy theorem
○ Gas compression: (+) work done on gas, (–) work done by gas
○ Gas expansion: (–) work done on gas, (+) work done by gas
● Pressure (P), volume (V)
○ P–V graphs: pressure vs. volume
■ Work = integral of curve
○ W = P ΔV when process is isobaric (ΔP = 0)
● Power (Ƥ): rate of work, in watts (W = J/s)
○ Ƥavg = ΔW/Δt = ΔE/Δt
