Reference shelf

Formula sheet

Every law the course reaches, grouped by module, each tagged definition, stated, or derived and linked to where it is built.

Written by Claude Fable 5

Every law the course actually reaches, in the order it reaches them. Come back to these cold and try to say what each one means, and where it comes from, before you read the caption. The tag tells you how the course earns it.

How to read the tags

A Definition just names a quantity, so it cannot be wrong, only useful or not. A Stated law is posited from experiment or taken as an axiom; the course asks you to trust it and then checks its consequences. A Derived result is built from something earlier on this page, so you could rebuild it yourself. Nothing here is derived from something the course has not already shown you.

Describing motion

v = \frac{Δ x}{Δ t} Velocity is the rate position changes: displacement over the time it took.

DefinitionPosition and velocity

a = \frac{Δ v}{Δ t} Acceleration is the rate velocity changes. It is what your body actually feels, not speed.

DefinitionAcceleration

Forces and motion

F_{net} = m a ⟺ a = \frac{F _{net}}{m} Newton's second law. The engine of the whole mechanics half of the course: acceleration is proportional to force and inversely proportional to mass.

StatedForce, mass, and acceleration

E = kinetic \frac{1}{2} m v^{2} + potential U (x) = constant The invariant a good integrator must not drift: total energy, kinetic plus potential, stays constant with no outside push.

DerivedSimulating motion

Momentum and collisions

p = m v Momentum: mass times velocity. A vector, so opposite momenta can cancel.

DefinitionMomentum and collisions

F = m a = m \frac{Δ v}{Δ t} = \frac{Δ p}{Δ t} Newton's second law rewritten: force is the rate momentum changes. This is the form Newton actually wrote.

DerivedMomentum and collisions

F Δ t = Δ p Impulse: a force acting over time delivers a change in momentum. Why a crumple zone, spreading the time, cuts the peak force.

DerivedMomentum and collisions

Δ p_{A} = - Δ p_{B} ⟹ p_{A} + p_{B} = constant Conservation of momentum, straight out of Newton's third law: what one body gains, the other loses, so the total holds through any collision.

DerivedMomentum and collisions

Energy and power

W = F d Work: force times the distance moved along it. The channel through which energy is transferred.

DefinitionEnergy, work, and conservation

W = F d = (ma) \cdot \frac{v ^{2}}{2 a} = \frac{1}{2} m v^{2} Kinetic energy, derived by pushing a mass from rest: the work done lands as one half m v squared.

DerivedEnergy, work, and conservation

PE = m g h, \frac{1}{2} m v^{2} + m g h = constant Gravitational potential energy is the work done lifting a mass to height h, ready to return as motion. Kinetic plus potential is conserved.

DerivedEnergy, work, and conservation

P = \frac{E}{t} 1 watt = 1 \frac{joule}{second} Power: energy per second. One watt is one joule per second.

DefinitionPower and efficiency

Efficiency: the fraction of input energy that becomes the output you wanted. The rest becomes heat.

DefinitionPower and efficiency

Rotation

τ = r \times F Torque: force times lever arm. The spinning twin of force, and why a longer wrench wins.

DefinitionRotation, torque, and angular momentum

I = m r^{2} Moment of inertia of a point mass: resistance to spin grows with the square of the distance from the axis. The spinning twin of mass.

DefinitionRotation, torque, and angular momentum

L = I ω, I_{in} ω_{in} = I_{out} ω_{out} Angular momentum is conserved with no outside torque, so pulling mass inward (smaller I) forces the spin to speed up. The skater and the falling cat.

DerivedRotation, torque, and angular momentum

Oscillations and waves

F = - k x Hooke's law: a spring pulls back in proportion to how far you stretch it. The one restoring force behind every oscillation here.

StatedSimple harmonic motion

x (t) = A cos (ω t), ω = \frac{k}{m} Feed Hooke's law into F = ma and the motion is a cosine. Its angular frequency is set by stiffness over mass, not by how hard you pull.

DerivedSimple harmonic motion

T = 2 π \frac{m}{k}, T_{pendulum} = 2 π \frac{L}{g} The period of a spring, and of a small-swing pendulum, whose period surprisingly does not depend on mass at all.

DerivedSimple harmonic motion

v = \frac{λ}{T} = λ f The wave equation of state: speed equals wavelength times frequency. A crest passes each cycle, f times a second, each one lambda apart.

DerivedWaves

s (t) = n = 1 \sum \infty a_{n} sin (2 π n f t + ϕ_{n}) Fourier's claim: any sound is a sum of pure sine tones at whole-number multiples of a fundamental. The math behind every spectrum and codec.

StatedSound, Fourier, and DSP

A (ω) = \frac{F _{0}}{( ω _{0}^{2} - ω ^{2} ) ^{2} + ( γ ω ) ^{2}}, ω_{0} = \frac{k}{m} Driven amplitude blows up when the drive frequency nears the natural one and damping is small. The bridge, the wine glass, the radio dial.

StatedResonance

Gravity and orbits

a = \frac{F _{grav}}{m} = \frac{m g}{m} = g Why everything falls at the same rate: the mass that pulls cancels the mass that resists, leaving g for all objects alike.

DerivedGravity

F = G \frac{m _{1} m _{2}}{r ^{2}}, W = m g Newton's law of gravitation: every mass pulls every other, falling off as the square of the distance. Weight is just this force, mg, at Earth's surface.

StatedGravity

\frac{GM}{r ^{2}} = \frac{v ^{2}}{r} ⟹ v = \frac{GM}{r} An orbit is free fall that keeps missing: set gravity equal to the centripetal pull and the orbital speed drops out, set by the mass below and the radius alone.

DerivedOrbits and GPS

d = c \times Δ t How GPS turns time into place: distance is the speed of light times the signal's travel time, and trilateration does the rest.

DefinitionOrbits and GPS

Heat and entropy

⟨ \frac{1}{2} m v^{2} ⟩ = \frac{3}{2} k T What temperature really is: the average kinetic energy of molecular jiggling, three halves k T per particle.

StatedTemperature and kinetic theory

P V = N k T The ideal gas law, derived by counting wall collisions and matching to the temperature relation above.

DerivedTemperature and kinetic theory

Δ T = P \cdot R_{thermal} Heat flow is Ohm's law in disguise: temperature difference is the power through a layer times its thermal resistance. Same math as V = I R.

StatedHeat transfer and throttling

S = k ln W Boltzmann's entropy: the log of the number of microscopic arrangements W a macrostate allows. Why systems drift toward overwhelming likelihood.

StatedEntropy and the second law

H = - x \sum p (x) lo g_{2} p (x) Shannon entropy: the same counting math, now the average surprise of a source in bits. Boltzmann counted molecules; Shannon counted messages.

StatedEntropy and the second law

Electricity and magnetism

F = k \frac{q _{1} q _{2}}{r ^{2}}, E = \frac{F}{q} Coulomb's law: the same inverse-square shape as gravity, but charge can repel as well as attract. The field is the force per unit charge.

StatedCharge and the electric field

V = I R Ohm's law: for many materials, current is proportional to voltage, with resistance the constant between them.

StatedCircuits: voltage, current, resistance

P = V I = (I R) I = I^{2} R Electrical power, and why resistors heat up: substitute Ohm's law and it becomes current squared times resistance.

DerivedCircuits: voltage, current, resistance

E = - \frac{d Φ}{d t} Faraday's law: a changing magnetic flux through a loop induces a voltage. The whole basis of generators, transformers, and wireless charging.

StatedMagnetism and induction

c = f λ Light is an electromagnetic wave; its speed c, wavelength, and frequency lock together. Radio and X-rays are the same wave at different pitches.

StatedElectromagnetic waves and light

Light and the modern world

Snell's law: light bends at a boundary by the ratio of the two refractive indices. Every lens and every camera is this rule, applied twice.

StatedOptics, lenses, and cameras

\frac{1}{f} = \frac{1}{d _{o}} + \frac{1}{d _{i}}, m = - \frac{d _{i}}{d _{o}} The thin-lens equation ties object distance, image distance, and focal length; the magnification follows. Focusing a camera is solving this for the image.

StatedOptics, lenses, and cameras

t = γ t_{0}, γ = \frac{1}{1 - v ^{2} / c ^{2}} Time dilation, built from one light-clock and the constant speed of light: a moving clock ticks slow by the factor gamma. GPS drifts 11 km a day without it.

DerivedRelativity, and why GPS needs it

E = h f Light comes in lumps: each photon's energy is set by its frequency alone. This one line explains the photoelectric effect, LEDs, and solar cells.

StatedThe quantum leap: photons, LEDs, solar

P = C V^{2} f A chip's dynamic power: capacitance times voltage squared times clock frequency. When voltage stopped falling, this became the power wall that ended the megahertz race.

StatedSemiconductors, CPUs, and the power wall

n qubits ⟶ 2^{n} amplitudes Where the classical picture ends: n qubits in superposition span 2 to the n amplitudes at once. The exponential the last course begins from.

StatedBridge to quantum computing

Two shapes recur on purpose. The inverse-square law $F \propto 1/ r^{2}$ governs both gravity and charge, and $Δ T = P R$ is $V = I R$ with heat in place of current. Physics reuses its own math, which is exactly why so little of it has to be memorized.