Every definition and working formula from the fourteen modules, gathered for revision. Fractions are stacked, exactly as you should write them on paper. Take g = 9.8 m/s² unless a problem says otherwise; axes and signs follow the conventions primer.
01 · Introduction to Physics
Dimensional formula — every mechanical quantity written as MᵃLᵇTᶜ. An equation must be dimensionally homogeneous: every added term carries the same dimensions.
Errors — absolute error Δa; fractional errors add for products and quotients, and powers multiply them. Significant figures survive multiplication as the least count of the inputs.
Vector — magnitude + direction + triangle law of addition. Unit vector â = A divided by its magnitude. Resolution: Aₓ = A cos θ, A_y = A sin θ.
Dot product gives a scalar (work); cross product a vector ⊥ to both (torque, L). Derivative = slope of a graph; integral = area under it.
A·B = AB cos θ|A×B| = AB sin θR = √A² + B² + 2AB cos θtan α = B sin θA + B cos θ
03 · Rest & Motion — Kinematics
Velocity — rate of change of displacement (slope of x–t); acceleration — rate of change of velocity (slope of v–t; area under a–t gives Δv). The constant-a equations below hold only when a is constant.
v = u + ats = ut + ½at²v² = u² + 2asR = u² sin 2θgH = u² sin²θ2gT = 2u sin θg
04 · The Forces
Four interactions — gravitational, electromagnetic, strong, weak. All contact forces (normal, friction, tension) are electromagnetic in origin.
Normal pushes ⊥ from a surface; tension pulls along a rope; a spring restores toward natural length. Weight W = mg acts at the centre of gravity.
W = mgHooke F = −kxseries 1k = 1k₁ + 1k₂parallel k = k₁ + k₂
05 · Newton's Laws of Motion
First law — no net force, no change of velocity (defines inertial frames). Second — ΣF = ma per axis. Third — forces come in equal-and-opposite pairs on different bodies.
Pseudo-force — in a frame accelerating at a₀, add −ma₀ to every body; only then does ΣF = ma hold inside that frame.
ΣF = maAtwood a = (m₁ − m₂)gm₁ + m₂T = 2m₁m₂ gm₁ + m₂lift scale N = m(g ± a)
06 · Friction
Static friction — self-adjusting, anything from 0 up to μₛN; it equals the applied force until slipping. Kinetic friction — fixed at μₖN (μₖ < μₛ), opposing sliding.
Angle of friction λ — resultant of N and f leans at tan λ = μ. Angle of repose θᵣ — steepest slope a block rests on: tan θᵣ = μₛ; independent of mass.
fₛ ≤ μₛNfₖ = μₖNtan λ = μtan θᵣ = μₛincline a = g(sin θ − μ cos θ)
07 · Circular Motion
Centripetal acceleration — velocity turns without changing magnitude; a points at the centre. Something real (tension, gravity, friction, normal) must supply mv² over r — centripetal force is a role, not a new force.
Banking — tilting the road lets the normal force's horizontal component turn the car with no friction at the design speed.
a = v²r = ω²rv = ωrbanking tan θ = v²rgtop of circle v_min = √gRbottom v_min = √5gR
08 · Work & Energy
Work–energy theorem — the net work of all forces equals the change in kinetic energy. Conservative force — work independent of path (gravity, spring); only then does potential energy exist.
Mechanical energy E = KE + PE is conserved when only conservative forces act; friction bleeds it into heat. Power — rate of doing work.
W = Fd cos θW_net = ΔKEU_g = mghU_spring = ½kx²P = dWdt = F·v
09 · Centre of Mass, Momentum & Collision
Centre of mass — mass-weighted mean position; external forces move the COM as if all mass sat there. Momentum conserved whenever ΣF_ext = 0 — the working law of every collision.
Coefficient of restitution e — separation speed over approach speed: e = 1 elastic (KE kept), e = 0 perfectly inelastic (bodies merge, max KE loss).
Torque — turning effect, force × lever arm. Moment of inertia — rotational mass; grows with distance² from the axis. Parallel-axis: I about any axis = I_cm + Md².
Angular momentum L = Iω conserved with no external torque (the skater's spin). Rolling — the k = I over MR² ratio decides who wins the race downhill; less spin energy, more speed.
τ = rF sin θ = IαI = Σmr²I = I_cm + Md²L = IωKE = ½Iω²rolling a = g sin θ1 + k, k = IMR²
11 · Gravitation
Universal law — every pair of masses attracts along the line joining them. g falls off as 1 over r² outside the Earth and grows linearly from zero at the centre.
Kepler — ellipses with the sun at a focus; equal areas in equal times (L conserved); T² ∝ a³. Orbit energy is negative — a bound satellite must be given energy to escape.
12 · Simple Harmonic Motion full module lives in Waves ↗
SHM — restoring acceleration proportional to displacement, a = −ω²x; the shadow of uniform circular motion. Period independent of amplitude. Velocity leads displacement by π/2.
x = A sin(ωt + φ)a = −ω²xspring T = 2π√m/kpendulum T = 2π√L/gE = ½kA²
13 · Fluid Mechanics
Pressure in a still fluid depends only on depth. Pascal — an applied pressure change reaches every point undiminished (the hydraulic press). Archimedes — upthrust = weight of fluid displaced; float when average density < fluid density.
Continuity — what flows in must flow out: Av constant. Bernoulli — along a streamline, pressure + kinetic + height energy per volume is constant: fast flow, low pressure.
Stress — restoring force per area; strain — fractional deformation; their ratio (the modulus) belongs to the material. Elastic below the elastic limit; permanent set beyond the yield point; fracture past the ultimate strength.
Surface tension — a surface costs energy T per area, so liquids minimise it; a wetting liquid climbs a capillary. Viscosity — fluid friction; a small falling sphere reaches terminal velocity when Stokes' drag balances net weight.
Y = FLAΔLu = ½·stress·straindrop ΔP = 2Trbubble ΔP = 4Trh = 2T cos θρgrStokes F = 6πηrvv_t = 2r²(ρ_s − ρ_f)g9η