ENERGY CONSERVATION

Loop-the-loop by the energy ledger — clear it only if released from h ≥ 2.5r, mass be damned.

Braking distance grows as v² — double your speed, quadruple the road; the v² law, priced.

½kx² becomes ½mv² becomes mgH — one joule in three costumes, and it is shm’s peak speed.

A circle’s shadow — the phasor shows why everything oscillates in cosines.

The period carries no m — mass cancels before the physics even starts.

The CM worldline runs straight through the wreck — that line IS momentum.

Δp = 0 while 98.7% of the KE dies — measure a bullet with a protractor.

ma + cv + kx = 0 at every drawn frame — ring, settle, or crawl.

T² = a³, equal areas proven as an identity — Newton’s arithmetic, checked.

Shell theorem, drawn: g climbs straight to the surface, then falls as 1/r². Zero at the core.

v_esc = √(2GM/R) — the energy line that just clears the rim. Mass and angle need not apply.

v²r = GM, E = −GMm/2r, and 2KE + PE = 0 to 1e-12 — the virial ledger, audited.

a = g·sinθ/(1+β) — shape alone runs the race. Mass and radius cancel, bitwise.

τ = Iα, θ = ½αt², KE = τθ — F = ma in a spinning alphabet, audited to 1e-12.

Pull in: I₁ω₁ = I₂ω₂ holds, KE jumps. L conserved, energy not — two different books.

Gravity turns the computed angular-momentum vector: full Euler–quaternion motion, exact regular precession, nutation, and an honestly bounded fast-top estimate.

Tf is the heat-capacity–weighted average; heat lost = heat gained, audited to 1e-12.

q = ΔT/(L₁/k₁+L₂/k₂) — R-values add in series. Ohm’s law for heat, drawn.

P = εσAT⁴ and λ_max = b/T — Stefan, Wien, and the Planck curve that broke classical physics.

V = IR, and the power VI = I²R = V²/R — three routes to the same watt, all of it heat. Series adds, parallel divides, and R_series × R_parallel = R₁R₂ to the bit.

Q = CV; the energy ½CV² is the triangle under the Q–V line. And the mirror of resistors — caps in parallel ADD, in series reciprocate, with C_series × C_parallel = C₁C₂.

RC·dV/dt + V = 0 at every frame — the exponential goodbye, audited.

A perfect ratio machine — until you connect the thing you built it for.

Lq″ + Rq′ + q/C = 0 — the damped oscillator’s own code, running in copper.

Faster means cheaper — the throat trades pressure for speed, exactly priced.

Four roads between two states — and the tolls differ. That difference is why engines exist.

W_by = ∫P dV, signed by direction; ΔU = Q − W_by closes while one computed piston state draws the entire ledger.

Four exact isobars and isochores close ΔU and ΔS_gas to zero while Q_net = W_net = the signed rectangular P–V area.

Q_h = W + Q_c to the last joule, η = 1 − T_c/T_h — the ceiling nobody beats.

E² − (pc)² = (mc²)², residual zero — and a collision that CREATES mass.

E = hf − φ — below threshold, a floodlight ejects nothing; above it, a glimmer.

Only the rungs exist — and the jumps between them are Balmer’s lines, to the nanometer.

Δλ = λ_C(1−cosθ) — and the recoil electron’s E²−p² is still m², to the bit.

Two 511 keV photons, exactly back-to-back — the 180° derived from p-conservation, residual 0.

One exact stationary wave, matched twice: density may decay inside the barrier while probability current crosses and R + T stays one.

T = 2πm/qB — no speed in it: five radii, one period, audited to 1e-15. The spiral obeys √k to the bit.

ε = BLv, and the ledger: Fv === I²R with residual 0 at every speed — work becomes heat, joule for joule.

Exact finite-field overlap drives Φ, ε, the full RL transient, signed magnetic force, and Wext = QR + UL on one clock.

At ω₀ the reactances cancel BITWISE and φ === 0 — driven Q equals free-ringing Q, one number.

V₁I₁ === V₂I₂ — the invoice, bitwise across the whole plane at 2:1. Volts down, amps up, never free.

Huygens’ tautochrone: beads from any height reach the bottom in the same instant — exact SHM in arc length.

Drop a stone through the Earth: it oscillates in 84 min — the exact period of a skimming satellite, and any straight tunnel keeps the same clock.

The gravity-train period depends ONLY on density, 2π√(3/4πGρ) — a whole planet or a marble of the same stuff take identical time.

Push a damped spring: displacement peaks BELOW ω₀, velocity peaks AT ω₀ — the exact curve acres draws for RLC current.

Two carts, two normal modes — ω_s=√(k/m) in step, ω_a=√((k+2k_c)/m) opposed. Start one and their motion envelopes trade prominence and return.

Two charges: the field vanishes at x = r/(1+√(q₂/q₁)) — but ∇²V=0 makes it a saddle, so nothing rests there. Earnshaw, drawn.

A toppling chimney rotates, so its tip accelerates at (3/2)g·sinθ — past 42° that beats free fall, and the mortar tears at L/3 up.

Match the periods from two pivots and g = 4π²L/T² — the moment of inertia cancels completely. Kater weighed gravity without knowing I.