Newton's Principia
50 Concepts of Classical Mechanics

Simple harmonic motion (SHM) occurs whenever a restoring force is proportional to and directed opposite to displacement from equilibrium (F = -kx). The resulting acceleration a = -(k/m)x leads to sinusoidal position variation. SHM is the most important type of periodic motion because any small-amplitude oscillation about a stable equilibrium is approximately SHM — regardless of the exact shape of the potential well. This is why the same mathematics describes springs, pendulums, molecular vibrations, electrical LC circuits, sound waves, and quantum harmonic oscillators. The total energy continuously converts between kinetic (maximum at centre) and potential (maximum at turning points).

Hooke's Law states that the restoring force of a spring is proportional to its extension or compression. The negative sign indicates the force always opposes the displacement — this is what makes it a restoring force and gives rise to oscillatory motion. The spring constant k characterises stiffness: a car suspension spring might have k ≈ 20 000 N/m; a watch spring, a few N/m. Critically, the period T = 2π√(m/k) is independent of amplitude (for ideal springs and small oscillations). Stiffer springs oscillate faster; heavier masses oscillate slower. Hooke's Law breaks down at large deformations (elastic limit exceeded).

The simple pendulum — a mass on a massless string swinging through small arcs — exhibits SHM because the restoring force component (mg·sinθ ≈ mgθ for small θ) is proportional to displacement. The remarkable result: the period depends only on length and local gravity, not on mass or amplitude. Galileo first noted this isochronism (allegedly by timing his pulse against a swinging chandelier in Pisa Cathedral). The pendulum clock, accurate to seconds per day, exploited this property and was the most precise timekeeping device for over 200 years. The period formula also enables measurement of local g: measure T and L, solve for g = 4π²L/T².

Resonance occurs when a periodic driving force is applied at a system's natural frequency. Energy is added constructively each cycle and amplitude builds dramatically — limited only by damping. Resonance is simultaneously one of the most useful and most dangerous phenomena in engineering. Useful: radio tuning circuits select signals by resonance; MRI scanners use nuclear magnetic resonance; musical instrument bodies amplify specific frequencies. Dangerous: the Tacoma Narrows Bridge collapse (1940) was caused by aerodynamic forcing at the bridge's structural resonance frequency; soldiers break step crossing bridges to avoid resonant forcing; earthquake damage is amplified when building natural frequencies match seismic frequencies.

Johannes Kepler (1609–1619) derived his three laws empirically from Tycho Brahe's meticulous observational data. Newton's supreme achievement was deriving all three laws mathematically from his Law of Universal Gravitation and Second Law — demonstrating that they are not independent facts but consequences of a single force law. Kepler's Second Law (equal areas in equal times) is equivalent to conservation of angular momentum; the Third Law (T² ∝ a³) follows from equating centripetal force to gravitational force. This unification of terrestrial physics (falling apple) with celestial mechanics (planetary orbits) was the scientific revolution's defining moment.

A circular orbit exists when gravitational force exactly provides the centripetal acceleration: GMm/r² = mv²/r. The orbital speed is entirely determined by the altitude — higher orbits are slower. At geostationary orbit (35 786 km above Earth's equator), the orbital period matches Earth's rotation, so the satellite appears stationary — the basis of communication and GPS satellites. Elliptical orbits generalise this: at periapsis (closest approach) speed is maximum; at apoapsis (farthest point) speed is minimum, consistent with conservation of energy and angular momentum. The total energy of an orbit: E = -GMm/(2a), negative indicating a bound system.

Escape velocity is the minimum speed at which an object must be launched to escape a gravitational field without further propulsion. Derived by setting total mechanical energy to zero: ½mv² - GMm/r = 0. Notably, v_esc = √2 · v_orbit — escape velocity is always √2 ≈ 1.414 times the circular orbital speed at the same radius. A black hole is formally defined (in classical terms) as an object for which the escape velocity at the surface equals or exceeds the speed of light — the Schwarzschild radius is r_s = 2GM/c². For Earth, r_s ≈ 9 mm; Earth would need to be compressed to a marble to become a black hole.

Tidal forces arise because gravity follows an inverse-square law: the near side of a body is pulled more strongly toward a massive object than the far side, stretching the body along the radial direction and squeezing it sideways. Earth's ocean tides are caused by the Moon's tidal force — the near-side ocean is pulled toward the Moon, the far-side ocean is "left behind," creating two tidal bulges. The Sun also contributes tides; spring tides (Moon and Sun aligned) and neap tides (90° apart) result from their combination. Tidal forces also cause orbital decay of nearby moons, the synchronous rotation of the Moon (same face always toward Earth), and the eventual destruction of objects inside the Roche limit.

Pressure is force per unit area, transmitted equally in all directions within a static fluid (Pascal's principle). Hydrostatic pressure increases linearly with depth. Archimedes' Principle (287–212 BC, predating Newton): the upward buoyant force on a submerged object equals the weight of the fluid it displaces. An object floats when its average density is less than the fluid; sinks when greater. Newton's contribution was explaining buoyancy quantitatively via pressure differences: the higher pressure on the bottom face minus the lower pressure on the top face yields the net upward buoyant force. This explains ships, hot-air balloons, submarines, and the hydrometer.

The continuity equation expresses conservation of mass for fluid flow: the volume flow rate (volume per unit time) is constant in a pipe of varying cross-section (for incompressible fluids). Where the pipe is narrow (A small), the fluid moves faster (v large), and vice versa. This is why a river speeds up at a narrow gorge and slows in a wide delta; why squeezing a garden hose end increases jet speed. For compressible fluids (gases), the mass flow rate ρAv = constant must be used instead. Continuity is the foundation of all fluid mechanics and is conservation of mass in differential form (the advection equation).

Bernoulli's equation is the energy conservation equation for an ideal (inviscid, incompressible, steady) fluid: along a streamline, the sum of static pressure, dynamic pressure, and hydrostatic pressure is constant. Where fluid speeds up, pressure drops. This explains: aircraft lift (faster flow over curved upper wing surface → lower pressure above → net upward force); the carburetor (fast airflow → low pressure → fuel drawn in); the Venturi meter; the curveball in baseball; the Coandă effect. Real fluids deviate from Bernoulli due to viscosity — addressed by the Navier-Stokes equations, one of the unsolved Millennium Prize Problems.

Dimensional analysis is the technique of tracking the physical dimensions of every quantity in an equation to verify consistency and derive relationships. Every valid physical equation must be dimensionally homogeneous — each term must have the same dimensions. Dimensional analysis can: check the consistency of derived equations; predict the form of unknown relationships (Buckingham π theorem); convert between unit systems; and catch algebraic errors instantly. The pendulum period must involve L (length), g (acceleration, [LT⁻²]) and possibly m (mass) — only T = f(L/g)^{1/2} gives the right dimensions of time, and experiment confirms the coefficient is 2π.

Newton invented calculus ("method of fluxions") between 1665–1666, simultaneously and independently with Leibniz (who published first, 1684). Without calculus, the Principia could not have been written — the laws of motion are inherently differential equations. Velocity is the derivative of position; acceleration is the derivative of velocity (or second derivative of position). Newton's Second Law becomes a second-order ordinary differential equation whose solution gives position as a function of time. The integral of a force over time gives impulse; over distance gives work. Modern physics — from quantum mechanics to general relativity — is entirely written in the language Newton invented to describe motion.

The principle of superposition states that when multiple forces act on a body, each force acts independently, and the net effect is the vector sum of all forces. This is Newton's implicit assumption that forces are additive — a body simultaneously experiencing gravity, normal force, friction, and tension simply experiences the vector resultant of all four. This linear principle fails for strong gravitational fields (General Relativity is nonlinear) and in nonlinear mechanical systems, but holds exactly within Newtonian mechanics. The practical method: resolve all forces into orthogonal components (typically x and y), sum each component independently, then find the resultant magnitude and direction.

Newtonian mechanics is not wrong — it is a limiting case of more complete theories, accurate when: speeds are much less than light (v ≪ c ≈ 3×10⁸ m/s); gravitational fields are weak; and the action of the system is much larger than Planck's constant (ℏ ≈ 10⁻³⁴ J·s). Special Relativity (Einstein, 1905) governs high-speed motion: mass increases with velocity, time dilates, lengths contract, and the speed of light is invariant. General Relativity (Einstein, 1915) replaces Newton's gravity with spacetime curvature — required for GPS satellites (gravitational and velocity time dilation corrections of ~38 microseconds/day). Quantum Mechanics governs atoms and subatomic particles: energy is quantised, particles have wave nature, and the Heisenberg uncertainty principle sets fundamental limits on simultaneous knowledge of position and momentum. For all engineering, ballistics, orbital mechanics, and most of everyday physics, Newton is exact to many decimal places.