Lesson 2: Forces

Kinematics describes motion; dynamics explains it. Why do things speed up, slow down, or change direction? Because forces act on them. Isaac Newton’s three laws (1687) unified earthly and celestial motion — the same gravity pulling an apple pulls the Moon. Chinese engineers used forces practically for millennia: levers, pulleys, crossbows, and the trebuchets that hurled gunpowder bombs. This lesson covers Newton’s laws, friction, gravity, and the problem-solving methods that make dynamics the most practical branch of physics.

Newton’s First Law — Inertia

An object keeps its velocity unless a net force acts — inertia. Passengers lurch forward when a bus brakes: their bodies maintain velocity while the bus slows. Seatbelts provide the stopping force. The law seems obvious now, but it overthrew 2,000 years of Aristotelian belief that rest was natural and motion needed pushing.

Inertia depends on mass: a loaded truck resists acceleration more than a bicycle. Mass IS inertia’s measure. In space, where friction vanishes, the first law is vividly clear — spacecraft coast for years on brief engine burns. China’s Chang’e probes coast to the Moon on inertia, firing thrusters only for course corrections.

Reference frames: Newton’s laws hold in inertial frames (non-accelerating). In a braking car (non-inertial frame), phantom "forces" seem to push you forward. Einstein generalized this insight into general relativity — inertia and gravity are deeply connected.

Newton’s Second Law — F = ma

Net force equals mass times acceleration: F = ma. Double the force, double the acceleration; double the mass, half the acceleration. A 1,000 kg car accelerating at 3 m/s² needs 3,000 N of net force. Units: 1 newton = 1 kg·m/s².

Problem-solving method: draw a free-body diagram (all forces as arrows), choose axes, write F_net = ma per axis, solve. A 5 kg block on a frictionless 30° incline: gravity component along slope = mg sin30° = 24.5 N; a = 24.5/5 = 4.9 m/s². Free-body diagrams are dynamics’ universal language — draw first, compute second.

Weight vs. mass: mass (kg) is inertia; weight (N) is gravity’s force: W = mg. On the Moon (g = 1.6), a 60 kg astronaut weighs 96 N but keeps 60 kg of mass — inertia unchanged. "Weightless" astronauts still have mass; they’re in free fall, not force-free.

Newton’s Third Law — Action and Reaction

Every force has an equal, opposite counterpart: you push the wall, the wall pushes you. Rockets work by expelling gas backward — the gas pushes the rocket forward. China’s Long March rockets are Newton’s third law made visible: tons of exhaust expelled downward, the rocket rising upward.

The subtlety: action-reaction pairs act on DIFFERENT objects. A book on a table: Earth pulls the book (gravity), the book pulls Earth (reaction); the table pushes the book (normal), the book pushes the table. Don’t cancel them — they act on different bodies!

Horse-and-cart paradox: "if the horse pulls the cart forward and the cart pulls the horse backward equally, how does anything move?" Resolution: the horse also pushes backward on the GROUND; the ground pushes the horse forward harder than the cart pulls back. Net force on the horse-cart system comes from external ground forces. Always identify the system.

Friction

Friction opposes relative motion (kinetic) or impending motion (static). f_k = μ_k N; f_s ≤ μ_s N (static adjusts up to its max!). A 10 kg block (μ_s = 0.5, μ_k = 0.3): max static friction = 49 N. Push with 30 N: it doesn’t move (static friction = 30 N, matching!). Push with 60 N: it slides with kinetic friction 29.4 N.

Friction enables walking, driving, writing — and causes wear, heat, inefficiency. Chinese high-speed rail minimizes it: smooth rails, aerodynamic trains, and (for maglev) eliminating contact entirely — Shanghai’s maglev floats on magnetic fields, reaching 430 km/h with no rail friction.

Lubrication, ball bearings, and air cushions reduce friction; sand, chains, and textured soles increase it. Engineering is applied friction management. The Mohists studied inclined planes and leverage — ancient Chinese tribology!

Gravity and Orbits

Newton’s universal gravitation: F = Gm₁m₂/r². The same law drops apples and holds the Moon. Cavendish measured G (1798); Chinese astronomers had tracked the Moon’s motion for millennia without the inverse-square law — data awaiting theory.

Orbits are continuous free fall: the Moon falls toward Earth constantly but moves sideways fast enough to miss — falling around the planet. Orbital velocity at Earth’s surface: ~7.9 km/s. The Tiangong space station orbits at ~7.7 km/s, circling Earth every 90 minutes — its crew in perpetual free fall, hence "weightless."

Weightlessness isn’t zero gravity (Earth’s pull at Tiangong’s altitude is ~90% of surface!) — it’s free fall. Einstein’s happiest thought: a falling observer feels no gravity. Orbits, tides (the Moon’s differential pull), and satellite geodesy all flow from F = Gm₁m₂/r².

Problem Solving — Dynamics Strategies

Atwood machine: masses m₁, m₂ over a pulley — the heavier accelerates down: a = (m₂−m₁)g/(m₁+m₂). Derive via F = ma on each mass with shared tension. Pulley problems are Newton’s laws in pure form.

Elevator problems: a 60 kg person in an elevator accelerating upward at 2 m/s² — scale reads N: N − mg = ma ⟹ N = 60(9.8+2) = 708 N ("heavier"). Accelerating down: 468 N ("lighter"). Apparent weight changes; mass doesn’t. This is why stomachs drop in elevators.

Strategy summary: (1) draw free-body diagrams; (2) choose axes (often along the acceleration); (3) write F_net = ma per axis; (4) solve the system; (5) check: units, signs, limiting cases (what if θ → 0? what if m → ∞?). Limiting-case checks catch most errors.

Deep Dive — Tides, Satellites, and Orbital Mechanics

Tides: the Moon’s gravity pulls Earth’s near side harder than its center, and the center harder than the far side — the DIFFERENCE stretches the oceans into two bulges (near and far). Hence two high tides daily! The Sun contributes (spring tides at new/full moon when aligned; neap tides at quarters). China’s Qiantang River bore — a 9-meter tidal wave — is tides made spectacular.

Kepler’s third law quantifies orbits: T² = (4π²/GM)a³. Geostationary satellites (T = 24 h) sit at 35,786 km — hovering over one spot, perfect for communications and weather (China’s Fengyun satellites). GPS satellites (T ≈ 12 h) orbit at ~20,200 km; BeiDou's medium-Earth-orbit satellites fly slightly higher, at ~21,500 km (T ≈ 12.9 h). Altitude determines period — one equation, all satellites.

Hohmann transfers: the fuel-efficient path between orbits — an elliptical transfer orbit tangent to both. Chang’e missions used transfer orbits to reach the Moon; Tianwen-1 cruised 7 months to Mars. Orbital mechanics is applied Newton: F = ma plus gravity, integrated over trajectories. Every space mission is a dynamics problem.

Lagrange points: five spots where Sun-Earth gravity balances — parking places for observatories (JWST at L2!). China’s Queqiao relay satellite sits at the Earth-Moon L2 point, relaying Chang’e-4’s far-side signals around the Moon’s bulk. Gravitational balance points as infrastructure — celestial mechanics serving engineering.

Escape velocity: 11.2 km/s from Earth’s surface (kinetic energy exceeding gravitational binding). The Long March rockets’ staging discards mass to achieve it — the rocket equation (Tsiolkovsky): Δv = v_e ln(m₀/mᵢ). Getting to orbit is an energy problem (Lesson 3!): most of a rocket is fuel because physics demands it.

Orbital decay and space junk: low orbits drag on the atmosphere; 30,000+ tracked debris objects threaten satellites (Kessler syndrome — cascading collisions). China’s space station dodges debris regularly. Dynamics doesn’t end at launch — it governs the entire space environment.

Conclusion — Why This Matters

Forces are the "why" behind motion’s "how." Newton’s three laws — inertia, F = ma, action-reaction — explain everything from a falling apple to the Chang’e lunar missions’ rocket launches. The ancient Chinese understood leverage, buoyancy puzzles, and magnetic force practically; Newton gave the universal theory. Together, kinematics and dynamics are classical mechanics: the foundation of all physics and engineering.

Going deeper — connected bodies: when forces are shared

Many Gaokao problems join objects together — blocks linked by a string, a locomotive pulling carriages. The method has two steps: treat the whole system as one mass to find the acceleration, then isolate one part to find the internal force.

Worked example. A horizontal force of 15 N pulls a 3.0 kg block across a frictionless floor; the block is linked by a light string to a 2.0 kg block behind it.

• Step 1 — the system. Total mass = 5.0 kg, so a = F/m = 15/5.0 = 3.0 m/s². Both blocks share this acceleration (the string keeps them together).
• Step 2 — isolate the rear block. The only horizontal force on the 2.0 kg block is the string's tension T, so T = ma = 2.0 × 3.0 = 6.0 N.
• Check with the front block. Forces on it: 15 N forward, tension 6.0 N backward (Newton's third law — the string pulls back). Net = 9.0 N = 3.0 kg × 3.0 m/s² ✓.

Notice the tension (6.0 N) is less than the applied force (15 N): internal forces are always found by isolating a part, never by guessing. The same two-step method solves Atwood machines, elevator cables and train couplings.

Practical focus — testing F = ma with an Atwood machine

Prediction. Masses 0.20 kg and 0.30 kg hang over a pulley. Net driving force = (0.30 − 0.20) × 9.8 = 0.98 N; total mass = 0.50 kg; predicted a = 0.98/0.50 = 1.96 m/s².

Measurement. The heavier mass falls s = 0.80 m in t = 0.95 s (mean of three). From s = ½at²: a = 2s/t² = 1.60/0.9025 ≈ 1.77 m/s² — about 10% below prediction.

Evaluation. The shortfall is systematic, not random: friction in the pulley and air resistance oppose the motion. A smarter analysis repeats with several mass differences and plots measured a against (m₂ − m₁)g: the slope should be 1/(m₁ + m₂), and any intercept reveals the friction force directly. Turning a discrepancy into a measurement is what separates an experiment from a demonstration.

Common mistakes

• Cancelling action–reaction pairs. They act on different bodies, so they never cancel on one free-body diagram.
• Thinking static friction always equals μ_s N. It equals the applied force, up to that maximum — push gently and it pushes back gently.
• Reporting weight in kilograms. Kilograms measure mass; weight is a force, in newtons.

Exam Experience

Question 1 (choice, 3 marks). When a crowded bus brakes suddenly, standing passengers lurch forward. The best explanation is:

• A. A force throws them forward
• B. Their inertia keeps them moving at the bus's previous velocity while the bus slows
• C. Gravity temporarily increases
• D. The floor's friction reverses direction

Answer: B. No forward force exists; the passengers' bodies continue at constant velocity (Newton's first law) while the bus decelerates beneath them. Seatbelts supply the backward force their bodies need.

Question 2 (choice, 3 marks). A book rests on a table. Which pair is a Newton's third-law action–reaction pair?

• A. Gravity on the book and the normal force on the book
• B. The book's weight and the book pushing down on the table
• C. The table pushing up on the book and the book pulling up on the Earth
• D. The Earth pulling the book down and the book pulling the Earth up

Answer: D. Third-law pairs act on *different* bodies and are the same type of force: Earth-on-book gravity pairs with book-on-Earth gravity. A lists two forces on the same body — they balance, but they are not a third-law pair.

Question 3 (structured, 6 marks). (a) A 60 kg student stands on a scale in a lift accelerating upwards at 1.5 m/s². Find the scale reading (the normal force). (b) A block slides down a 30° slope with kinetic friction coefficient μ = 0.20. Find its acceleration.

Marking scheme. (a) N − mg = ma [1]; N = 60 × (9.8 + 1.5) = 678 N [1]. (b) Component of weight along slope: mg sin30° [1]; friction: μmg cos30° opposing [1]; a = g(sin30° − μ cos30°) = 9.8 × (0.500 − 0.173) ≈ 3.2 m/s² [1 value, with m cancelling shown or stated 1].