I. Mechanics

Introduction

Mechanics is the oldest and most foundational branch of physics. It asks three connected questions about any object: where is it, how fast is it moving, and why does its motion change? Every other part of physics — electromagnetism, thermodynamics, quantum theory — eventually needs the language mechanics builds: vectors, forces, energy, and momentum. A candidate who is fluent in mechanics finds the rest of the syllabus far easier, because concepts such as energy conservation and equilibrium reappear in circuits, fluids, waves and nuclear reactions.

The chapter has a natural logical order. Vectors come first because force, velocity and acceleration cannot be handled correctly without them. Newton’s laws come next because they connect force to motion. Conservation laws (energy, momentum, angular momentum) follow because they are consequences of Newton’s laws that let us solve problems without tracking every instant of motion. Rigid-body rotation extends the same ideas to spinning objects. Gravitation applies the whole framework to orbits and satellites. Special relativity closes the chapter by showing where Newtonian mechanics stops being accurate — at speeds close to the speed of light — and what replaces it.

1. Vectors: scalar and vector products, gradient, divergence and curl

Why vectors matter. A scalar (mass, temperature, energy) is fully described by a single number. A vector (force, velocity, electric field) needs a magnitude and a direction. Physics equations that ignore direction give wrong answers the moment more than one force or velocity acts at an angle to another, so resolving vectors into components (usually along x, y, z axes) is the first practical skill.

Scalar (dot) product: A·B = |A||B|cosθ. It returns a scalar and measures how much one vector “acts along” another. Physically, work done by a force, W = F·d, is a dot product — only the component of force along the displacement does work; a force perpendicular to motion (like the normal force on a sliding block) does zero work.

Vector (cross) product: A×B = |A||B|sinθ n̂, where n̂ is perpendicular to both A and B (direction given by the right-hand rule). It returns a vector and is used wherever rotation or a perpendicular effect is involved: torque τ = r×F, angular momentum L = r×p, and the magnetic force F = qv×B.

Triple products. The scalar triple product A·(B×C) gives the volume of the parallelepiped formed by the three vectors, and is zero if the vectors are coplanar — a quick test for linear dependence. The vector triple product A×(B×C) = B(A·C) − C(A·B) is mainly a computational identity used to simplify rotational-dynamics expressions.

Gradient, divergence and curl. These are vector-calculus operators applied to fields (quantities defined at every point in space).

• The gradient (∇f) of a scalar field points in the direction of its steepest increase, with magnitude equal to the rate of increase. Example: the gradient of a temperature field points from cold toward hot, and heat flows opposite to it.
• The divergence (∇·F) of a vector field measures whether a point acts as a source (positive divergence, field lines spreading out, as around a positive electric charge) or a sink (negative divergence).
• The curl (∇×F) measures the tendency of a field to circulate around a point, like the field around a current-carrying wire.

These three operators reappear constantly in Paper II (Gauss’s law uses divergence; Ampère’s and Faraday’s laws use curl), so mastering their physical meaning here pays off later.

2. Newtonian mechanics

Newton’s three laws form the operating system of classical mechanics.

• First law (inertia): an object continues in its state of rest or uniform motion unless acted on by a net external force. This defines what a force actually is — something that changes motion, not something needed to sustain it.
• Second law: F = ma (more precisely, F = dp/dt, the rate of change of momentum). This is the working equation for almost every mechanics problem: identify all forces, sum them as vectors, and set the result equal to mass times acceleration.
• Third law: every force has an equal and opposite reaction force, acting on the other body. This is often misunderstood — the two forces act on different objects and therefore do not cancel each other in any single object’s equation of motion.

Method for solving a mechanics problem: draw a free-body diagram of the object, identify every force acting on it (gravity, normal force, tension, friction, applied force), resolve each into convenient axes, apply F = ma along each axis separately, and solve the resulting equations. Friction deserves special care: static friction (up to μₛN) prevents motion from starting, while kinetic friction (μₖN, usually slightly smaller) acts once sliding begins, always opposing relative motion.

3. Conservation of energy, linear momentum and angular momentum

A conservation law states that some quantity’s total value does not change over time in an isolated system, even though the system’s configuration changes drastically. Conservation laws are powerful because they let us compare only the “before” and “after” states of a system, skipping the complicated details of what happened in between.

Energy conservation: total mechanical energy (kinetic + potential) is conserved when only conservative forces (like gravity or an ideal spring) do work. Kinetic energy KE = ½mv² is the energy of motion; potential energy is stored energy that depends on position (gravitational PE = mgh near Earth’s surface). Example: a ball dropped from height h converts all its potential energy to kinetic energy just before impact, so ½mv² = mgh gives its speed without needing to track the fall moment by moment.

Linear momentum conservation: total momentum (p = mv, summed over all objects) is conserved in a system with no net external force — most usefully applied to collisions. In a perfectly elastic collision, both momentum and kinetic energy are conserved; in a perfectly inelastic collision, the objects stick together, momentum is conserved but kinetic energy is not (some converts to heat and sound).

Angular momentum conservation: L = Iω (moment of inertia times angular velocity) is conserved when no net external torque acts. This explains why a spinning ice skater speeds up when pulling their arms in — reducing the moment of inertia I forces ω to increase so that L stays constant.

4. Rigid-body motion, rotational dynamics, spin, precession and the gyroscope

A rigid body is an idealised object whose shape does not change under the forces applied to it, which allows rotational motion to be studied separately from deformation. The rotational analogues of familiar linear quantities are:

Moment of inertia depends not just on mass but on how that mass is distributed relative to the rotation axis (I = Σmr²) — a mass farther from the axis contributes disproportionately more, which is why flywheels are built with mass concentrated at the rim.

Precession is the slow rotation of the spin axis itself, seen in a spinning top or gyroscope: gravity exerts a torque on the tilted spinning top, but instead of simply falling over, the top’s angular momentum vector rotates sideways because torque changes the direction of L, not (in this geometry) its magnitude. This is why gyroscopes are used for navigation — a fast-spinning gyroscope resists changes to its orientation and precesses only slowly under small disturbing torques, making its axis a stable reference direction.

5. Gravitation, planetary motion, satellites, Kepler’s laws

Newton’s law of universal gravitation states that every pair of masses attracts with force F = Gm₁m₂/r², where G is the gravitational constant. This single law explains both falling apples and orbiting planets — the difference is only in the geometry of the resulting motion.

Kepler’s three laws (originally empirical, later derived from Newton’s law):

• Planets move in ellipses with the Sun at one focus.
• A line from the Sun to a planet sweeps out equal areas in equal times (a consequence of angular momentum conservation — planets move faster when closer to the Sun).
• The square of the orbital period is proportional to the cube of the semi-major axis (T² ∝ r³).

Satellites and centripetal force: a satellite in a circular orbit is in continuous free fall around the Earth — gravity supplies exactly the centripetal force needed to keep it on its curved path: GMm/r² = mv²/r, which gives the orbital speed v = √(GM/r). Geostationary satellites are placed at the specific radius where the orbital period equals 24 hours, so they appear stationary relative to a point on Earth’s surface — essential for communication satellites.

6. Special relativity

By the late 19th century, physicists expected light to travel through a medium (the “luminiferous ether”), and expected its measured speed to depend on the observer’s motion through that ether. The Michelson–Morley experiment (1887) used an interferometer to look for this expected difference in the speed of light along different directions as the Earth moved through the supposed ether — and found none. This null result was one of the key pieces of evidence that eventually forced physics to abandon the ether concept.

Einstein’s postulates (1905):

• The laws of physics are the same in all inertial (non-accelerating) reference frames.
• The speed of light in vacuum, c, is the same for all observers, regardless of the motion of the source or observer.

These two postulates, taken seriously, force several counter-intuitive but experimentally confirmed consequences:

• Time dilation: a clock moving relative to an observer runs slow, by a factor γ = 1/√(1 − v²/c²). Time itself is not absolute; two observers in relative motion can disagree on how much time has passed between two events.
• Length contraction: an object moving relative to an observer appears shortened along its direction of motion, by the same factor γ.
• Mass–energy relation: E = mc², perhaps the most famous equation in physics, states that mass and energy are interconvertible — a small amount of mass corresponds to an enormous amount of energy, which is why nuclear reactions (Chapter Nuclear Physics) release far more energy per unit mass than chemical reactions.

These effects are negligible in everyday life because v/c is tiny for ordinary speeds, but they are measurable and essential in particle accelerators, GPS satellite clock corrections, and nuclear physics.

Key Takeaways

• Vectors, not scalars, are needed whenever direction matters; the dot product measures alignment (work), the cross product measures perpendicular/rotational effect (torque, magnetic force).
• Newton’s second law, F = ma, is the master equation of mechanics; solving problems means correctly identifying and resolving every force.
• Conservation of energy, momentum, and angular momentum let you compare “before” and “after” states without tracking the full motion — indispensable for collisions and rotational problems.
• Rotational dynamics mirrors linear dynamics with moment of inertia replacing mass and torque replacing force; precession arises because torque changes the direction, not just the magnitude, of angular momentum.
• Gravitation and Kepler’s laws together explain planetary and satellite motion as continuous free fall under a centripetal force.
• Special relativity replaces Newtonian absolute time and length with observer-dependent quantities once speeds approach c, and reveals mass and energy as interconvertible via E = mc².