Mechanics

1. Kinematics

Kinematics describes motion without worrying about its causes.

• Displacement (vector, has direction) vs distance (scalar, total path length).
• Average velocity: v = Δs/Δt (displacement over time).
• Acceleration: a = Δv/Δt (rate of change of velocity).
• Uniformly accelerated motion — three key equations (v₀ = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement):
  1. v = v₀ + at
  2. s = v₀t + ½at²
  3. v² = v₀² + 2as
• Free fall: a special case where a = g (use g = 10 m/s² unless stated otherwise), and v₀ = 0 if the object is dropped from rest.

Common mistakes: confusing distance traveled with displacement magnitude when direction reverses; forgetting v₀ = 0 only applies when starting from rest (not when "thrown down" or "thrown up"); using the wrong equation when time is not given (use v² = v₀² + 2as instead).

2. Newton's Laws of Motion

• First law (inertia): an object stays at rest or in uniform motion unless acted on by a net external force.
• Second law: F_net = ma. Net force determines acceleration, not velocity directly.
• Third law: for every action force, there is an equal and opposite reaction force, acting on the other object (not canceling out on the same object).
• Applications: free-body diagrams, friction (f = μN), inclined planes (resolve gravity into components parallel/perpendicular to the surface), connected objects via strings/pulleys (same tension, same magnitude of acceleration if inextensible string).

Common mistakes: applying F=ma with the wrong (non-net) force; thinking action-reaction pairs act on the same object and therefore cancel; ignoring the normal force changes when there's a vertical component of applied force.

3. Momentum and Impulse

• Momentum: p = mv (vector, same direction as velocity).
• Impulse: J = FΔt = Δp (impulse equals change in momentum).
• Law of conservation of momentum: in an isolated system (no external net force), total momentum before an interaction equals total momentum after: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'.
• Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum only (some KE is lost to heat/deformation/sound).

Common mistakes: forgetting momentum is a vector — signs matter for objects moving in opposite directions; assuming kinetic energy is always conserved in collisions (only true for elastic collisions).

4. Work and Energy

• Work: W = Fs·cos(θ), where θ is the angle between the force and displacement. Work is zero if force is perpendicular to displacement.
• Kinetic energy: KE = ½mv².
• Gravitational potential energy: PE = mgh (relative to a reference height).
• Work-energy theorem: net work done on an object equals its change in kinetic energy: W_net = ΔKE.
• Conservation of mechanical energy: when only conservative forces (gravity, ideal springs) act, KE + PE = constant.

Common mistakes: confusing distance fallen with speed (a very common trap — e.g., in free fall, the distance formula ½gt² gives a distance in metres, not a speed in m/s); forgetting energy conservation only holds without friction/air resistance; using v instead of v² in kinetic energy.

5. Circular Motion and Universal Gravitation

• Centripetal acceleration: a = v²/r, directed toward the center.
• Centripetal force: F = mv²/r = mω²r (this is not a new force — it's whatever net force, e.g. tension/gravity/friction, happens to point toward the center).
• Angular velocity: ω = 2π/T = 2πf, where T is the period and f is the frequency.
• Newton's law of universal gravitation: F = Gm₁m₂/r², where G ≈ 6.67×10⁻¹¹ N·m²/kg².
• Application: for a satellite in circular orbit, gravity provides the centripetal force: GMm/r² = mv²/r, giving orbital speed v = √(GM/r).

Common mistakes: treating "centripetal force" as an additional force to add to a free-body diagram (it's the net force, not an extra one); forgetting r in gravitation is the distance between centers, not the altitude above a surface.

6. Simple Harmonic Motion (SHM) and Mechanical Waves

• SHM condition: restoring force proportional to displacement and directed opposite to it: F = −kx (ideal spring, Hooke's Law).
• Period of a mass-spring system: T = 2π√(m/k).
• Period of a simple pendulum (small angle): T = 2π√(L/g) — independent of mass and amplitude (for small angles).
• Mechanical waves: transverse (particle motion perpendicular to wave travel, e.g. wave on a string) vs longitudinal (particle motion parallel to wave travel, e.g. sound).
• Wave equation: v = fλ (wave speed = frequency × wavelength).

Common mistakes: thinking pendulum period depends on mass (it doesn't, for small angles); confusing wave speed (property of the medium) with the speed of individual particles in the medium.