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Chapter 01 · Part II · Waves
Revised on ____________________
Chapter 01 · Simple Harmonic Motion

One equation, and everything that obeys it

2 sheets
1 diagram
32 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

The defining equation

x IS MEASURED FROM EQUILIBRIUM
I.Definition  a = − ω² x restoring, proportional to displacement
II.Displacement  x = A sin(ω t + φ)
III.Velocity  v = ω √(A² − x²) maximum ωA at the centre
IV.Acceleration  |a|max = ω² A at the extremes, where v = 0
V.Period  T = 2πω = 2π √( mk )
VI.Amplitude from a launch  A = √( x0² + v0²ω² ) use energy whenever both are non-zero
VII.Released from rest  A = x0 rest off-centre means a turning point
VIII.Phase  check the sign of v at t = 0 moving outward ⇒ φ negative
tx(t)v(t)a(t)v leads x by a quarter period · a = −ω²x (opposite to x)
II

Energy

TWICE THE FREQUENCY OF THE MOTION
I.Total  E = 12 m ω² A² = 12 k A² constant, independent of x
II.Kinetic  K = 12 m ω² (A² − x²)
III.Potential  U = 12 m ω² x²
IV.Equal shares at  x = ± A√2
V.Time averages  ⟨K⟩ = ⟨U⟩ = E2
VI.Energy frequency  2 f K and U cycle twice per oscillation
Both K and U vary as cos² or sin², so each completes two cycles per period. A question about the energy frequency is a question about 2f.
III

Every standard oscillator

ALWAYS ω = √(keff / meff)
I.Spring  T = 2π √( mk )
II.Vertical spring  δ = m gk same T; oscillates about the stretched position
III.Springs in parallel  keff = k1 + k2
IV.In series  1keff = 1k1 + 1k2
V.Simple pendulum  T = 2π √( Lg ) small angles only
VI.Physical pendulum  T = 2π √( Im g d ) d = pivot to centre of mass
VII.Torsional  T = 2π √( IC )
VIII.Liquid in a U-tube  T = 2π √( L2 g ) L = total column length
Seconds pendulum
L = 0.994 m, T = 2 s
g from a pendulum
g = 4π²L / T²
Pendulum at 30°
T is 1.7 % longer
Spring, doubled m
T grows by √2
Halved amplitude
E falls to E/4
Resonance amplitude
∝ 1 / b
IV

Damping and resonance

THE REAL WORLD
I.Damped motion  x = A e−bt/2m cos ω′t
II.Shifted frequency  ω′ = √( ω0² − ( b2m )² ) always below ω₀
III.Energy decay  E = E0 e−bt/m twice the amplitude rate
IV.Critical damping  b = 2 √(m k) fastest return without overshoot
V.Resonance  ωdrive ≈ ω0 sharper the lighter the damping
VI.Quality factor  Q = ω0 mb cycles before the energy falls by e
V

Two harmonic motions at once

AND SHM FROM ANY WELL
I.Same line, phase δ  A = √(A1² + A2² + 2A1A2 cos δ)
II.Its phase  tan φ = A2 sin δA1 + A2 cos δ
III.Perpendicular, equal ω  a line at δ = 0, a circle at δ = 90° with equal A
IV.SHM from a potential well  ω = √( U″(x0)m ) any minimum is harmonic close enough in
Where marks are lost in this chapter
1Taking the amplitude as the initial position. With both x0 and v0 non-zero, get A from energy.
2Measuring x from the spring's natural length on a vertical spring. It oscillates about the stretched equilibrium.
3Using k1 + k2 for springs in series — that is the parallel result.
4Assuming T is amplitude-independent for a pendulum at 30° or 60°. The θ₀²/16 correction adds 0.4 % at 15°, 1.7 % at 30° and about 7 % at 60°.