Oscillations
Repeating motion back and forth around an equilibrium — SHM.
Simple harmonic motion (SHM) is defined by: acceleration is proportional to displacement from equilibrium and always directed back toward equilibrium: a = −ω²x. The minus sign is essential — it shows the restoring nature of the force. F = −kx (Hooke's Law for a spring) leads directly to SHM. The condition a ∝ −x is both necessary and sufficient for SHM.
Amplitude A: maximum displacement from equilibrium (m). Period T: time for one complete oscillation (s). Frequency f: oscillations per second (Hz). Angular frequency ω = 2π/T = 2πf (rad/s). All three are related: T = 1/f = 2π/ω. These describe the timing of oscillation without reference to the restoring force.
A mass m on a spring with spring constant k oscillates with period T = 2π√(m/k). Key insight: period depends only on m and k — NOT on amplitude. Doubling A does not change T. To increase period: increase mass or decrease spring stiffness. The spring constant k has units N/m.
A simple pendulum of length L oscillates with period T = 2π√(L/g). Valid only for small angles (< ~10°). Period depends only on L and g — NOT on mass, NOT on amplitude (for small angles). To increase period: increase length or go to a weaker gravitational field. On the Moon, a pendulum swings more slowly.
Total mechanical energy E = ½kA² = constant (no damping). KE and PE continuously exchange. At equilibrium (x = 0): KE is maximum, PE = 0, speed = v_max = ωA. At amplitude (x = ±A): KE = 0, PE is maximum, speed = 0. KE = ½m(ωA)² − ½mω²x² = ½mω²(A²−x²). PE = ½kx² = ½mω²x².
Damping removes energy from the oscillator each cycle. Underdamped: amplitude decreases gradually, oscillations continue (e.g. a car suspension). Critically damped: system returns to equilibrium as fast as possible without oscillating (e.g. door closer). Overdamped: system returns to equilibrium slowly without oscillating. All have the same equilibrium position — only the return speed differs.
Resonance occurs when the driving frequency equals the natural frequency of the system → amplitude reaches maximum. In lightly damped systems, the amplitude at resonance is very large. Damping: (1) reduces amplitude at all frequencies, (2) shifts the resonance peak slightly lower, (3) broadens the resonance curve. Examples: Tacoma Narrows Bridge, tuning a radio.
Starting from x = A·cos(ωt): Displacement: x = A·cos(ωt). Velocity: v = −Aω·sin(ωt). Acceleration: a = −Aω²·cos(ωt) = −ω²x. Phase differences: v leads x by 90° (π/2 rad) — when x is at maximum, v = 0; when x = 0, v is maximum. a is 180° (π rad) out of phase with x — when x is maximum positive, a is maximum negative. This is why acceleration always points back toward equilibrium (restoring). Energy: KE = ½mω²(A² − x²), PE = ½mω²x², Total E = ½mω²A² = ½kA² = constant.
x–t Graph (Displacement vs. Time)
A cosine wave if starting at maximum displacement. Shows how the object oscillates between +A and −A. Period T is the time between identical points on the wave.
v–t Graph (Velocity vs. Time)
A negative sine wave. Lags x–t by 90°. Maximum when x = 0 (equilibrium), zero when x = ±A (amplitude).
a–t Graph (Acceleration vs. Time)
An inverted cosine wave. Opposite phase to x–t (180° difference). Maximum magnitude when x = maximum. Always points toward equilibrium.
a–x Graph (Acceleration vs. Displacement)
slope: Slope = −ω²A straight line through the origin with a NEGATIVE slope. Slope = −ω². This graph immediately shows whether motion is SHM — if a vs. x is linear with negative slope, it is SHM.
Energy vs. Displacement Graphs
KE = ½mω²(A²−x²): an inverted parabola (max at x=0, zero at x=±A). PE = ½mω²x²: an upright parabola (zero at x=0, max at x=±A). Total E = constant horizontal line. The two curves mirror each other symmetrically.
Every particle in a travelling wave oscillates in SHM. The wave's frequency and period equal the oscillator's frequency and period.
💡Exam tip: f comes from the source (oscillations). λ comes from wave geometry. Then v = fλ.
SHM is the projection of uniform circular motion onto one axis. Both topics use ω (angular frequency) with the same formula: ω = 2π/T.
💡Exam tip: Memorise ω = 2π/T once — it applies to both circular motion and SHM.
Click any formula to see symbol definitions.
All formulas for this topic are in BINAS BINAS 35B1. In the exam you don't need to memorise the equations — but you must know which table to open and what every symbol means.