Simple harmonic motion — A/L Combined Mathematics

Simple Harmonic Motion (SHM) is rectilinear oscillatory motion where the acceleration ẍ of a particle is directly proportional to its displacement x from a fixed equilibrium centre O and always directed towards O: ẍ = -ω² x (where ω is a positive constant). The general kinematic solutions are: (1) Displacement: x = a cos(ωt + ε) or x = a sin(ωt + φ), where a is the amplitude; (2) Velocity-Displacement Relation: v² = ω²(a² - x²); (3) Periodic Time: T = 2π / ω (independent of amplitude); (4) Frequency: f = 1/T = ω / (2π); (5) Maximum Speed: v_max = aω occurs at the centre (x = 0); (6) Maximum Acceleration: f_max = aω² occurs at the extreme amplitudes (x = ±a).

Check before you answer (A/L Applied Maths Paper 2)

1. Standard Differential Form: Establishing that the acceleration satisfies ẍ = -ω² x before citing SHM properties.
2. Velocity-Displacement Formula: Applying v² = ω²(a² - x²) to find speed at given displacements or to evaluate amplitude a.
3. Elapsed Time Integrals: Using t = (1/ω) cos⁻¹(x/a) or angular phasor diagrams to calculate the exact time taken to travel between two given points.