Simple harmonic motion on a horizontal line — A/L Combined Mathematics

Horizontal Simple Harmonic Motion arises when a particle of mass m moves on a smooth horizontal table attached to the ends of elastic strings or springs governed by Hooke's Law: T = λx / l (where λ is modulus of elasticity, l is natural length, and x is extension). When displaced by extension x from equilibrium centre O, the restoring tension force exerts T = λx / l towards O. Applying Newton's Second Law: m ẍ = -T = -(λ / l) x ⟹ ẍ = -[λ / (ml)] x. This strictly matches the standard SHM equation ẍ = -ω² x with angular frequency ω = √(λ / ml) and periodic time T = 2π √(ml / λ).

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

1. Hooke's Law Formulation: Stating T = λx / l accurately with correct extension expressions.
2. Equilibrium Identification: Defining the equilibrium position O where opposing tensions balance before measuring displacement x.
3. String Slackening Dynamics: Identifying the exact point where the string becomes slack (x = 0), after which the particle moves under constant velocity or separate dynamics.