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

NIE competency: Describes the nature of a simple harmonic motion on a horizontal line

Aligned with the NIE syllabus

What must I remember about Simple harmonic motion on a horizontal line?

Hooke's Law for elastic strings and springs: Tension T is directly proportional to extension x beyond natural length l: T = λ * x / l, where λ is the modulus of elasticity (units of force, N).
Particle attached to single horizontal elastic string: Particle of mass m on smooth horizontal table attached to fixed peg by string of natural length l and modulus λ: when extended by x, restoring force is F = -T = - (λ/l)x.
Horizontal SHM proof: By F = ma => - (λ/l)x = m * x'' => x'' = - (λ / (ml)) * x; of the standard SHM form x'' = -ω² x with angular frequency ω = √(λ / (ml)) and period T = 2π √(ml / λ).
Two opposite elastic strings: Particle attached between two fixed pegs distance 2L apart by two identical strings of natural length l and modulus λ; at displacement x from midpoint, restoring force is net tension difference: F = -(T1 - T2) = -2(λ/l)x => ω = √(2λ / (ml)).
Composite motion when string goes slack: An elastic string CANNOT support thrust; when particle passes natural length (x < 0), string goes SLACK (T = 0) and particle travels at constant velocity v_max under zero net force until string tightens on opposite side.

Time calculation for composite string-slack motion

Total periodic time = (Time in SHM where string is taut) + (Time in constant speed where string is slack): T_total = 2 * (2 * t_SHM) + 2 * (distance / v_max).
Elastic Potential Energy

Work done in stretching an elastic string from natural length to extension x is E_e = (1/2) * (λ / l) * x² = (1/2) * T * x.
Energy Conservation in horizontal SHM

(1/2)m v² + (1/2)(λ/l)x² = (1/2)(λ/l)a² = (1/2)m v_max².

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 6