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

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

Aligned with the NIE syllabus

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

Vertical Elastic String/Spring system: Particle of mass m hangs in static equilibrium from fixed support by an elastic string of natural length l and modulus λ.
Static Equilibrium position e: Weight balances static tension: T0 = mg => λ * e / l = mg => equilibrium extension e = mgl / λ.
Equation of Motion about Equilibrium: Measure downward displacement x from the static equilibrium position: total extension is (e + x); restoring force F = mg - T = mg - (λ/l)(e + x) = mg - (λ/l)e - (λ/l)x = - (λ/l)x.
Proof of Vertical SHM: F = m*x'' => - (λ/l)x = m*x'' => x'' = - (λ / (ml)) * x; simple harmonic motion centered at the static equilibrium position with ω = √(λ / (ml)) and period T = 2π √(ml / λ) = 2π √(e / g).
String slackness threshold in vertical motion: The string becomes slack if the upward displacement exceeds the equilibrium extension: x < -e (i.e., if amplitude a > e); when string is slack, particle moves freely under gravity as a vertical projectile with downward acceleration g.

Complete vertical oscillation without slackness

The particle executes pure SHM for its entire cycle if and only if amplitude a ≤ e = mgl / λ.
Maximum height reached when string goes slack (a > e)

When particle rises above slack point (x = -e), it has upward speed v_slack = ω√(a² - e²); additional height climbed as free projectile under gravity is h_add = v_slack² / (2g) = (a² - e²)/(2e).
Total Mechanical Energy in vertical SHM

Sum of kinetic energy, gravitational potential energy, and elastic potential energy is conserved throughout the entire motion.

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