Oscillations

Defining simple harmonic motion

SHM is motion under a restoring force proportional to displacement: F = −kx. The equation of motion ma = −kx leads to x(t) = A cos(ωt + φ), where ω = √(k/m), period T = 2π/ω, and frequency f = 1/T. The hallmark of SHM is constant period independent of amplitude (within the small-angle limit). Velocity v(t) = −Aω sin(ωt+φ), maximum at the equilibrium position; acceleration a(t) = −ω²x, maximum at extremes.

Mass on a spring

For a horizontal mass m attached to a spring of stiffness k, T = 2π√(m/k). A 0.5 kg mass on a 200 N/m spring oscillates at T = 2π√(0.0025) ≈ 0.314 s. Adding mass increases T as √m; stiffer spring decreases T. In a vertical spring system gravity simply shifts the equilibrium; the period is unchanged.

Simple pendulum

For small angles (sin θ ≈ θ), the simple pendulum has T = 2π√(L/g). A 1 m pendulum at g = 9.8 m/s² has T ≈ 2.007 s — historically used to define the "seconds pendulum." T is independent of mass and (for small angles) amplitude. The compound pendulum about a pivot has T = 2π√(I/mgd), where d is the distance from pivot to centre of mass.

Energy in SHM

Total mechanical energy E = ½kA² is constant. KE = ½k(A² − x²) and PE = ½kx²; KE is maximum at x = 0 and PE is maximum at x = ±A. At x = A/2, PE = ¼·½kA² = ¼E, so KE = ¾E. The kinetic and potential energies each oscillate at twice the frequency of the displacement — a result frequently exploited in MDCAT MCQs.

Damped and forced oscillations, resonance

Real oscillators experience damping: amplitude decays as e^−bt/2m. Light, critical, and heavy damping are distinguished by whether the system oscillates before reaching equilibrium. Driving a damped oscillator at its natural frequency produces resonance — amplitude peaks sharply when ω_drive ≈ ω₀. The Tacoma Narrows bridge collapse and the operation of MRI scanners are two examples cited in HRW Chapter 15. The FSc Chapter 7 covers all of this with worked spring and pendulum problems.