Oscillation

What these notes cover

These notes cover oscillations for Cambridge International A Level Physics (9702): the terms displacement, amplitude, period, frequency and angular frequency, the defining equation of simple harmonic motion, displacement, velocity and acceleration graphs, kinetic, potential and total energy in SHM, the period of a pendulum and of a mass on a spring, free and damped oscillations, and forced oscillations and resonance with its uses and problems. They include worked examples and past-paper questions.

Key formulas and definitions

• Simple harmonic motion: the acceleration is proportional to the displacement from a fixed point and is always directed towards that point: a = −ω²x.
• Period T is the time for one complete oscillation; frequency f = 1 / T; angular frequency ω = 2πf = 2π / T. The amplitude x₀ is the maximum displacement.
• Displacement: x = x₀ sin ωt (starting from the equilibrium position) or x = x₀ cos ωt (starting from maximum displacement).
• Velocity: v = ±ω√(x₀² − x²), so the maximum speed v₀ = ωx₀ occurs at the equilibrium position. Maximum acceleration = ω²x₀ occurs at maximum displacement.
• Energy in SHM: total energy E = ½mω²x₀² stays constant. Eₖ = ½mω²(x₀² − x²), and kinetic and potential energy are equal when x = x₀ / √2.
• Simple pendulum: T = 2π√(l / g). Mass on a spring: T = 2π√(m / k).
• Damping: resistive forces remove energy, so the amplitude decreases. Light damping: the amplitude falls gradually (exponentially) and the period stays almost constant. Critical damping: the system returns to equilibrium in the shortest time without oscillating. Heavy damping: it returns slowly without oscillating.
• Resonance: when the driving frequency of a forced oscillation equals the natural frequency of the system, the amplitude is a maximum. Damping lowers and broadens the resonance peak. Uses include microwave ovens, MRI and lithotripsy; resonance must be avoided in structures and car suspensions.