Oscillations

NUST NET Physics · Physics · Oscillations, waves and light. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

In simple harmonic motion the acceleration is proportional to the displacement from the mean position and always points back towards it: a = −ω²x. The period does not depend on the amplitude.

At the mean position the speed is greatest and the acceleration is zero. At the ends the speed is zero and the acceleration is greatest. Energy swaps between kinetic and potential, while the total stays constant and is proportional to the amplitude squared.

Damping (friction) removes energy, so the amplitude dies away. Resonance happens when a driving force has the same frequency as the natural frequency of the system: the amplitude becomes very large.

Rules to remember

a = −ω²x; ω = 2πf = 2π/T
Mass on a spring: T = 2π√(m/k); for a load that stretches the spring by e, T = 2π√(e/g)
Simple pendulum: T = 2π√(l/g), independent of the mass of the bob; a seconds pendulum has T = 2 s
Starting from the mean position: x = x₀ sin ωt; v = ω√(x₀² − x²)
Maximum speed = ωx₀ (at the centre); maximum acceleration = ω²x₀ (at the ends)
Total energy = ½kx₀² = ½mω²x₀²; PE = ½kx²; KE = ½k(x₀² − x²)
KE = PE when x = x₀/√2; resonance when driving frequency = natural frequency

Common mistake

Treating the period as proportional to length or mass. T depends on the square root, so a pendulum four times as long has twice the period, and a heavier bob changes nothing.

Worked example

A 0.50 kg mass on a spring of spring constant 200 N m⁻¹ oscillates with amplitude 4.0 cm. What is its maximum speed?

ω = √(k/m) = √(200/0.50) = √400 = 20 rad s⁻¹.
Maximum speed = ωx₀, with x₀ = 0.040 m.
= 20 × 0.040 = 0.80.

Answer: 0.80 m s⁻¹

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