- Define and explain Simple Harmonic Motion in your own words
- Use key terms such as simple harmonic motion accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on Simple Harmonic Motion: modelling oscillations with a = −ω²x and the mass–spring period.
Modelling oscillations with a = −ω²x and the mass–spring period.
Key ideas
SHM describes smooth back-and-forth oscillation
In simple harmonic motion the restoring force grows with displacement: a = −ω²x, the minus sign showing acceleration always points back to equilibrium. A mass on a spring has period T = 2π√(m/k); a pendulum has T = 2π√(l/g). Maximum speed occurs at the centre of the swing, maximum acceleration at the extremes.
Circular motion needs a constant centre-seeking force
An object circling at steady speed is accelerating because its velocity direction keeps changing. The centripetal acceleration is v²/r towards the centre, so the required force is F = mv²/r. Remove the force — cut the string — and the object flies off along a tangent, which is why mud flings off a spinning wheel.
Key term — simple harmonic motion: Oscillation where acceleration is proportional to displacement and always towards equilibrium: a = −ω²x.
A 0.2 kg mass on a spring (k = 80 N/m) oscillates. Calculate the period.
√(0.2 ÷ 80) = √0.0025 = 0.05, so T = 2π × 0.05 ≈ 0.31 s.
Answer: √(0.2 ÷ 80) = √0.0025 = 0.05, so T = 2π × 0.05 ≈ 0.31 s.
- Using T = 2π√(m/k) for a pendulum That formula is for a mass on a spring; a pendulum uses T = 2π√(l/g).
- Inventing an outward 'centrifugal force' In an inertial frame there is only the inward centripetal force — the outward feeling is just inertia resisting the turn.
Practice
(1200 × 400) ÷ 50 = 480,000 ÷ 50 = 9600 N.
Δp = 0.5 × 6 = 3 kg m/s; F = 3 ÷ 0.2 = 15 N.
Before: 2 × 3 = 6 kg m/s. After: 3 kg × v = 6, so v = 2 m/s.
They increase the time over which momentum changes, so the average force on the passenger is smaller.
Quick check
Which of these best defines "simple harmonic motion"?
In SHM, where are speed and acceleration greatest?
- Simple Harmonic Motion: modelling oscillations with a = −ω²x and the mass–spring period.
- SHM describes smooth back-and-forth oscillation: In simple harmonic motion the restoring force grows with displacement: a = −ω²x, the minus sign showing acceleration always points back to equilibrium.
- momentum: Mass × velocity (p = mv), a vector quantity measured in kg m/s.
- Watch out for: using T = 2π√(m/k) for a pendulum