Oscillations

Learning objectives

• Use acceleration proportional and opposite to displacement.
• Relate period to angular frequency.
• Explain damping and resonance.

The big idea

In simple harmonic motion, acceleration is proportional to displacement and points towards equilibrium. Velocity is greatest at equilibrium and zero at the extremes; acceleration has the opposite pattern in magnitude.

Damping transfers energy away from an oscillator. A periodic driving force can produce resonance near the natural frequency, with the size and sharpness of the response depending on damping. Ideal SHM equations assume the restoring relationship remains linear. Kinetic and potential energy exchange; total mechanical energy stays constant only without damping or driving.

1. Restoring acceleration defines SHM

Simple harmonic motion requires acceleration proportional to displacement from equilibrium and directed back towards it: a = −ω²x. The minus sign states direction, not a negative acceleration magnitude everywhere. At positive displacement acceleration is negative; at negative displacement it is positive. A system that merely repeats its motion is not automatically harmonic, because its restoring relationship may be non-linear.

2. Position, speed and acceleration do not peak together

For x = A cos ωt, maximum speed is ωA and occurs at equilibrium. At either extreme, speed is zero while acceleration magnitude is greatest. The period is 2π/ω. Choosing sine rather than cosine changes the initial phase, not the underlying period or physical model. Set the phase from the specified starting position and direction rather than memorising one time-zero picture.

3. Energy moves between stores

For an ideal mass–spring oscillator, total mechanical energy is ½kA², also equal to ½mω²A². At an extreme it is elastic potential energy; at equilibrium it is kinetic. Doubling amplitude quadruples total energy while leaving the ideal period unchanged. A pendulum approximates SHM only at small angles; larger swings do not obey the same simple amplitude-independent period approximation exactly.

4. Damping and driving change the story

A resistive interaction transfers mechanical energy to other stores, reducing free-oscillation amplitude. A periodic driver supplies energy, and the steady response can become large near resonance. Damping limits the amplitude and broadens the frequency response. The frequency giving maximum displacement can shift with damping, so do not insist that every experimental resonance peak is exactly the undamped natural frequency.

A method you can reuse

1. Locate equilibrium and define displacement from it.
2. Check whether the restoring relationship is linear.
3. Find ω, period and phase from the stated conditions.
4. Use position or energy relationships for the required instant.
5. State whether damping, driving or large-angle effects invalidate the ideal approximation.

Equations and relationships

Investigate and explain

Time several oscillations for two masses on the same simulated spring. Compare T² with mass, keeping amplitude in the linear range.

Compare spring and pendulum oscillations while changing one quantity. Keep pendulum angles small when testing the SHM approximation, and use damping controls to distinguish free from driven behaviour.

Open the interactive investigation

Check your understanding