Oscillations
Describe oscillations, use the equations of simple harmonic motion and their graphs, follow the energy in SHM, and explain damping, forced oscillations and resonance.
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Before you begin
Pendulums, springs, guitar strings and swaying bridges all oscillate. The lessons start by describing free oscillations with amplitude, period, frequency and phase, then define simple harmonic motion by a = -ω²x and use its equations and graphs. They follow the energy in SHM, and finish with damping, forced oscillations and resonance.
Be comfortable with: angular velocity, because angular frequency ω = 2π f works the same way.
Learning goals
- Use oscillation quantities and describe free oscillations and their investigation.
- Relate displacement, velocity, acceleration and phase in simple harmonic motion.
- Identify and analyse simple harmonic motion using its defining equation and sinusoidal solutions.
- Describe kinetic–potential energy interchange in ideal simple harmonic motion.
- Compare light, critical and heavy damping and explain critical-damping applications.
- Distinguish free and forced oscillations, natural frequency and driving frequency.
- Interpret resonance response curves, damping effects and practical applications.
Syllabus statements covered
- describe simple examples of free oscillations, where particles periodically return to an equilibrium position without gaining energy from or losing energy to the environment
- investigate the motion of an oscillator using experimental and graphical methods
- show an understanding of and use the terms amplitude, period, frequency, angular frequency, phase and phase difference and express the period in terms of both frequency and angular frequency
- show an understanding that a = −ω²x is the defining equation of simple harmonic motion, where acceleration is (directly) proportional to displacement from an equilibrium position and acceleration is always directed towards the equilibrium position
- recognise and use x = x₀ sin ωt as a solution to the equation a = −ω²x
- recognise and use the equations v = v₀ cos ωt and v = ±ω√(x₀² − x²)
- describe, with graphical illustrations, the relationships between displacement, velocity and acceleration during simple harmonic motion
- describe the interchange between kinetic and potential energy during simple harmonic motion
- describe practical examples of damped oscillations, with particular reference to the effects of the degree of damping (light/under, critical, heavy/over), and to the importance of critical damping in applications such as a car suspension system
- describe graphically how the amplitude of a forced oscillation changes with driving frequency, resulting in maximum amplitude at resonance when the driving frequency is close to or at the natural frequency of the system
- show a qualitative understanding of the effects of damping on the frequency response and sharpness of the resonance
- describe practical examples of forced oscillations and resonance, and show an appreciation that there are some circumstances in which resonance is useful, and other circumstances in which resonance should be avoided
Lessons
Work through them in order.
Core lessons
- Free oscillations, investigation and quantitiesDescribe a free oscillation by its amplitude, period, frequency, angular frequency and phase, and measure them in the laboratory.
- The SHM restoring law and motion equationsUse the defining equation a = −ω²x, the solution x = x₀ sin ωt and v = ±ω√(x₀² − x²) to describe simple harmonic motion.
- Reading SHM graphs and phaseDraw and read displacement, velocity and acceleration graphs for SHM, and find phase differences between oscillations.
- Energy interchange in ideal SHMDescribe the interchange of kinetic and potential energy in SHM and calculate each at any displacement.
- Damped motion and response timeDescribe light, critical and heavy damping, and explain why critical damping matters in a car suspension.
- Forced oscillations, resonance and responseExplain forced oscillations and resonance, how damping changes the resonance curve, and when resonance is useful or harmful.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| period and frequency | f = 1/T | |
| angular frequency | ω = 2π f = 2π/T | rad s⁻¹ |
| phase difference | Δφ = (2πΔ t)/T | compare matching points |
| defining equation of SHM | a = -ω²x | acceleration towards equilibrium |
| displacement | x = x₀ sin ω t | starts at equilibrium |
| velocity | v = v₀ cos ω t, v₀ = ω x₀ | leads x by π/2 |
| velocity at displacement x | v = ±ω square root of (x₀²-x²) | sign from the direction of motion |
| total energy | E = (1/2)mω²x₀² | constant without damping |
Core knowledge to remember
- In a free oscillation, a body returns periodically to its equilibrium position without gaining energy from, or losing energy to, its surroundings.
- Simple harmonic motion: the acceleration is directly proportional to the displacement from the equilibrium position and is always directed towards that position.
- In SHM, kinetic energy is greatest at equilibrium and potential energy is greatest at the extremes; with no damping, their sum is constant.
- Critical damping returns a system to equilibrium in the shortest time without oscillating; light damping oscillates with decreasing amplitude, heavy damping returns slowly.
- Resonance is the maximum amplitude of a forced oscillation, when the driving frequency is close to or at the natural frequency. More damping makes the peak lower and broader.
Common mistakes
- Thinking any repeating motion is SHM; the acceleration must be proportional to -x.
- Putting the maximum speed at the maximum displacement; the body is at rest there.
- Thinking halving the displacement halves the potential energy; it quarters it.
- Thinking more damping always gives a quicker return to equilibrium.
- Saying a forced oscillator vibrates at its natural frequency; it follows the driving frequency.