The shape of the lesson
By the end of the lesson, learners can
- Free, damped, driven: A real oscillator loses energy to resistive forces and its amplitude decays: it is damped.
- Damping types and resonance: Light damping lets the oscillations continue with a slowly decaying amplitude; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping returns it slowly without oscillating.
- Defining condition: a = −ω²x, with ω = 2πf = 2π / T.
- Velocity: v = ±ω√(x₀² − x²), greatest (ωx₀) at the centre and zero at the extremes.
- Energy: total E = ½mω²x₀², shared between KE (greatest at the centre) and PE (greatest at the extremes).
One picture every idea returns to
Resistive forces damp an oscillation, removing energy and reducing amplitude. A driven oscillator shows resonance, a maximum amplitude, when the driving frequency equals the natural frequency; damping lowers and broadens the resonance peak.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Damping, driving, and resonance.. Real oscillators lose energy and can be driven. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Free, damped, driven | A real oscillator loses energy to resistive forces and its amplitude decays: it is damped. Project tab 3 of the lab (Damping and resonance) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Damping types and resonance | Light damping lets the oscillations continue with a slowly decaying amplitude; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping returns it slowly without oscillating. | Cooperative Concept Mapping |
| 30 to 43 min | Check and plenary | The quick-check questions from the student lesson, answered on mini whiteboards one at a time; re-teach the one most get wrong, then a random call on the last. | Individual, then whole class |
| 43 to 45 min | Exit and homework | Set the worksheet below (3 questions, 34 marks) from the topic practice set. Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Damping types and resonance, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
No examiner trap is recorded for this lesson. Use the quiz key below to spot the distractor most of the class picks.
Damping and resonance: topic worksheet
3 questions, 34 marks. Written in the style of the papers; no past paper question is reproduced. These questions cover the whole topic, so sibling lessons share them.
An oscillating system can be damped, and can also be driven by an external periodic force. Fig. 3.1 shows how the amplitude of the driven oscillations varies with driving frequency for two amounts of damping.
(a) Explain what is meant by damping. [2]
(b) Describe the displacement–time behaviour for light damping, critical damping and heavy damping. [3]
(c) State what is meant by the natural frequency of an oscillator. [1]
(d) Explain what resonance is and state the condition for it to occur. [2]
(e) Using Fig. 3.1, state and explain the effect of increasing the damping on the resonance peak. [2]
(f) Give one situation in which resonance is useful and one in which it is a problem. [2]
This question is about free and forced oscillations, and damping.
(a) Distinguish between free and forced oscillations. [2]
(b) Explain what is meant by damping. [2]
(c) Describe the difference between light, critical and heavy damping. [3]
(d) State what happens to the total energy of a lightly damped oscillator over time. [1]
(e) Give one practical situation where critical damping is useful, and say why. [2]
A mass on a spring is driven by an external periodic force of variable frequency. Fig. 10.1 shows the response. This question links the SHM, its energy and resonance.
(a) State what is meant by the natural frequency of the system. [1]
(b) The mass is 0.25 kg and the spring gives ω = 20 rad s⁻¹. Calculate the natural frequency. [2]
(c) Explain what is meant by resonance and state the driving frequency at which it occurs. [2]
(d) At resonance the amplitude is 6.0 cm. Calculate the total energy of the oscillation. [3]
(e) Describe and explain how increasing the damping changes the resonance peak, as in Fig. 10.1. [2]
(f) State one situation where resonance is useful and one where it must be avoided. [2]
Answer key and marking notes
- (a)Damping is the action of resistive forces ✓ that remove energy from the oscillator, so its amplitude decreases with time ✓
- (b)Light damping: oscillations continue but the amplitude decays gradually ✓; critical damping: returns to equilibrium in the shortest time without oscillating ✓; heavy damping: returns to equilibrium slowly without oscillating ✓
- (c)The frequency at which the system oscillates freely when displaced and released ✓
- (d)Resonance is a maximum amplitude response ✓, occurring when the driving frequency equals the natural frequency of the system ✓
- (e)Increasing the damping lowers the peak amplitude and broadens it ✓; the peak also shifts to a slightly lower frequency, because more energy is dissipated each cycle ✓
- (f)Useful: tuning a radio receiver, or magnetic resonance imaging ✓; problem: a bridge or building driven by wind or an earthquake, which can be damaged ✓
- (a)A free oscillation is at the natural frequency with no external periodic force ✓; a forced oscillation is driven by an external periodic force ✓
- (b)A resistive force removes energy from the oscillator, so the amplitude decreases with time ✓✓
- (c)Light: the amplitude decays gradually over many oscillations ✓; critical: returns to equilibrium in the shortest time without oscillating ✓; heavy: returns slowly without oscillating ✓
- (d)It decreases, being dissipated mainly as thermal energy ✓
- (e)Car suspension or a door closer: it returns to equilibrium quickly without overshooting or oscillating ✓✓
- (a)The frequency at which the system oscillates freely when displaced and released ✓
- (b)f₀ = ω/2π = 20/(2π) ✓ = 3.2 Hz ✓
- (c)The amplitude becomes very large when the driving frequency equals the natural frequency ✓, when energy transfer to the oscillator is greatest; this occurs at f₀ ✓
- (d)E = ½mω²x₀² = ½ × 0.25 × 20² × 0.060² ✓✓ = 0.18 J ✓
- (e)The peak becomes lower and broader ✓, with the maximum amplitude reduced (and shifted to a slightly lower frequency) ✓
- (f)Useful: tuning a radio or a musical instrument; to be avoided: bridges or buildings driven by wind or earthquakes ✓✓
The lesson's own checks as a quiz
The quick-check and exam-style questions from the student lesson, printable with the key.
Damping of an oscillation is caused by:
- Aresistive forces that remove energy from the system
- Ban increase in the natural frequency
- Cthe driving force
- Da larger amplitude
Critical damping is the case in which the system:
- Areturns to equilibrium in the shortest time without oscillating
- Boscillates with a slowly decaying amplitude
- Cnever returns to equilibrium
- Dreturns to equilibrium very slowly
Resonance, the maximum amplitude of a driven oscillator, occurs when the driving frequency is:
- Aequal to the natural frequency
- Bas low as possible
- Cas high as possible
- Dtwice the natural frequency
Increasing the damping of a driven system makes the resonance peak:
- Alower and broader
- Btaller and sharper
- Cunchanged
- Dshift to a higher frequency
Quiz key
Damping is due to resistive (dissipative) forces that remove energy, so the amplitude decreases with time.
Critical damping gives the quickest return to equilibrium with no overshoot or oscillation.
Resonance happens when the driving frequency matches the natural frequency, so energy is transferred most effectively.
More damping dissipates more energy each cycle, lowering and broadening the peak and shifting it slightly to a lower frequency.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.