Oscillations: SHM, damping and resonance
Key idea: Connect measured periodic motion to the defining SHM model, its phase and energy relationships, then distinguish damping from driven resonance.
Before you start: Circular Motion objective chainEnergy & Fields objective chain
By the end, you can
- Describe and investigate free oscillations, and use amplitude, period, frequency, angular frequency, phase and phase difference.
- Recognise SHM from a = −ω²x and use its displacement, velocity, acceleration and graphical relationships.
- Describe kinetic–potential energy interchange and compare light, critical and heavy damping in practical systems.
- Interpret forced-response curves, damping effects and useful or harmful resonance.
Starting-point self-check
1. Check your starting point
Attempt all four groups without notes and mark the first definition, phase sign, energy location or response-curve feature you could not justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Free oscillations, investigation and quantities 9(a)–(c)
Question 1
Describe an ideal free oscillation and outline how a student should obtain a reliable period for a vertical spring–mass oscillator.
Check the model response
After one displacement and release, the mass repeatedly returns through equilibrium without periodic driving and, in the ideal model, without energy exchange with the environment. Time many complete oscillations between the same directional crossing, repeat, divide each total time by the number of oscillations and compare the periods.
Question 2
Forty oscillations take 52.0 s. Find period, frequency and angular frequency. What phase difference corresponds to a delay of one quarter-period?
Check the model response
T = 52.0/40 = 1.30 s, f = 1/T = 0.769 Hz and ω = 2πf = 4.83 rad s⁻¹. A delay of T/4 corresponds to 2π(T/4)/T = π/2 rad or 90°.
repair
2. Repair the eight common breaks
Use only the repair matching an error, then redraw the equilibrium line, phase cycle, energy bars or response axes before retrying.
Free oscillations, investigation and quantities 9(a)–(c)
Check this idea
Misconception: Free oscillation means no force acts on the oscillator.
Repair: A restoring force must act. Free means no periodic driving; the ideal syllabus model also exchanges no energy with the environment.
Check this idea
Misconception: Timing one oscillation repeatedly is as precise as timing many oscillations.
Repair: The start–stop uncertainty is a smaller fraction of a longer multi-cycle interval. Repeats then reveal random scatter.
worked example
3. Follow four worked models
Follow how each solution fixes the timing reference, equilibrium direction, energy boundary and driving frequency before calculating.
Free oscillations, investigation and quantities 9(a)–(c)
Model 1
An oscillator reaches consecutive positive maxima at 0.35 s and 1.15 s. Find T, f and ω. A second oscillator reaches its maximum 0.20 s later; find its phase lag.
Check the model response
T = 1.15 − 0.35 = 0.80 s, f = 1.25 Hz and ω = 2π/T = 7.85 rad s⁻¹. The lag is 2π(0.20/0.80) = π/2 rad.
guided practice
4. Guided practice
Use each hint only to select a time-to-phase conversion, SHM equation, energy location or response-curve comparison.
Free oscillations, investigation and quantities 9(a)–(c)
Question 1
A student records 25 cycles in 18.5 s, 18.8 s and 18.6 s. Find the mean period and angular frequency, and state why the method is preferable to one cycle.
Hint: Average the total times before dividing by 25, then use ω = 2π/T.
Check the model response
Mean total time = 18.63 s, so T = 18.63/25 = 0.745 s and ω = 2π/T = 8.43 rad s⁻¹. Multi-cycle timing reduces fractional start–stop uncertainty and repeats expose scatter.
independent practice
5. Independent practice
Solve without repair notes and state the equilibrium reference, phase convention and energy or driving assumptions used.
Free oscillations, investigation and quantities 9(a)–(c)
Question 1
Two equal-frequency oscillators have period 1.60 s. Oscillator B passes equilibrium in the positive direction 0.60 s after A does so. Find their frequency, angular frequency and B's phase lag.
Check the model response
f = 1/1.60 = 0.625 Hz, ω = 2π/1.60 = 3.93 rad s⁻¹ and the lag is 2π(0.60/1.60) = 0.75π = 2.36 rad.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Free oscillations, investigation and quantities 9(a)–(c)
Question 1
Define amplitude and phase difference. An oscillator completes 36 cycles in 45.0 s; calculate T, f and ω and give one defensible improvement to a single stopwatch timing.
Check the model response
Amplitude is maximum displacement from equilibrium. Phase difference states how far one oscillation leads or lags another within a cycle. T = 45.0/36 = 1.25 s, f = 0.800 Hz and ω = 5.03 rad s⁻¹. Repeat timings of many cycles from the same directional reference and average.
Re-test practice
7. Delayed re-test practice
Return after at least three days and solve these fresh contexts without reopening earlier responses. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Free oscillations, investigation and quantities 9(a)–(c)
Question 1
A free oscillator has frequency 2.50 Hz. Find T and ω. A second trace reaches the same directional equilibrium crossing 0.060 s later; find its phase lag.
Check the model response
T = 1/f = 0.400 s, ω = 2πf = 15.7 rad s⁻¹ and the lag is 2π(0.060/0.400) = 0.30π = 0.942 rad.