Learning objectives
- I can explain and apply: Simple harmonic motion.
- I can explain and apply: Period and phase.
- I can explain and apply: Energy and turning points.
Panhellenic Exams / Physics / Two-unit revision pilot
Oscillations
Oscillations: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
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completed
Estimated active study time
156 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- T=2π√(m/k) for an ideal mass–spring system.
- vmax=Aω and amax=Aω².
- Doubling amplitude quadruples energy.
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
Understanding the concept
In ideal simple harmonic motion, restoring force is proportional and opposite to displacement: F=−Dx. For an ideal horizontal frictionless spring, D=k and ω=√(k/m). Measure displacement from equilibrium.
Method and conditions
For x=A·sin(ωt+φ), velocity is Aω·cos(ωt+φ) and acceleration is −ω²x. Period T=2π/ω does not depend on amplitude in the ideal linear model. Numerical phase exercises here use φ=0.
Checking and coverage boundaries
Total mechanical energy is ½DA². At turning points velocity is zero but force and acceleration magnitudes are greatest. At equilibrium speed is greatest. This pilot does not include damped or driven oscillations.
Original pilot material — human educator review pending
Written specifically for YourFavTeacher with AI assistance, without copying past-paper prompts. Technical and numerical checks do not replace review by a human educator. It is not an official paper or endorsed by an examination body.
Physics
Worked examples
Follow the method step by step and check why every step is valid.
Worked example 1
An ideal horizontal spring system has m=1 kg and k=36 N/m. Find ω and T.
- 1For an ideal spring, ω=√(k/m) and T=2π/ω.
- 2ω=√36=6 rad/s; T=2π/6 s
- 3Period is independent of amplitude in the ideal model.
ω=6 rad/s; T=2π/6 s
Worked example 2
For SHM with ω=6 rad/s and x=0.1 m, find acceleration.
- 1Acceleration has the opposite sign to displacement.
- 2a=−ω²x=−6²·0.1=−3.6 m/s²
- 3The negative sign points towards equilibrium.
−3.6 m/s²
Worked example 3
At an oscillation endpoint, velocity is momentarily zero. Is the restoring force also zero?
- 1F=−kx
- 2At an endpoint |x|=A, so |F|=kA.
- 3The force towards equilibrium reverses the motion.
No; the force magnitude is maximal.
Oscillations
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency6 minutes
An ideal horizontal spring system has m=1 kg and k=9 N/m. Find ω and T.
Hint
For an ideal spring, ω=√(k/m) and T=2π/ω.
Answer guide
For an ideal spring, ω=√(k/m) and T=2π/ω. ω=√9=3 rad/s; T=2π/3 s Period is independent of amplitude in the ideal model. ω=3 rad/s; T=2π/3 s
Core fluency6 minutes
Simple harmonic motion has A=0.2 m and ω=3 rad/s. Find maximum speed.
Hint
Maximum speed occurs at equilibrium: vmax=ωA.
Answer guide
Maximum speed occurs at equilibrium: vmax=ωA. υmax=3·0.2=0.6 m/s Speed is zero at the endpoints, not maximum. 0.6 m/s
Core fluency6 minutes
A spring with k=100 N/m oscillates with amplitude A=0.02 m. Find total energy.
Hint
In the ideal system E=kA²/2.
Answer guide
In the ideal system E=kA²/2. E=0.5·100·(0.02)²=0.02 J Amplitude is in metres and is squared. 0.02 J
Application6 minutes
At which positions in simple harmonic motion is acceleration magnitude greatest?
Hint
a=−ω²x
Answer guide
a=−ω²x Its magnitude increases with |x|. The greatest |x| is amplitude A. At the endpoints x=±A, directed towards equilibrium.
Application6 minutes
For SHM with ω=3 rad/s and x=0.1 m, find acceleration.
Hint
Acceleration has the opposite sign to displacement.
Answer guide
Acceleration has the opposite sign to displacement. a=−ω²x=−3²·0.1=−0.9 m/s² The negative sign points towards equilibrium. −0.9 m/s²
Application6 minutes
Double the amplitude of an ideal spring without changing mass or stiffness. How do period and energy change?
Hint
T=2π√(m/k)
Answer guide
T=2π√(m/k) E′=k(2A)²/2=4E The model is ideal, lossless, and obeys Hooke's law. Same period, four times the energy.
Reasoning6 minutes
In SHM, x=0.2sin(3t) in SI. Find position and direction of motion at t=0.
Hint
Velocity is the derivative of position.
Answer guide
Velocity is the derivative of position. x(0)=0; υ(t)=0.6cos(3t); υ(0)=0.6>0 x=0 does not mean the body is stationary. x(0)=0 m; υ(0)=0.6 m/s > 0
Reasoning6 minutes
At an oscillation endpoint, velocity is momentarily zero. Is the restoring force also zero?
Hint
F=−kx
Answer guide
F=−kx At an endpoint |x|=A, so |F|=kA. The force towards equilibrium reverses the motion. No; the force magnitude is maximal.
Common mistakes
- Confusing instantaneous and maximum velocity.
- Assuming zero force at a turning point because v=0.
- Using centimetres in an SI-unit calculation.
Mastery check
- I can solve independently and check: Simple harmonic motion.
- I can solve independently and check: Period and phase.
- I can solve independently and check: Energy and turning points.
Oscillations
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
An ideal horizontal spring system has m=1 kg and k=16 N/m. Find ω and T.
Hint
For an ideal spring, ω=√(k/m) and T=2π/ω.
Answer guide
For an ideal spring, ω=√(k/m) and T=2π/ω. ω=√16=4 rad/s; T=2π/4 s Period is independent of amplitude in the ideal model. ω=4 rad/s; T=2π/4 s
Core8 minutes
Simple harmonic motion has A=0.2 m and ω=4 rad/s. Find maximum speed.
Hint
Maximum speed occurs at equilibrium: vmax=ωA.
Answer guide
Maximum speed occurs at equilibrium: vmax=ωA. υmax=4·0.2=0.8 m/s Speed is zero at the endpoints, not maximum. 0.8 m/s
Stretch8 minutes
A spring with k=100 N/m oscillates with amplitude A=0.03 m. Find total energy.
Hint
In the ideal system E=kA²/2.
Answer guide
In the ideal system E=kA²/2. E=0.5·100·(0.03)²=0.045 J Amplitude is in metres and is squared. 0.045 J
Stretch8 minutes
At which positions in simple harmonic motion is acceleration magnitude greatest?
Hint
a=−ω²x
Answer guide
a=−ω²x Its magnitude increases with |x|. The greatest |x| is amplitude A. At the endpoints x=±A, directed towards equilibrium.
Challenge8 minutes
For SHM with ω=4 rad/s and x=0.1 m, find acceleration.
Hint
Acceleration has the opposite sign to displacement.
Answer guide
Acceleration has the opposite sign to displacement. a=−ω²x=−4²·0.1=−1.6 m/s² The negative sign points towards equilibrium. −1.6 m/s²
Challenge8 minutes
Double the amplitude of an ideal spring without changing mass or stiffness. How do period and energy change?
Hint
T=2π√(m/k)
Answer guide
T=2π√(m/k) E′=k(2A)²/2=4E The model is ideal, lossless, and obeys Hooke's law. Same period, four times the energy.
45 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Show working, units and a final check. Equivalent correct methods are accepted. Timing is a revision guide, not official examination conditions.
1. An ideal horizontal spring system has m=1 kg and k=25 N/m. Find ω and T.
5 marksAnswer guide
For an ideal spring, ω=√(k/m) and T=2π/ω. ω=√25=5 rad/s; T=2π/5 s Period is independent of amplitude in the ideal model. ω=5 rad/s; T=2π/5 s
2. Simple harmonic motion has A=0.2 m and ω=5 rad/s. Find maximum speed.
5 marksAnswer guide
Maximum speed occurs at equilibrium: vmax=ωA. υmax=5·0.2=1 m/s Speed is zero at the endpoints, not maximum. 1 m/s
3. A spring with k=100 N/m oscillates with amplitude A=0.04 m. Find total energy.
5 marksAnswer guide
In the ideal system E=kA²/2. E=0.5·100·(0.04)²=0.08 J Amplitude is in metres and is squared. 0.08 J
4. At which positions in simple harmonic motion is acceleration magnitude greatest?
5 marksAnswer guide
a=−ω²x Its magnitude increases with |x|. The greatest |x| is amplitude A. At the endpoints x=±A, directed towards equilibrium.
5. For SHM with ω=5 rad/s and x=0.1 m, find acceleration.
5 marksAnswer guide
Acceleration has the opposite sign to displacement. a=−ω²x=−5²·0.1=−2.5 m/s² The negative sign points towards equilibrium. −2.5 m/s²
6. Double the amplitude of an ideal spring without changing mass or stiffness. How do period and energy change?
5 marksAnswer guide
T=2π√(m/k) E′=k(2A)²/2=4E The model is ideal, lossless, and obeys Hooke's law. Same period, four times the energy.
7. In SHM, x=0.2sin(5t) in SI. Find position and direction of motion at t=0.
5 marksAnswer guide
Velocity is the derivative of position. x(0)=0; υ(t)=1cos(5t); υ(0)=1>0 x=0 does not mean the body is stationary. x(0)=0 m; υ(0)=1 m/s > 0
8. At an oscillation endpoint, velocity is momentarily zero. Is the restoring force also zero?
5 marksAnswer guide
F=−kx At an endpoint |x|=A, so |F|=kA. The force towards equilibrium reverses the motion. No; the force magnitude is maximal.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Physics
Strict methodology by question: diagram, principles/laws, equations, numerical accuracy, and qualitative explanation.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Simple harmonic motion
Period and phase
Energy and turning points
Where to focus
The areas that usually create mistakes or need extra revision.
I can solve independently and check: Simple harmonic motion.
I can solve independently and check: Period and phase.
I can solve independently and check: Energy and turning points.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
An ideal horizontal spring system has m=1 kg and k=9 N/m. Find ω and T.
Simple harmonic motion has A=0.2 m and ω=3 rad/s. Find maximum speed.
A spring with k=100 N/m oscillates with amplitude A=0.02 m. Find total energy.
At which positions in simple harmonic motion is acceleration magnitude greatest?
For SHM with ω=3 rad/s and x=0.1 m, find acceleration.
Double the amplitude of an ideal spring without changing mass or stiffness. How do period and energy change?
In SHM, x=0.2sin(3t) in SI. Find position and direction of motion at t=0.
At an oscillation endpoint, velocity is momentarily zero. Is the restoring force also zero?
Mocks and progress checks
How to measure progress in this chapter and when it enters a cumulative mock.
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.