Describe the quantities of simple harmonic motion
Define displacement, amplitude, period, frequency, angular frequency and phase difference, then connect the three measures of cycle timing.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 17. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use displacement, velocity, acceleration, force, energy and trigonometric functions with SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is amplitude in an oscillation?
What you will learn
- Use displacement and amplitude for oscillations about an equilibrium position
- Relate period and frequency
- Use angular frequency and phase difference in oscillations
Measure motion from its equilibrium position
Displacement x is the signed distance from the fixed equilibrium position. Amplitude x₀ is the maximum magnitude of that displacement; it is a positive measure of the largest excursion. An oscillator passes through equilibrium repeatedly, so its displacement changes sign.
The period T is the time for one complete cycle. Frequency f is the number of cycles per second, so f = 1/T. Use seconds for T and hertz for f.
Angular frequency and phase describe cycle timing
Angular frequency ω measures phase angle per second. One complete cycle is 2π radians, so ω = 2πf = 2π/T, in rad s⁻¹. A larger frequency therefore means a larger angular frequency and a shorter period.
Phase difference compares where two oscillations are in their cycles. It may be expressed in radians or degrees; a difference of 2π rad or 360° is a full cycle. State whether a phase leads or lags when comparing timing.
Pause and explain
An oscillator has period 0.50 s. What is its frequency?
Worked example
An oscillator completes one cycle in 0.40 s. Find its frequency and angular frequency.
Show the worked solution
- Use f = 1/T.
- Use ω = 2πf or 2π/T.
- Attach Hz and rad s⁻¹ to the results.
Answer f = 2.5 Hz and ω = 5π rad s⁻¹ ≈ 15.7 rad s⁻¹.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
An oscillator has period 0.25 s. Calculate its frequency and angular frequency.
Give me a hint
Find the reciprocal of T, then multiply f by 2π.
Compare my reasoning
- f = 1/T = 1/0.25 = 4.0 Hz.
- ω = 2πf = 8π rad s⁻¹.
- Give the angular frequency as about 25.1 rad s⁻¹ if a decimal is needed.
f = 4.0 Hz; ω = 8π rad s⁻¹ ≈ 25.1 rad s⁻¹.
Look for these in your work
- Period reciprocated
- Angular relation used
- Units stated
Two oscillations have phase difference π/2 rad. Express this as a fraction of a cycle and in degrees.
Give me a hint
A full cycle is 2π rad or 360°.
Compare my reasoning
- The cycle fraction is (π/2)/(2π) = 1/4.
- The angle in degrees is (π/2)(180°/π).
- State both equivalent measures.
The phase difference is one quarter of a cycle, or 90°.
Look for these in your work
- Full-cycle phase used
- Cycle fraction calculated
- Degree conversion correct
A student says a pendulum with amplitude 8 cm always has displacement 8 cm. Explain the error.
Give me a hint
Compare the maximum excursion with the signed position at one instant.
Compare my reasoning
- Amplitude is the maximum displacement magnitude.
- The pendulum passes through equilibrium, where displacement is zero, and moves to both sides.
- Its instantaneous displacement varies between −8 cm and +8 cm.
The amplitude is 8 cm, but displacement changes through the cycle from −8 cm to +8 cm.
Look for these in your work
- Amplitude defined
- Instantaneous displacement distinguished
- Range and sign stated
Common mistakes
- Confusing amplitude with the current displacement
- Using f = T instead of f = 1/T
- Giving angular frequency in hertz rather than rad s⁻¹
- Comparing phase without identifying which oscillation leads
What can you explain now?
What is amplitude in an oscillation?
Compare with the explanation
The maximum displacement magnitude from equilibrium
Amplitude is the largest distance from the fixed equilibrium position.
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Make it stick
How do period, frequency and angular frequency describe the same cycle on different scales?
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