Periodic Motion

1. Definitions (Must Know)

A. Periodic motion

Periodic motion is motion that repeats itself after a fixed time interval.

B. Period, T (s)

The period, T, is the time taken for one complete cycle.

C. Frequency, f (Hz)

The frequency, f, is the number of cycles per second.

Relationship: T = 1/f and f = 1/T

D. Angular frequency, ω (rad s⁻¹)

Angular frequency, ω, is the rate of change of phase: ω = 2π f = 2π/T

E. Displacement and amplitude

• displacement, x (m): distance from equilibrium (can be positive or negative)
• amplitude, x₀ (m): maximum magnitude of displacement

F. Phase and phase difference

• phase, φ (rad): where you are within a cycle
• phase difference, Δφ (rad): how far one oscillation leads/lags another

G. Free oscillation

An ideal free oscillation repeats about equilibrium with no periodic driving and no energy gained from or lost to the environment. A restoring force still acts; “free” does not mean force-free.

Phase difference (two oscillations)

Two sine waves with the same frequency. Oscillation B lags oscillation A by a constant phase difference.

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Values for Phase difference (two oscillations)

Phase angle, ωt (rad)	A: sin(ωt)	B: sin(ωt − π/3) (lags by π/3)
0	0	-0.866
0.524	0.5	-0.5
1.047	0.866	0
1.571	1	0.5
2.094	0.866	0.866
2.618	0.5	1
3.142	0	0.866
3.665	-0.5	0.5
4.189	-0.866	0
4.712	-1	-0.5
5.236	-0.866	-0.866
5.76	-0.5	-1
6.283	0	-0.866

Values for Phase difference (two oscillations)

Phase angle, ωt (rad)	A: sin(ωt)	B: sin(ωt − π/3) (lags by π/3)
0	0	-0.866
0.524	0.5	-0.5
1.047	0.866	0
1.571	1	0.5
2.094	0.866	0.866
2.618	0.5	1
3.142	0	0.866
3.665	-0.5	0.5
4.189	-0.866	0
4.712	-1	-0.5
5.236	-0.866	-0.866
5.76	-0.5	-1
6.283	0	-0.866

2. Key Ideas (What Earns Marks)

• Always include units:
  • T in seconds, f in hertz, ω in rad s⁻¹.
• Convert “revolutions per second” or “cycles per second” directly to frequency.
• Use ω = 2π f when you see sine/cosine SHM forms like sin(ω t).
• Measure several cycles and divide by the number of cycles to reduce the percentage timing uncertainty in T.

3. Detailed Explanations

A. Why ω = 2π f

One complete cycle corresponds to a phase change of 2π radians.

If there are f cycles per second, the phase changes by 2π f radians per second: ω = 2π f

B. Where these terms show up in SHM

In simple harmonic motion, displacement is often written in the form: x = x₀ sin(ω t + φ₀)

You do not need to memorise every variant. You mainly need:

• x₀ is the amplitude
• ω sets the period ( T = 2π/ω)
• φ₀ sets the starting point

C. Investigating an oscillator

Use a motion sensor or video tracking to record displacement against time. Read T between equivalent points such as successive maxima, or time N complete cycles and calculate T = t/N. Repeat the measurement and compare the periods; amplitude should be kept within any approximation stated for the oscillator.

The displacement–time graph establishes periodicity. To establish SHM, you must additionally show that acceleration is proportional to -x, for example with a straight-line a–x graph through the origin.

4. Common Mistakes

• Using T = 2π f (wrong; it is ω = 2π f and T = 1/f).
• Mixing up f (Hz) and ω (rad s⁻¹).
• Forgetting that phase must be in radians in expressions like sin(ω t).
• Calling a motion SHM merely because it repeats.

5. Exam Tips

• If you are given x = x₀ sin(ω t), read off ω directly and find T using T = 2π/ω.
• If you are given “rpm”, convert to Hz first: f = rpm/60.
• In an investigation, time several complete cycles rather than one whenever practical.

6. Worked Examples

7. Mind Stretchers