Count full cycles: period and frequency
You will be able to: Calculate period and frequency from a count of complete cycles.
What does a stopwatch tell you about an oscillation?
A spring block returns to its rightmost point 10 times in 20 seconds, starting at that point. Ten complete cycles took 20 seconds: each cycle took 2 seconds. That is a frequency of half a cycle per second.
A useful starting point: Find equilibrium before measuring displacement →
Words and symbols before equations
- Cycle
- One complete repeat of position and direction of motion.
- Period T
- Time for one full cycle, in seconds (s).
- Frequency f
- Cycles per second, in hertz (Hz), equivalent to 1/s.
- Amplitude A
- Maximum displacement magnitude from equilibrium, in m for a spring block.
What this picture assumes
Exactly 10 complete cycles, no counting or timing error in this calculation. The graph shows an illustrative 8 s segment at the inferred period with amplitude 0.3 m, not the entire timing interval.
Connect the picture to the physics
For N complete cycles timed over Δt, T=Δt/N and f=N/Δt. Therefore f=1/T. A larger period means a smaller frequency: the same motion is described as time per cycle or cycles per time.
Passing the center twice does not automatically mean two cycles. Successive crossings in opposite directions are half a period apart. Start and stop at the same recognizable phase, such as successive rightmost positions or center crossings in the same direction.
Timing several complete cycles spreads the effect of start/stop reaction uncertainty over more cycles. Repeat the timing and compare results. This reduces some random timing uncertainty; it does not remove a consistent mistake in counting half-cycles or a systematic clock error.
| Quantity | Meaning | Unit |
|---|---|---|
| Period T | Time per full cycle | s |
| Frequency f | Full cycles per second | Hz = 1/s |
A worked example, step by step
A pendulum completes 12 full cycles in 24 s. Find T and f, then find the time between successive center crossings in opposite directions.
- T=Δt/N=24/12=2 s.
- f=N/Δt=12/24=0.5 Hz.
- Opposite-direction center crossings occur half a cycle apart.
- Their time separation is T/2=1 s. A same-direction return takes the full 2 s.
Frequency is not period. Check whether the requested unit is seconds or hertz, and count complete cycles.
How many full cycles occur in 6 s if T=2 s?
Compare with an explanation
Three: N=Δt/T=6/2.
Predict. Change one thing. Explain.
Change the total measured time while keeping the count at 10 full cycles. Predict how both T and f change. Then explain why a 20 s measurement of 10 half-cycles would give a different result.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
10 cycles in 20 s: T=2 s; f=0.5 Hz. Successive opposite-direction equilibrium crossings are 1 s apart. The plotted window is 8 s.
Exactly 10 complete cycles, no counting or timing error in this calculation. The graph shows an illustrative 8 s segment at the inferred period with amplitude 0.3 m, not the entire timing interval.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA block completes 15 full cycles in 30 s. (a) Find T. (b) Find f. (c) Predict the time for 25 cycles. (d) Explain why timing multiple cycles can improve the period estimate.
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Compare with the answer and four-point rubric
- 1 point: T=2 s.
- 1 point: f=0.5 Hz.
- 1 point: 25T=50 s.
- 1 point: The same start/stop timing uncertainty is divided by a larger cycle count; repeats help assess variability, while systematic errors still need attention.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Period unit?
Seconds per cycle, usually written s.
RECALL 2Frequency unit?
Hz, or cycles per second.
RECALL 3Relation between f and T?
f=1/T.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Count full cycles: period and frequency
- T=Δt/N; f=N/Δt=1/T.
- Repeat the same position and direction to count a full cycle.
Remember: Frequency is not period. Check whether the requested unit is seconds or hertz, and count complete cycles.
Conditions: Exactly 10 complete cycles, no counting or timing error in this calculation. The graph shows an illustrative 8 s segment at the inferred period with amplitude 0.3 m, not the entire timing interval.
Refresh Kid · Unit 7 · Objectives 7.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.2, objectives 7.2.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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