Choose a sine or cosine model that matches the start
You will be able to: Interpret amplitude and frequency in a sinusoidal position equation and match the initial state.
Why can two equations describe the same kind of oscillation?
One spring block starts at its rightmost point; another starts at the center moving right. Their amplitudes and periods can be identical, yet their position graphs start at different places in the cycle. A cosine or sine can describe these different starts.
A useful starting point: Read the same oscillation on three graphs →
Words and symbols before equations
- Amplitude A
- Maximum displacement magnitude; never a signed starting position.
- Angular frequency ω
- Phase change per time, ω=2πf=2π/T in rad/s; the block need not physically rotate.
- Phase
- The location within the repeating cycle.
- Sine and cosine
- Functions that repeat after their argument increases by 2π radians.
What this picture assumes
A=0.4 m, T=4 s, ω=π/2 rad/s. Choose cosine for a rightmost release or sine for a center start moving right. Ideal undamped motion. This changes phase, not the amplitude or period.
Connect the picture to the physics
For release at x=+A with v=0 at t=0, choose x(t)=A cos(2πft)=A cos(ωt). At t=0 the cosine is 1; at T/4 it is zero; at T/2 it is −1. This matches the sequence of the released block.
For a start at equilibrium moving right, x(t)=A sin(ωt) matches x(0)=0 and initially increasing x. Changing the start changes phase, not the period or amplitude. More general initial states require a phase shift; the two simple forms here cover two common choices.
The coefficient inside the sine or cosine is angular frequency, not ordinary frequency. If x=0.20 cos(4πt) m, then ω=4π rad/s, f=2 Hz and T=0.5 s. Use radian mode when evaluating the argument. For a spring, a=−ω²x with ω²=k/m.
A worked example, step by step
An oscillator follows x(t)=0.40 cos[(π/2)t] m, with t in s. Find A, f, T and x at t=1 s, then describe the initial state.
- A=0.40 m and ω=π/2 rad/s.
- f=ω/(2π)=1/4 Hz; T=1/f=4 s.
- At t=1 s, the argument is π/2 and x=0.
- At t=0, x=+0.40 m and v=0: a rightmost release. At 1 s it crosses equilibrium moving left.
The coefficient of t in cos(ωt) is angular frequency in rad/s, not f in Hz.
Does changing from cosine to sine automatically change the frequency?
Compare with an explanation
No. With the same coefficient of t, the period is unchanged; the starting phase changes.
Predict. Change one thing. Explain.
Switch between a rightmost release and an equilibrium start moving right. Keep A=0.40 m and T=4 s. Move time to 0, 1 and 2 s; compare the phase shift without changing amplitude or period.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
A=0.4 m; ω=π/2 rad/s; f=0.25 Hz; T=4 s. At t=1 s: x=0 m, v=-0.628 m/s, a=0 m/s². The start is at +A at rest.
A=0.4 m, T=4 s, ω=π/2 rad/s. Choose cosine for a rightmost release or sine for a center start moving right. Ideal undamped motion. This changes phase, not the amplitude or period.
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 questionAn oscillator follows x=0.20 cos(πt) m. (a) Identify A and ω. (b) Find f and T. (c) Find x at 0.5 s and 1 s. (d) Describe the acceleration sign at t=0.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: A=0.20 m and ω=π rad/s.
- 1 point: f=0.5 Hz and T=2 s.
- 1 point: x(0.5)=0; x(1)=−0.20 m.
- 1 point: Acceleration is negative because x>0 and a=−ω²x.
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 1What does A set in a position equation?
Maximum displacement magnitude.
RECALL 2How are angular frequency and frequency related?
ω=2πf.
RECALL 3What changes when sine replaces cosine with the same A and ω?
The phase or starting state, not amplitude or period.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Choose a sine or cosine model that matches the start
- Rightmost release: x=A cos(2πft).
- Equilibrium start moving right: x=A sin(2πft).
- ω=2πf=2π/T; for an ideal spring, a=−ω²x.
Remember: The coefficient of t in cos(ωt) is angular frequency in rad/s, not f in Hz.
Conditions: A=0.4 m, T=4 s, ω=π/2 rad/s. Choose cosine for a rightmost release or sine for a center start moving right. Ideal undamped motion. This changes phase, not the amplitude or period.
Refresh Kid · Unit 7 · Objectives 7.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.3, objectives 7.3.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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