More amplitude: more energy, same ideal period
You will be able to: Compare amplitude, maximum speed, maximum acceleration and total energy under controlled conditions.
Why does pulling a spring twice as far not double its period?
Release the same ideal spring block from 0.10 m and then from 0.20 m. The second trial travels twice as far each cycle, but it also moves faster. Both have the same period, while the second trial starts with four times the mechanical energy.
A useful starting point: Find speed from position using energy →
Words and symbols before equations
- Controlled comparison
- Change one input while keeping the stated other quantities fixed.
- Maximum acceleration a_max
- Largest acceleration magnitude, (k/m)A.
- External work W_ext
- Energy transferred to the selected system by an external interaction.
- Dissipation
- Conversion of mechanical energy into other forms, such as thermal energy, through resistance.
What this picture assumes
Separate ideal horizontal spring trials with m=1 kg and k=20 N/m held fixed. Reference amplitude is 0.2 m. Each trial receives its own starting energy; the slider does not inject work during an ongoing trajectory.
Connect the picture to the physics
At fixed m and k, T=2π√(m/k) does not depend on A. Maximum speed is A√(k/m), and maximum acceleration magnitude is (k/m)A. Both scale directly with amplitude, whereas E=½kA² scales with its square.
An amplitude change between trials requires a different energy input or initial condition. To increase amplitude from A_1 to A_2 without dissipation, the net added mechanical energy is ½k(A_2²−A_1²). Amplitude does not grow spontaneously in an isolated ideal oscillator.
Real oscillations often decay as resistance converts mechanical energy to thermal energy. Then the isolated constant-energy SHM model no longer describes the complete process. For this lesson, compare ideal trials separately and keep m and k fixed; changing mass or stiffness at the same time changes the timing too.
| Quantity | Dependence on A | Change |
|---|---|---|
| Period | Independent of A in ideal SHM | Unchanged |
| Maximum speed | Proportional to A | Doubles |
| Maximum acceleration | Proportional to A | Doubles |
| Total mechanical energy | Proportional to A² | Quadruples |
A worked example, step by step
A 1 kg block on a 20 N/m spring is released first from A=0.10 m and then from A=0.20 m. Compare energy, maximum speed and period.
- E_1=½(20)(0.10²)=0.10 J; E_2=0.40 J: four times larger.
- v_max,1=0.10√20≈0.447 m/s; v_max,2≈0.894 m/s: twice as large.
- Both periods are 2π√(1/20)≈1.405 s.
- The second release needs 0.30 J more starting mechanical energy. The longer path is traveled with proportionally larger speeds.
A constant ideal period does not mean unchanged speed, force or energy. Always specify what is held fixed.
To triple amplitude at fixed k, must energy triple?
Compare with an explanation
No. E∝A², so energy must become nine times as large.
Predict. Change one thing. Explain.
Change amplitude while holding m=1 kg and k=20 N/m fixed. Compare each state with the 0.20 m reference. Predict the factors for maximum speed, maximum acceleration, energy and period before moving the slider.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
A=0.2 m (1× reference); E=0.4 J (1×). v_max=0.894 m/s; a_max=4 m/s². T=1.405 s stays fixed. Both maxima scale by 1×.
Separate ideal horizontal spring trials with m=1 kg and k=20 N/m held fixed. Reference amplitude is 0.2 m. Each trial receives its own starting energy; the slider does not inject work during an ongoing trajectory.
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 ideal horizontal oscillator’s amplitude increases from 0.10 m to 0.30 m with k=40 N/m and the same mass. (a) Find initial and final energy. (b) Find the added energy. (c) Give the maximum-speed factor. (d) State the period change and justify it.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: E_i=0.20 J and E_f=1.80 J.
- 1 point: Added energy=1.60 J.
- 1 point: Maximum speed triples because it is proportional to A at fixed m,k.
- 1 point: Period is unchanged because T=2π√(m/k) and neither m nor k changed.
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 1At fixed m,k, which quantities double when A doubles?
Maximum speed and maximum acceleration magnitude.
RECALL 2How does energy change when A doubles?
It quadruples.
RECALL 3How can a real oscillation’s amplitude decay without destroying energy?
Mechanical energy is converted into thermal or other energy forms.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
More amplitude: more energy, same ideal period
- Fixed m,k: T independent of A; v_max∝A; a_max∝A; E∝A².
- Ideal added energy=½k(A_2²−A_1²).
Remember: A constant ideal period does not mean unchanged speed, force or energy. Always specify what is held fixed.
Conditions: Separate ideal horizontal spring trials with m=1 kg and k=20 N/m held fixed. Reference amplitude is 0.2 m. Each trial receives its own starting energy; the slider does not inject work during an ongoing trajectory.
Refresh Kid · Unit 7 · Objectives 7.3.A, 7.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.4, objectives 7.3.A, 7.4.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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