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AP Physics 1 — Algebra-Based
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Lesson 16 of 1910 minutes

Restoring Forces and Periodic Motion

Start with the lesson question, connect the representations, and test the model with evidence.

restoring forcesimple harmonic motionspringpendulumperiodoscillation energy

Learning objectives

  • Identify conditions for simple harmonic motion.
  • Connect force, energy, and motion representations.
  • Test how period depends on system parameters.
Lesson flowHook, model, explanationShow guidance

Inspect the opening phenomenon

Predict what changes, then name the evidence.

Apply in the lab

Name the evidence before reading the answer.

Read only what helps

Then use the lab and recall check.

More when needed

Transcript and resources stay available below.

Course progress

AP Physics 1 — Algebra-Based · Oscillations · Lesson 16

Restoring Forces and Periodic Motion

In progress

Decision challenge

Observe the phenomenon. Then connect the representations.

Use the opening example to make a prediction, identify evidence, and explain which model supports it.

Zero Speed, Maximum Acceleration | AP Physics 1

At maximum positive displacement, predict the directions and relative magnitudes of velocity, force, and acceleration.

Predict the acceleration at a turning point and test whether amplitude changes an ideal spring's period.

Before

At maximum positive displacement, predict the directions and relative magnitudes of velocity, force, and acceleration.

During

Track how force, acceleration, speed, spring potential energy, and kinetic energy change from endpoint to equilibrium.

After

Explain why doubling amplitude does not double the period of an ideal mass-spring oscillator.

Reference drawerTranscript, source notes, scripts, and package status stay tucked away until you need them.7 files

Lesson reading

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10 min

Video script

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Transcript fallback

available

courses/ap-physics-1/modules/07-oscillations/lessons/01-restoring-forces-and-periodic-motion/video-transcript.md

Test the Spring-Oscillator Period Model

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1 hr 25 min

Mastery check

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7 questions / 15 min

Book section:courses/ap-physics-1/modules/07-oscillations/lessons/01-restoring-forces-and-periodic-motion/book-section.md
Transcript for accessibility and fallback

# Accessible transcript: Zero Speed, Maximum Acceleration At a spring's turning point, speed is zero—but acceleration is greatest. At maximum positive displacement, the spring pulls hardest toward equilibrium: force equals negative k x. The block speeds up toward center. At equilibrium, force and acceleration are zero, but speed is maximum. Spring potential energy at the endpoint transforms into kinetic energy at the center. Quick check: if ideal amplitude doubles, does period double? Pause. No. Ideal spring period depends on mass and spring constant, not amplitude. Learn oscillations free at EduQuest AI. ## Visual description A block begins at maximum rightward extension with a large leftward restoring-force arrow, moves through equilibrium at maximum speed, and transfers spring potential energy into kinetic energy. A final prompt compares amplitude and period.

Reading lab

Core explanation

Connect the lesson's words, diagrams, graphs, evidence, and equations.

Driving question

How do restoring interactions produce repeated motion?

Before learning: endpoint prediction

A block on a spring reaches its rightmost point. Rank the magnitudes of displacement, velocity, acceleration, spring force, and kinetic energy at that instant. Then predict their values at equilibrium.

Hook: at the turning point, acceleration is greatest

At an oscillation endpoint, velocity is zero for an instant—but the restoring force and acceleration are largest in magnitude. Confusing velocity with acceleration hides the mechanism that sends the object back.

Periodic motion and simple harmonic motion

Periodic motion repeats after a period TT; frequency is

f=1T.f=\frac{1}{T}.

Simple harmonic motion (SHM) occurs when the net restoring force is proportional to and opposite displacement from equilibrium:

Fx=−kx.F_x=-kx.

Newton's second law gives

ax=−kmx=−ω2x,a_x=-\frac{k}{m}x=-\omega^2x,

where ω=k/m\omega=\sqrt{k/m}. The minus sign means acceleration points toward equilibrium.

Spring-mass restoring force at left, center, and right positions

Motion representations

One sinusoidal model is

x(t)=Acos⁡(ωt+ϕ).x(t)=A\cos(\omega t+\phi).

Then

v(t)=−Aωsin⁡(ωt+ϕ),a(t)=−ω2x(t).v(t)=-A\omega\sin(\omega t+\phi),\qquad a(t)=-\omega^2x(t).

Position and acceleration are opposite in phase. Velocity is shifted by one-quarter cycle.

Aligned position, velocity, and acceleration graphs for one SHM cycle

At x=±Ax=\pm A, speed is zero and acceleration magnitude is maximum. At x=0x=0, acceleration is zero and speed magnitude is maximum.

Spring period

For an ideal mass-spring oscillator,

T=2πmk.T=2\pi\sqrt{\frac{m}{k}}.

Period increases with the square root of mass and decreases with the square root of spring constant. In the ideal model, period does not depend on amplitude.

Worked example

A 0.50 kg0.50\text{ kg} mass is attached to a 200 N/m200\text{ N/m} spring:

T=2π0.50200=0.314 s.T=2\pi\sqrt{\frac{0.50}{200}}=0.314\text{ s}.

Quadrupling mass doubles the period. Quadrupling kk halves it.

Energy in an ideal spring oscillator

Mechanical energy is

E=12mv2+12kx2=12kA2.E=\frac12mv^2+\frac12kx^2=\frac12kA^2.

Kinetic, spring potential, and total energy versus position

At endpoints, energy is entirely spring potential. At equilibrium, it is entirely kinetic. With damping, mechanical energy decreases and amplitude shrinks.

Speed from energy

From energy conservation,

v=±ωA2−x2.v=\pm\omega\sqrt{A^2-x^2}.

The two signs correspond to passing the same position in opposite directions.

Pendulum approximation

For a simple pendulum of length LL at small angles,

T≈2πLg.T\approx2\pi\sqrt{\frac{L}{g}}.

The restoring torque is approximately proportional to angular displacement only when sin⁡θ≈θ\sin\theta\approx\theta in radians. The ideal small-angle period does not depend on bob mass and is approximately amplitude independent. At larger amplitudes, the approximation weakens.

During learning: representation audit

  1. Mark equilibrium and define positive displacement.
  2. Identify the restoring interaction and test whether F∝−xF\propto-x.
  3. Connect the signs of xx, vv, and aa at a stated instant.
  4. Use force for acceleration questions and energy for speed questions.
  5. Check ideal assumptions: negligible damping, linear spring, small pendulum angle.

Misconception clinic

“Acceleration is zero at an endpoint.” Velocity is zero; acceleration magnitude is maximum.

“The oscillator moves at constant speed.” Speed changes continuously, reaching maximum at equilibrium.

“Larger amplitude always changes period.” Not for an ideal spring or small-angle ideal pendulum.

“The pendulum period depends on bob mass.” Mass cancels in the ideal small-angle model.

“Any repeating motion is SHM.” SHM specifically requires acceleration proportional to negative displacement.

After learning: retrieval and transfer

  1. Where are speed and acceleration magnitude greatest?
  2. How does period change if spring mass becomes nine times larger?
  3. Why is a large-angle pendulum not exactly SHM?
  4. Sketch energy versus position over −A-A to +A+A.
  5. Describe evidence that damping is present.

AP-style evidence routine

  1. Define equilibrium, coordinate, system, and interval.
  2. Test the restoring-force condition.
  3. Select force, graph, energy, or period representation.
  4. State idealizations before applying a period formula.
  5. Solve symbolically and check units, phase, limits, and energy bounds.

Key takeaway

SHM is produced by a linear restoring force. Force, phase, energy, and period are mutually consistent views of the same oscillation.

Further learning and alignment

Practice labTest the Spring-Oscillator Period ModelOpen this when you are ready to apply the model, collect evidence, and check your explanation.1 hr 25 min

Lab: Test the Spring-Oscillator Period Model

Objective

Does the measured spring-mass period follow T=2πm/kT=2\pi\sqrt{m/k}, and is it independent of small amplitude within uncertainty?

Safety

Work under teacher supervision. Secure the support, wear eye protection if required, inspect the spring, use small masses, keep faces and feet away from hanging masses, add/remove mass only while supported, and place a catch tray below. Never exceed the spring's approved elastic range.

Materials

  • secured spring and stand;
  • small mass hanger and approved masses;
  • balance and meterstick;
  • stopwatch, motion sensor, or fixed video;
  • catch tray and high-contrast marker.

Low-cost alternative: a teacher-approved elastic element characterized over a verified linear range.

Simulation alternative: an approved spring simulation with exported data; identify omitted damping and measurement effects.

Steps

  1. Measure masses and instrument resolution.
  2. Determine kk from at least five static force-extension measurements within the linear range.
  3. Choose at least five oscillating masses. Include the hanger and a justified fraction of spring mass if required by the model.
  4. Displace each mass slightly, release without pushing, and time at least ten cycles.
  5. Repeat each condition at least three times.
  6. For one mass, repeat at three safe amplitudes to test amplitude independence.
  7. Preserve raw times, extensions, anomalies, and excluded trials.
  8. Plot T2T^2 versus mm and fit a line.

Expected Result

T2T^2 should be approximately proportional to mass, with slope near 4π2/k4\pi^2/k. Period should not change systematically with small amplitude within uncertainty.

Analysis

  • Calculate TT from multi-cycle timing and its trial spread.
  • Compare dynamic slope-derived kk with static kk.
  • Evaluate intercept and uncertainty.
  • Discuss reaction time, damping, spring mass, support motion, nonlinear extension, and amplitude measurement.
  • Distinguish random scatter from systematic bias.

Reflection Questions

  1. Why time many cycles rather than one?
  2. What graph linearizes the period model?
  3. Did amplitude affect period within uncertainty?
  4. What evidence shows the spring remained in its linear range?

Claim-evidence-reasoning conclusion

Claim whether data support the period model. Cite fitted slope, intercept, uncertainties, and amplitude trials, then connect evidence to the restoring-force model.

Accessibility

Offer safety, tactile setup, release, timing, recording, graphing, uncertainty, and narration roles. Use high-contrast/tactile markers and screen-reader-friendly tables. Learners may analyze shared data without handling masses.

Extension Challenge

Use the fitted line to predict the period for one new safe mass, measure it, and evaluate agreement with uncertainty.