FOCUSED REVIEW

Focused Review: Simple Harmonic Motion — Calculus-Based

Reinforce the highest-leverage ideas and problem-solving tools for Simple Harmonic Motion.

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TIME

15–20 minutes

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BEST FOR

Focused reinforcement

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FINISH WITH

Greater confidence

After this focused review, you’ll be able to...

explain the essential relationships, apply the key methods, and continue with greater confidence.

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Course Alignment

This bundle follows the approved independent Physics Sensei unit specification. Use it to reinforce key concepts, prepare for coursework, or review before an assessment.

RESOURCE: Independent Physics Sensei Unit Review

UNIT: Mechanics • Unit MEC-U09

TOPIC: Simple Harmonic Motion

COURSE LEVEL: Calculus-Based

BEST USED

✓ After studying the unit

✓ Before starting homework

✓ Before a quiz or exam

Your Review Plan

Complete these six focused stages in order. Each stage reinforces the highest-leverage ideas and prepares you for a final confidence check.

6 Stages • Approximately 15–20 minutes

①

Warm-Up Check

Refresh key ideas.

②

Core Concepts

Reinforce the essentials.

③

Guided Practice

Strengthen key skills.

④

Confidence Check

Confirm your understanding.

⑤

Summary

Remember the essentials.

⑥

Next Step

Choose your next step.

Warm-Up Check

Distance is center-to-center and field follows an inverse-square rule. Gravity remains substantial in orbit; the support force is nearly absent.

WARM-UP 1

Key Ideas

Recall the defining SHM rule.

Write the defining SHM differential equation.

Reveal Answer

x¨ + ω²x = 0.

Why it works: The essential test is a restoring acceleration proportional to −displacement.

WARM-UP 2

Common Mistakes

Check the cycle quantities.

Differentiate x = A cos(ωt + φ) once and twice.

Reveal Answer

v = −Aω sin(ωt + φ); a = −Aω² cos(ωt + φ).

Why it works: Displacement, velocity, and acceleration have fixed phase relationships in ideal SHM.

WARM-UP 3

Quick Application

Recall the energy and pendulum limits.

State the approximation that makes a simple pendulum SHM.

Reveal Answer

sin θ ≈ θ for small θ in radians.

Why it works: Energy and the small-angle approximation provide two fast ways to check an SHM model.

Ready to strengthen your understanding?

You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?

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Continue to Core Concepts →

Core Concepts

Start with source, center-to-center distance, and direction; then choose force, field, energy, or orbit relationships.

KEY CONCEPT 1

SHM model and cycle relationships

Start from x¨ + ω²x = 0. Its sinusoidal solution has v = dx/dt and a = −ω²x, fixing the phase relations among x, v, and a.

x¨ + ω²x = 0; v = −Aω sin(ωt + φ); a = −ω²x

Example: For x = 0.020 cos(5t), v = −0.100 sin(5t) and a = −0.500 cos(5t).

Sensei Note: Do not treat amplitude, velocity, and acceleration as quantities that peak at the same location in the cycle.

KEY CONCEPT 2

Energy and standard oscillators

The spring energy integral follows from x¨ + (k/m)x = 0. The pendulum becomes linear only when sin θ ≈ θ, giving θ¨ + (g/L)θ = 0.

d/dt[½m ẋ² + ½kx²] = 0; θ¨ + (g/L)θ ≈ 0

Example: Near θ = 0, U(θ) = mgL(1 − cos θ) ≈ ½mgLθ², a quadratic potential.

Sensei Note: Use the small-angle pendulum formula only after checking that the angular amplitude is sufficiently small for the intended accuracy.

Ready to apply these ideas?

You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?

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Start Guided Practice →

Guided Practice

Apply the two highest-leverage methods in short activities. Reveal each solution only after attempting the problem.

PRACTICE 1

Guided Example

Solve one complete spring-cycle model.

For x = 0.040 cos(10t + π/4), find v(0) and a(0).

Reveal Answer

v(0) = −0.40 sin(π/4) ≈ −0.283 m/s; a(0) = −4.0 cos(π/4) ≈ −2.83 m/s².

Why it works: Use the restoring law first, then cycle relations; do not mix turning-point and equilibrium properties.

PRACTICE 2

Independent Check

Use energy or the small-angle pendulum relation.

Show how θ¨ = −(g/L)sin θ becomes SHM for small angle.

Reveal Answer

Use sin θ ≈ θ: θ¨ + (g/L)θ = 0, so ω = √(g/L).

Why it works: Energy is often the fastest route when position and speed are related; the pendulum requires an explicit approximation.

Ready to check your understanding?

You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?

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Start the Confidence Check →

Confidence Check

Complete these two short checks without looking back. Then use the scoring guide to decide your next step.

CONFIDENCE CHECK 1

SHM readiness check

Answer without a calculator.

If a = −25x, what is ω?

Reveal Answer

ω = 5 rad/s.

Why it works: Use a = −ω²x and T ∝ √m for an ideal spring oscillator.

CONFIDENCE CHECK 2

Energy and pendulum check

State one reason with each answer.

Why is a large-angle pendulum not exactly SHM?

Reveal Answer

Because sin θ is nonlinear and cannot be replaced accurately by θ at large angles.

Why it works: Turning points have v = 0; ideal linear-spring frequency depends on k and m; pendulum SHM requires small-angle linearization.

How did it go?

I answered ___ of 2 questions correctly.

1 correct: You’re ready to continue. 1 correct: Review the missed idea, then continue. 0 correct: Revisit Core Concepts or choose more practice.

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