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Representing and Analyzing SHM

AP Course

AP Physics C: Mechanics

Updated for the AP Physics C: Mechanics framework effective fall 2024, with calculus-based derivations, labs, quizzes, and practice.

Build calculus-based mechanics with original derivations, experiments, quizzes, and practice sets.

AP Physics C: Mechanics · Unit 7 · Topic 7.3

Representing and Analyzing SHM

Differential equations and energy methods describe free, damped, and driven oscillations and resonance.

Experimental relationship · Representing and Analyzing SHMUse the overall pattern, spread, uncertainty, and possible outliers when evaluating the proposed relationship.

1. Topic Lens

Representing and Analyzing SHM is studied through oscillations. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[m\frac{d^2x}{dt^2}=-kx\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. Substitute Hooke's law into Newton's second law to form x''+(k/m)x=0.
  2. Test a sinusoidal solution; its second derivative is -ω² times itself.
  3. Matching coefficients gives ω²=k/m, while initial conditions determine A and φ.
\[x(t)=A\cos(\omega t+\phi),\quad\omega=\sqrt{\frac{k}{m}}\]

3. Detailed Visual Model

Representing and Analyzing SHMTopic-matched fine pixel-art model · use the lesson's equations and labeled diagrams for exact scientific meaning.

4. Worked Example and Lab Link

For k=9 N/m and m=1 kg, find the natural angular frequency.

Answer: ω=√(9/1)=3 rad/s.

Investigation idea: Fit oscillation data to a sinusoid, compare fitted angular frequency with √(k/m), and inspect residuals.

Common trap: Resonance amplitude is limited by damping; the driving frequency need not equal instantaneous oscillation phase.

Checkpoint · Topic 7.3

Explain how representing and analyzing shm supports or limits this conclusion: ω=√(9/1)=3 rad/s.

Official curriculum reference: College Board AP Physics C: Mechanics course page. The explanation and worked example are independently written for this study site.