IInsomniaStudyroom
Focus music
Pomodoro
25:00
BETAInsomnia StudyroomFrequency and Period of SHM
Frequency and Period of 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 7.2

Frequency and Period of SHM

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

Experimental relationship · Frequency and Period of SHMUse the overall pattern, spread, uncertainty, and possible outliers when evaluating the proposed relationship.

1. Topic Lens

Frequency and Period of 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

Frequency and Period of 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 7.2

Explain how frequency and period of 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.