IInsomniaStudyroom
Focus music
Pomodoro
25:00

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.

Lessons
Scalars and VectorsDisplacement, Velocity, and AccelerationRepresenting MotionReference Frames and Relative MotionMotion in Two or Three DimensionsSystems and Center of MassForces and Free-Body DiagramsNewton’s Third LawNewton’s First LawNewton’s Second LawGravitational ForceKinetic and Static FrictionSpring ForcesResistive ForcesCircular MotionTranslational Kinetic EnergyWorkPotential EnergyConservation of EnergyPowerLinear MomentumChange in Momentum and ImpulseConservation of Linear MomentumElastic and Inelastic CollisionsRotational KinematicsConnecting Linear and Rotational MotionTorqueRotational InertiaRotational Equilibrium and Newton’s First Law in Rotational FormNewton’s Second Law in Rotational FormRotational Kinetic EnergyTorque and WorkAngular Momentum and Angular ImpulseConservation of Angular MomentumRollingMotion of Orbiting SatellitesDefining Simple Harmonic Motion (SHM)Frequency and Period of SHMRepresenting and Analyzing SHMEnergy of Simple Harmonic OscillatorsSimple and Physical Pendulums
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics C: Mechanics · Unit 7 · Topic 7.4

Energy of Simple Harmonic Oscillators

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

1. Topic Lens

Energy of Simple Harmonic Oscillators 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

Pixel diagram for Energy of Simple Harmonic OscillatorsOriginal schematic connecting Energy of Simple Harmonic Oscillators to Oscillations.
Energy of Simple Harmonic Oscillators: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

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 7.4

Explain how energy of simple harmonic oscillators 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.