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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 2 · Topic 2.8

Spring Forces

Vector force models and differential equations determine center-of-mass motion in constrained and curving systems.

1. Topic Lens

Spring Forces is studied through force and translational dynamics. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[\sum\vec F=\frac{d\vec p}{dt}\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. For constant mass, momentum p=mv gives dp/dt=m dv/dt.
  2. When force is position-dependent, apply the chain rule dv/dt=(dv/dx)(dx/dt).
  3. Since dx/dt=v, the equation becomes F=mv dv/dx and can be integrated.
\[\sum F_x=m\frac{dv}{dt}=mv\frac{dv}{dx}\]

3. Detailed Visual Model

Pixel diagram for Spring ForcesOriginal schematic connecting Spring Forces to Force and Translational Dynamics.
Spring Forces: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

4. Worked Example and Lab Link

A constant net force of 8 N acts on a 2 kg mass. Find acceleration.

Answer: a=8/2=4 m/s².

Investigation idea: Measure force and acceleration for a changing-mass cart system and compare a numerical differential-equation model.

Common trap: Centripetal force is the inward net force, not an extra force to add to the free-body diagram.

Checkpoint · Topic 2.8

Explain how spring forces supports or limits this conclusion: a=8/2=4 m/s².

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