checkpoint 1

units/mechanics/unit-1.tex

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+\section{Kinematics}
+
+Kinematics describes motion by first choosing a reference frame and then tracking how an object's position vector $\vec{r}(t)$ changes with time. In AP Physics C: Mechanics, the focus is on motion in inertial frames, with 1D and 2D models that connect physical situations to equations, graphs, and vector components.
+
+This unit develops the core chain of ideas for motion: define position and displacement, differentiate locally to obtain $\vec{v}$ and $\vec{a}$, interpret motion graphs, use constant-acceleration relationships when applicable, and extend the same framework to relative motion and projectile motion with negligible air resistance.
+
+\input{concepts/mechanics/u1/m1-1-scalars-vectors}
+
+\input{concepts/mechanics/u1/m1-2-position-displacement}
+
+\input{concepts/mechanics/u1/m1-3-velocity-acceleration}
+
+\input{concepts/mechanics/u1/m1-4-motion-graphs}
+
+\input{concepts/mechanics/u1/m1-5-constant-acceleration}
+
+\input{concepts/mechanics/u1/m1-6-relative-projectile}

units/mechanics/unit-2.tex

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+\section{Force and Translational Dynamics}
+
+Dynamics explains changes in motion by relating the net external force on a chosen system to its acceleration through Newton's laws. In AP Physics C: Mechanics, this unit stays within inertial frames and emphasizes careful system definition, free-body reasoning, and consistent vector notation for quantities such as $\vec{F}$, $\vec{a}$, $\vec{g}$, $\vec{T}$, and $\vec{N}$.
+
+The flow of the unit begins with Newton's laws and center-of-mass ideas, then develops common force models such as gravitation, normal force, tension, friction, and springs. It closes with velocity-dependent forces and radial dynamics, including circular motion and circular orbits, with idealized strings, pulleys, and springs used throughout where appropriate.
+
+\input{concepts/mechanics/u2/m2-1-newton-laws}
+
+\input{concepts/mechanics/u2/m2-2-center-of-mass}
+
+\input{concepts/mechanics/u2/m2-3-gravitation}
+
+\input{concepts/mechanics/u2/m2-4-normal-tension}
+
+\input{concepts/mechanics/u2/m2-5-friction}
+
+\input{concepts/mechanics/u2/m2-6-springs}
+
+\input{concepts/mechanics/u2/m2-7-drag-terminal}
+
+\input{concepts/mechanics/u2/m2-8-circular-orbital}

units/mechanics/unit-3.tex

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+\section{Work, Energy, and Power}
+
+In this unit, we develop the energy viewpoint for mechanics. We begin with work as the accumulated effect of a force acting through a displacement, using the line-integral form $W=\int \vec{F}\cdot d\vec{r}$ as the calculus backbone and then connecting net work to changes in kinetic energy $K$.
+
+We then shift to conservative forces and potential energy $U$, which lets us track mechanical energy with $E_{\mathrm{mech}}=K+U$. With that accounting framework in place, we finish by defining power $P$ as the rate at which energy is transferred or transformed. The emphasis throughout stays on AP-level mechanical energy, including one-dimensional potential-energy graphs and clear bookkeeping of energy changes.
+
+\input{concepts/mechanics/u3/m3-1-work.tex}
+
+\input{concepts/mechanics/u3/m3-2-work-energy.tex}
+
+\input{concepts/mechanics/u3/m3-3-potential-energy.tex}
+
+\input{concepts/mechanics/u3/m3-4-energy-conservation.tex}
+
+\input{concepts/mechanics/u3/m3-5-power.tex}

units/mechanics/unit-4.tex

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+\section{Linear Momentum and Collisions}
+
+This unit develops the momentum viewpoint for interactions between objects and systems. We begin by defining linear momentum $\vec{p}$ for a particle and total momentum $\vec{P}$ for a system, using consistent vector notation and emphasizing that momentum conservation is applied componentwise in both one and two dimensions.
+
+We then connect external forces to momentum transfer through impulse $\vec{J}$, use isolated-system reasoning to establish conservation of momentum, and apply those ideas to collisions, recoil, and explosions. Throughout, the AP focus is on classifying collisions correctly, recognizing that kinetic energy is not always conserved, and relating system motion to the center-of-mass velocity $\vec{v}_{\mathrm{cm}}$ when helpful.
+
+\input{concepts/mechanics/u4/m4-1-linear-momentum.tex}
+
+\input{concepts/mechanics/u4/m4-2-impulse.tex}
+
+\input{concepts/mechanics/u4/m4-3-momentum-conservation.tex}
+
+\input{concepts/mechanics/u4/m4-4-collisions.tex}
+
+\input{concepts/mechanics/u4/m4-5-recoil-explosions.tex}

units/mechanics/unit-5.tex

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+\section{Torque and Rotational Dynamics}
+
+This unit extends translational mechanics to fixed-axis rotational motion. We begin with angular position $\theta$, angular velocity $\omega$, and angular acceleration $\alpha$, then connect those quantities to the corresponding linear quantities for points on a rigid body.
+
+We next develop the rotational analogs of force and mass through torque $\vec{\tau}$ and moment of inertia $I$, apply equilibrium ideas to situations with no angular acceleration, and conclude with the rotational dynamics relation $\sum \tau = I\alpha$ for planar rigid-body rotation in AP Physics scope.
+
+\input{concepts/mechanics/u5/m5-1-angular-kinematics.tex}
+
+\input{concepts/mechanics/u5/m5-2-linear-rotational-link.tex}
+
+\input{concepts/mechanics/u5/m5-3-torque.tex}
+
+\input{concepts/mechanics/u5/m5-4-moment-of-inertia.tex}
+
+\input{concepts/mechanics/u5/m5-5-rotational-equilibrium.tex}
+
+\input{concepts/mechanics/u5/m5-6-rotation-dynamics.tex}

units/mechanics/unit-6.tex

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+\section{Angular Momentum and Rolling Motion}
+
+This unit continues fixed-axis rotational mechanics by extending the work-energy ideas of Unit 5 to rotational kinetic energy and the rotational work-energy theorem. We then introduce angular momentum $\vec{L}$ and angular impulse, emphasizing how torque $\vec{\tau}$ changes rotational motion in the same way that force changes linear momentum.
+
+We next focus on conservation of angular momentum for isolated systems, apply that reasoning to rolling without slipping, and conclude with the AP Physics treatment of circular Newtonian orbits, including orbit speed and energy. Throughout, the emphasis is on fixed-axis rotation, rolling motion, and standard circular orbit results within AP scope.
+
+\input{concepts/mechanics/u6/m6-1-rotational-energy.tex}
+
+\input{concepts/mechanics/u6/m6-2-angular-momentum.tex}
+
+\input{concepts/mechanics/u6/m6-3-angular-momentum-conservation.tex}
+
+\input{concepts/mechanics/u6/m6-4-rolling.tex}
+
+\input{concepts/mechanics/u6/m6-5-orbits.tex}

units/mechanics/unit-7.tex

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+\section{Oscillations}
+
+This unit develops the mechanics of oscillatory motion about stable equilibrium. We begin with the defining linear restoring-force model for simple harmonic motion (SHM), write its governing differential equation, and connect the motion to sinusoidal solutions, angular frequency, period, and frequency.
+
+We then apply that framework to spring-mass oscillators, interpret SHM through energy, and finish with pendulum models. In AP Physics C: Mechanics, the pendulum results here are used in the small-angle regime, and the focus remains on undamped, unforced oscillations rather than broader circuit or driven-oscillation extensions.
+
+\input{concepts/mechanics/u7/m7-1-shm.tex}
+
+\input{concepts/mechanics/u7/m7-2-spring-oscillator.tex}
+
+\input{concepts/mechanics/u7/m7-3-shm-energy.tex}
+
+\input{concepts/mechanics/u7/m7-4-simple-pendulum.tex}
+
+\input{concepts/mechanics/u7/m7-5-physical-pendulum.tex}