AP Physics

AP Physics C: Mechanics Complete Study Guide & Formula Reference

Overview

• AP Physics C: Mechanics focuses on calculus-based derivations and conceptual explanations.

• The course covers 7 major units: Kinematics, Newton's Laws, Energy, Momentum, Rotation, Oscillations, and Gravitation.

• Exam weight: Approximately 50% of the AP Physics C: Mechanics score.

• Calculator policy: Calculators are permitted in Section II (Free Response).

• A formula sheet is provided; however, fluency with each formula is essential.

• Key prerequisite knowledge includes single-variable calculus, specifically derivatives and integrals.

Contents

• Part I: Complete Review Guide
    - Unit 1: Kinematics
    - Unit 2: Newton's Laws of Motion
    - Unit 3: Work, Energy, and Power
    - Unit 4: Systems of Particles and Linear Momentum
    - Unit 5: Rotation
    - Unit 6: Oscillations (Simple Harmonic Motion)
    - Unit 7: Gravitation
    - AP Exam Strategy

• Part II: Must-Know Formula Reference
    - Kinematics Formulas
    - Newton's Laws and Forces
    - Work, Energy, and Power
    - Momentum and Impulse
    - Rotation
    - Oscillations
    - Gravitation
    - How to choose the appropriate approach

Unit 1: Kinematics

• Definition: Describing motion with the tools of calculus, focusing on how motion is quantified rather than the reasons behind it.

• Fundamental Calculus Relationships:
    - Position, velocity, and acceleration are interconnected through differentiation and integration.
    - The relationships are:
      - Velocity from position: $v(t) = rac{dx}{dt}$ (differentiate position w.r.t. time)
      - Acceleration from velocity: $a(t) = rac{dv}{dt} = rac{d^2x}{dt^2}$ (differentiate velocity; a second derivative of position)
      - Position from velocity: $x(t) = x_0 + \int v(t) \, dt$ (integrate velocity, apply initial condition)
      - Velocity from acceleration: $v(t) = v_0 + \int a(t) \, dt$ (integrate acceleration)

• Key AP Tip: If acceleration or force is not constant, integration is necessary. The kinematic equations for constant acceleration do not apply.

Constant Acceleration Equations

• When acceleration is constant, integral evaluations yield the following equations:
    - Velocity: $v = v_0 + at$
    - Position: $x = x_0 + v_0 t + \frac{1}{2} at^2$
    - Velocity squared: $v^2 = v_0^2 + 2a \Delta x$
    - Average velocity: $x_{avg} = x_0 + \frac{1}{2} (v_0 + v) t$

2D Kinematics and Projectile Motion

• In two dimensions:
    - Horizontal component is independent (constant velocity, no acceleration): $x(t) = x_0 + v_{0x} t$
    - Vertical component includes gravitational acceleration: $y(t) = y_0 + v_{0y} t - \frac{1}{2} gt^2$
    - At the peak of projectile motion: $v_{y} = 0$ (this does not imply zero total velocity).
    - The range of projectile motion (from the same elevation): $R = rac{v_0^2 \sin(2\theta)}{g}$

Graphical Kinematics

• Slope and area interpretations:
    - Slope of $x-t$ graph = instantaneous velocity.
    - Slope of $v-t$ graph = instantaneous acceleration.
    - Area under $v-t$ graph = displacement (which can be signed).
    - Area under $a-t$ graph = change in velocity.

• AP Tip: The exam often provides graphs of $v(t)$ or $a(t)$ and asks for displacement or changes in velocity, which requires interpreting areas under the curves.

Unit 2: Newton's Laws of Motion

• Overview: Focuses on forces, free-body diagrams, friction, and circular motion.

Newton's Three Laws

1. First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force, defining an inertial reference frame.

2. Second Law: $F_{net} = ma = rac{dp}{dt}$. Apply $F = ma$ for constant mass and use $dp/dt$ for situations with varying mass (e.g., rockets).

3. Third Law: For every action, there is an equal and opposite reaction. Forces A and B act on different objects, meaning they do not cancel out in free-body diagrams (FBDs).

Free-Body Diagrams (FBDs)

• An FBD must depict all forces acting on an object in isolation. Establish a coordinate system oriented along the direction of acceleration.
    - Types of Forces:
      - Weight: $W = mg$ (acts downward toward Earth's center)
      - Normal Force: $N$ (perpendicular to contact surface)
      - Friction: $f$ (opposes relative motion)
      - Tension: $T$ (along string/ rope)
      - Spring Force: $F = -kx$ (toward equilibrium)
    - On inclined planes: adjust axes so one aligns with the slope.

• Important AP Tip: Normal force is not always $mg$; on a slope, it is given by $N = mg \cos(\theta)$, or adjust for vertical applied forces using $\Sigma F_y = ma_y$.

Friction

• Static Friction:
    - Variable, up to maximum: $f_s \leq \mu_s N$
    - Acts to prevent motion; utilize maximum when on verge of slipping.

• Kinetic Friction:
    - Constant while sliding: $f_k = \mu_k N$
    - This value is fixed; usually \mu_k < \mu_s.

Circular Motion

• Objects in circular motion require a net inward (centripetal) force, which arises from existing forces directed toward the center:
    - Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$
    - Centripetal force: $F_c = \frac{mv^2}{r} = m \omega^2 r$
    - Formulate net inward force using $F_{net} = rac{mv^2}{r}$; DO NOT confuse this with a new type of force.

• For the top and bottom of a vertical loop:
    - Top: $N + mg = \frac{mv^2}{r}$ (minimum speed when $N = 0$ is $v_{min} = \sqrt{gr}$)
    - Bottom: $N - mg = \frac{mv^2}{r}$; maximum normal force occurs here.

• For banked curves (no friction): $\tan(\theta) = \frac{v^2}{rg}$.

• Conical Pendulum: Torque balance provides:${T\cos(\theta) = mg; T\sin(\theta) = \frac{mv^2}{r} }$.

Unit 3: Work, Energy, and Power

• A scalar approach avoiding forces directly — often the fastest solution.

Work by a Variable Force

• Defined by integrating force; the constant force formula $W = Fd \cos(\theta)$ is a special case (F is constant).

• Work by variable force:
    $W = \int F \cdot dx = \int F \cos(\theta) dx$

• Evaluate limits between two positions (like areas under graphs):
    - Work by Gravity: $W_{gravity} = mg(h_f - h_i)$ (positive when an object falls).
    - Work by a Spring: $W_{spring} = -\Delta(\frac{1}{2}kx^2)$ (negative when storing energy).
    - Work by Normal Force is always zero if perpendicular to displacement.
    - Work by Friction is negative (it opposes displacement).

Work-Energy Theorem

• Stated as:
    $W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$

• Derived using work done=integral approach leading to the change in kinetic energy.

Potential Energy and Conservation

• For conservative forces, potential energy (U) defined by:
    $F = -\frac{dU}{dx}$
    - Gravitational Potential Energy (near surface): $U_{gravity} = mgh$; reference height arbitrary, be consistent.
    - Gravitational Potential Energy (General): $U = -\frac{GMm}{r}$ for orbital problems.
    - Spring Potential Energy: $U_{spring} = \frac{1}{2}kx^2$.

Conservation of Mechanical Energy

• Conservation formula valid only when no non-conservative forces do work:
    $KE_i + PE_i = KE_f + PE_f$

• If friction or other non-conservative forces work e.g. friction, derive using:
    $W_{non-conservative} = \Delta KE + \Delta PE$

• For friction specifically: $W = -f \cdot d$ (energy loss).

Energy Diagrams

• Plots of potential energy (U) corresponding with position (x), noting:
    - Equilibrium occurs where $\frac{dU}{dx} = 0$ (force equates zero).
    - Stable Equilibrium: local minimum (i.e., \frac{d^2U}{dx^2} > 0) — returns to the same position when pushed away.
    - Unstable Equilibrium: local maximum (i.e., \frac{d^2U}{dx^2} < 0) — moves away if disturbed.   - Total energy (E) remains constant across x; exists only where E > U(x).
    - Turning points occur where $E = U$.

Power

• Defined as:
    $P = \frac{dW}{dt} = F\cdot v$ (F vector and velocity vector).

• Units of power: Watts = $\frac{J}{s}$ (Joules per second).

Unit 4: Systems of Particles and Linear Momentum

• Focus on momentum, collisions, center of mass, impulse.

Center of Mass (CM)

• For discrete systems:
    $x_{cm} = \frac{\Sigma m_ix_i}{M}$

• For continuous distributions:
    $x_{cm} = \frac{1}{M}\int x \, dm$ (substitute dm in terms of coordinates).

• The center of mass accelerates according to:
    $F_{net} = Ma_{cm}$; internal forces do not affect CM motion.

• In explosions, CM continuity maintains the parabolic trajectory.

Momentum and Impulse

• Linear Momentum: $p = mv$ (vector direction aligns with velocity).

• General form of Newton's 2nd Law in momentum terms:
    $F_{net} = \frac{dp}{dt}$; reduce to $F = ma$ when mass is constant.

• Impulse-Momentum Theorem:
    $J = \Delta p = \int F \, dt$; area under the F-t graph offers change in momentum.

Conservation of Momentum and Collisions

• Conservation of Momentum:
   $p_{initial} = p_{final}$ when the sum of external forces is zero.
    - Apply separately to x and y components in 2D scenarios.
    - Elastic Collision: Both momentum and kinetic energy conserved.
    - One-dimensional Elastic Collision:
      $v_1 = \frac{(m_1 - m_2)v_1 + 2m_2v_2}{m_1 + m_2}$
    - If masses are equal, speeds switch.

• Perfectly Inelastic Collision: Masses stick together:
    - Only momentum conserved, not kinetic energy.
    - Expressed as: $m_1v_1 + m_2v_2 = (m_1 + m_2)v$, where KE is lost to other forms (heat, sound).

Unit 5: Rotation

• Covers rotational kinematics, torque, moment of inertia, and angular momentum.

Rotational Kinematics

• Rotational motion parallels linear motion:
    - Linear Position to Angular Position: $(x \leftrightarrow \theta)$
    - Linear Velocity to Angular Velocity: $(v = \frac{dx}{dt} \rightarrow \omega = \frac{d\theta}{dt})$
    - Linear Acceleration to Angular Acceleration: $(a = \frac{dv}{dt} \rightarrow \alpha = \frac{d\omega}{dt})$

• Key equations:
    - Rotational Equations of Motion:
      - $\omega = \omega_0 + \alpha t$
      - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
      - $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$

Moment of Inertia (I)

• Defined for discrete and continuous objects:
    - $I_{discrete} = \Sigma m_ir_i^2$
    - $I_{continuous} = \int r^2 \, dm$

• Apply Parallel Axis Theorem:
    $I = I_{cm} + Md^2$ where $d$ is the distance between the axis and center of mass.

• Common moments of inertia:
    - Solid Disk/Cylinder: $\frac{1}{2}MR^2$
    - Thin Hoop/Ring: $MR^2$
    - Solid Sphere: $\frac{2}{5}MR^2$
    - Thin Spherical Shell: $\frac{2}{3}MR^2$
    - Thin Rod: $\frac{1}{12}ML^2$ (center), $\frac{1}{3}ML^2$ (end).

Torque and Rotational Dynamics

• Torque defined by:
    $|\tau| = rF \sin(\theta)$ (measure of force's influence about a pivot).
    - Where $r$ is the moment arm, defined as the distance from the pivot to the line of action of the force.

• Newton's Second Law for Rotation: $\tau_{net} = I\alpha$ (analogous to linear $F = ma$). Choose an axis before applying equations.

Rolling, Angular Momentum, and Energy

• Rolling without slipping provides relationships:
    - $v = R\omega$
    - $a = R\alpha$ (applies only if there is no friction)

• Total Kinetic Energy for rolling objects combines:
    $KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• Angular Momentum (L):
    - For rigid bodies: $L = I\omega$
    - For point particles: $L = r \times p$ (where $p = mv$)
    - Conservation: When net torque is zero, $L_{initial} = L_{final}$ (popular applications include ice skaters and orbital motion).

Unit 6: Oscillations (SHM)

• Concerns harmonic motion defined by restoring forces.

Simple Harmonic Motion (SHM)

• Defined condition: The net restoring force is proportional to displacement from equilibrium:
    $F = -kx$
    - Derive acceleration: $a = -\frac{k}{m}x = -\omega^2 x$ (with $\omega = \sqrt{\frac{k}{m}}$).

Differential Equation of SHM

• $\frac{d^2x}{dt^2} = -\omega^2 x$; solutions yield sinusoidal behavior.

General Solution

• $x(t) = A \cos(\omega t + \phi)$ with:
    - A = amplitude (maximum displacement)
    - $\omega$ = angular frequency
    - $\phi$ = phase constant (initialized)

Velocity and Acceleration

• Velocity:
    $v(t) = -A\omega \sin(\omega t + \phi)$; maximum @ $x=0$.

• Acceleration:
    $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2x$; maximum @ $x=\pm A$.

Total Energy in SHM

• Constant: $E = \frac{1}{2}kA^2$; directly proportional to amplitude squared.

Simple Pendulum

• Period derived by $T = 2\pi \sqrt{\frac{L}{g}}$ (independent of mass).

• Physical Pendulum:
    $T = 2\pi \sqrt{\frac{I}{Mgd}}$ (M = total mass; d = distance from pivot to CM).

Unit 7: Gravitation

• Explores fundamental principles of gravitational forces and motion in space.

Newton's Law of Universal Gravitation

• Gravitational formula:
    $F = \frac{GMm}{r^2}$

• Gravitational field strength:
    $g = \frac{GM}{r^2}$; at Earth's surface, $g \approx 9.8 \, m/s^2$.

• Gravitational Potential Energy:
    $U = -\frac{GMm}{r}$ (negative, bound objects).

Escape Velocity

• Derived using total energy consideration set to zero:
    $v_{esc} = \sqrt{\frac{2GM}{R}}$ (approximation for Earth: 11.2 km/s).

Orbital Mechanics

• Circular orbital speed:
    $v = \sqrt{\frac{GM}{r}}$

• Period of orbit:
    $T = 2\pi \sqrt{\frac{r^3}{GM}}$ (Link to Kepler's Third Law).

Energy in Orbits

• Total orbital energy:
    $E = -\frac{GMm}{2r}$ (always negative for bound systems).

Kepler's Laws

1. First Law: Planets move in elliptical orbits with the Sun at one focus.

2. Second Law: Equal areas in equal times (derived from angular conservation).

3. Third Law: $T^2 \propto r^3$, comparisons between planets around the same sun.

Shell Theorem and Interior Gravity

• Outside uniformly, gravity acts as if all mass exists at a point.

• Inside a shell, gravitational force is zero everywhere.

• Inside a solid sphere, force varies linearly with radius helping visualize gravitational behaviors below Earth's surface.

AP Exam Strategy

• General Exam Tips:
    - Identify the governing principle before calculations commence.
    - Work conceptually, substituting numbers last.
    - Utilize dimensional analysis to validate solutions.
    - Fill in all answers on multiple-choice; there are no penalties for guessing.

Multiple Choice Tips:

• Utilize symbolism until numerical evaluation is absolutely necessary.

• For graphical problems, recall that slope = derivative and area = integral of plotted quantities.

Free Response Tips:

• Mark your coordinate system and positive direction before solving.

• Always draw free-body diagrams (FBDs) before establishing equations.

• State applicable laws or theorems explicitly.

• Show all calculus procedure with limits; always include units in calculations.

Method Selection Summary

• Use F = ma when focusing on forces/acceleration.

• Work-Energy theorem for connections between speeds at various points under variable forces.

• Conservation of energy for problems with no friction for quickest solutions.

• Handle collisions/explosions/momentum with conservation laws explicit to nature.

Common Errors to Avoid

• Do not confuse net displacement with total distance traveled (signed vs. non-signed).

• Remember friction's force and effects in all related calculations.

• Avoid neglecting rotational kinetic energy terms in calculations involving rolling objects.

Must-Know Formula Reference

• Thoroughly referenced formulas throughout entire course summary.