AP Physics C - Mechanics Practice Workbook

Copyright Notice

• The content is copyrighted by the College Entrance Examination Board from 1973-2012.
• Copies for student distribution are explicitly allowed for exam preparation.
• Trademarks mentioned include College Board, Advanced Placement Program, AP, AP Central, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, PSAT/NMSQT, Educational Testing Service, and ETS.

Table of Contents

• Includes chapters on Kinematics, Dynamics, Work and Energy, Center of Mass and Momentum, Rotation, Gravitation, and Oscillations.
• Each chapter contains multiple-choice questions, free-response questions, and answers.

General Information

• This book is a compilation of problems published by College Board in AP Physics C, organized by topic.
• Problems vary in difficulty and type, serving as a resource for practice and review.
• Doing these problems reinforces ideas and concepts.
• Solutions presented are not the only methods; physics teachers may use different approaches.
• Solutions should be read with an open mind to expand problem-solving skills.
• Typographical or formatting errors should be reported for corrections.

Table of Information and Equation Tables

• A table of information and equation tables is provided during AP Physics Exams.
• Students may not bring their own copies but can use them in classes.
• The latest versions are available on AP Central.

Table of Information Details

• Printed in the multiple-choice section and on the green insert for the free-response section.
• Tables are identical for Physics B and Physics C exams, except for one convention.

Equation Tables Details

• Printed only on the green insert provided with the free-response section.
• May be used during the free-response sections but not the multiple-choice sections.
• Equations express frequently encountered relationships but do not include all possible equations.
• Students must understand the physical principles and conditions for each equation's applicability.
• Tables are grouped by major content category, with variable symbols defined within each section.
• Symbols may represent different quantities in different tables, and conventions may vary from textbooks.

Notation Used in Equation Tables

1. Physical constants use the same symbols as in the Table of Information.
2. Bold face symbols represent vector quantities.
3. Subscripts represent special cases of variables.
4. $\Delta$ before a variable indicates a change in its value (final minus initial).
5. Various symbols (e.g., d, r, s, h, ${\Box}$) represent linear dimensions like length.

Constants and Conversion Factors

• Proton mass: $m_p = 1.67 \times 10^{-27} \text{ kg}$
• Neutron mass: $m_n = 1.67 \times 10^{-27} \text{ kg}$
• Electron mass: $m_e = 9.11 \times 10^{-31} \text{ kg}$
• Electron charge magnitude: $e = 1.60 \times 10^{-19} \text{ C}$
• One electron volt: $1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$
• Speed of light: $c = 3.00 \times 10^8 \text{ m/s}$
• Avogadro’s number: $N_0 = 6.02 \times 10^{23} \text{ mol}^{-1}$
• Universal gravitational constant: $G = 6.67 \times 10^{-11} \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2}$
• Universal gas constant: $R = 8.31 \frac{\text{J}}{\text{mol} \cdot \text{K}}$
• Acceleration due to gravity: $g = 9.8 \frac{\text{m}}{\text{s}^2}$
• Boltzmann’s constant: $k_B = 1.38 \times 10^{-23} \frac{\text{J}}{\text{K}}$
• Unified atomic mass unit: $1 \text{ u} = 1.66 \times 10^{-27} \text{ kg} = 931 \frac{\text{MeV}}{\text{c}^2}$
• Planck’s constant: $h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} = 4.14 \times 10^{-15} \text{ eV} \cdot \text{s}$
• $hc = 1.99 \times 10^{-25} \text{ J} \cdot \text{m} = 1.24 \times 10^{-6} \text{ eV} \cdot \text{m}$
• Vacuum permittivity: $\epsilon_0 = 8.85 \times 10^{-12} \frac{\text{C}^2}{\text{N} \cdot \text{m}^2}$
• Coulomb’s law constant: $k = \frac{1}{4\pi\epsilon_0} = 9.0 \times 10^9 \frac{\text{N} \cdot \text{m}^2}{\text{C}^2}$
• Vacuum permeability: $\mu_0 = 4\pi \times 10^{-7} \frac{\text{T} \cdot \text{m}}{\text{A}}$
• Magnetic constant: $k<em>m = \frac{\mu</em>0}{4\pi} = 1 \times 10^{-7} \frac{\text{T} \cdot \text{m}}{\text{A}}$
• Atmospheric pressure: $1 \text{ atm} = 1.0 \times 10^5 \frac{\text{N}}{\text{m}^2} = 1.0 \times 10^5 \text{ Pa}$

Unit Symbols

• meter (m)
• kilogram (kg)
• second (s)
• ampere (A)
• kelvin (K)
• mole (mol)
• hertz (Hz)
• newton (N)
• coulomb (C)
• volt (V)
• ohm (Ω)
• degree Celsius (°C)
• pascal (Pa)
• joule (J)
• tesla (T)
• watt (W)
• farad (F)
• electron-volt (eV)
• henry (H)

Prefixes

• giga (G): $10^9$
• mega (M): $10^6$
• kilo (k): $10^3$
• centi (c): $10^{-2}$
• milli (m): $10^{-3}$
• micro (\mu): $10^{-6}$
• nano (n): $10^{-9}$
• pico (p): $10^{-12}$

Values of Trigonometric Functions for Common Angles

• 0°: sin = 0, cos = 1, tan = 0
• 30°: sin = 1/2, cos = $\frac{\sqrt{3}}{2}$, tan = $\frac{\sqrt{3}}{3}$
• 37°: sin = 3/5, cos = 4/5, tan = 3/4
• 45°: sin = $\frac{\sqrt{2}}{2}$, cos = $\frac{\sqrt{2}}{2}$, tan = 1
• 53°: sin = 4/5, cos = 3/5, tan = 4/3
• 60°: sin = $\frac{\sqrt{3}}{2}$, cos = 1/2, tan = $\sqrt{3}$
• 90°: sin = 1, cos = 0, tan = $\infty$

General Conventions

• Unless stated, assume the frame of reference is inertial.
• Electric current direction is the flow of positive charge (conventional current).
• Electric potential for an isolated charge is zero at infinite distance.
• For mechanics and thermodynamics, W represents work done on a system. (Note: Thermodynamics is not a Physics C topic)

Mechanics Equations

• $x = x<em>0 + v</em>0 t + \frac{1}{2} a t^2$
• $v = v_0 + at$
• $v^2 = v<em>0^2 + 2 a (x - x</em>0)$
• $\vec{F}_{net} = m \vec{a}$
• $J = \int F dt$
• $p = mv$
• $F = \frac{dp}{dt}$
• $F_{fric} \le \mu N$
• $K = \frac{1}{2} m v^2$
• $W = \int F \cdot d\vec{r}$
• $P = F \cdot v$
• $\Delta U = mgh$
• $K = \frac{1}{2} k x^2$
• $F = -kx$
• $P = \frac{dW}{dt}$
• $x<em>{cm} = \frac{\sum m</em>i r<em>i}{\sum m</em>i}$
• $v = r \omega$
• $a_c = \frac{v^2}{r} = r \omega^2$
• $\tau = r \times F = I \alpha$
• $I = \sum m<em>i r</em>i^2$
• $\vec{L} = \vec{r} \times \vec{p} = I \vec{\omega}$
• $K = \frac{1}{2} I \omega^2$
• $\omega = \omega_0 + \alpha t$
• $\theta = \theta<em>0 + \omega</em>0 t + \frac{1}{2} \alpha t^2$
• $\tau = r F sin(\theta)$
• $U = \frac{1}{2} k x^2$
• $T = \frac{2\pi}{\omega}$
• $T = 2\pi \sqrt{\frac{m}{k}}$
• $F = \frac{G m<em>1 m</em>2}{r^2}$
• $U = - \frac{G m<em>1 m</em>2}{r}$

Electricity and Magnetism Equations

• $F = \frac{1}{4\pi\epsilon<em>0} \frac{q</em>1 q_2}{r^2}$
• $E = \frac{F}{q}$
• $\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}$
• $E = - \frac{dV}{dr}$
• $V = \frac{1}{4\pi\epsilon<em>0} \sum \frac{q</em>i}{r_i}$
• $C = \frac{Q}{V}$
• $C = \frac{\epsilon_0 A}{d}$
• $U = qV$ = $\frac{1}{2}QV = \frac{1}{2}CV^2$
• $I = \frac{dQ}{dt}$
• $E = J \rho$
• $V = IR$
• $P = IV$
• $I = nqvA$
• $C<em>s = \sum C</em>i$
• $\frac{1}{C<em>p} = \sum \frac{1}{C</em>i}$
• $R<em>s = \sum R</em>i$
• $\frac{1}{R<em>p} = \sum \frac{1}{R</em>i}$
• $F = qvB \sin \theta$
• $F = ILB \sin \theta$
• $B = \frac{\mu_0 I}{2\pi r}$
• $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$
• $B = \mu_0 nI$
• $\Phi = \int \vec{B} \cdot d\vec{A}$
• $\varepsilon = - \frac{d\Phi}{dt}$
• $\varepsilon = BLv$
• $U = \frac{1}{2}LI^2$
• $\varepsilon = - L \frac{dI}{dt}$
• $X_L = \omega L$
• $X_C = \frac{1}{\omega C}$
• $V<em>{rms} = I</em>{rms} Z$

Where the impedance Z is given by:

• Series RLC Circuit: $Z = \sqrt{R^2 + (X<em>L - X</em>C)^2}$

Geometry and Trigonometry

• Rectangle Area: $A = bh$ (b = base, h = height)
• Triangle Area: $A = \frac{1}{2}bh$
• Circle Area: $A = \pi r^2$ (r = radius)
• Circle Circumference: $C = 2 \pi r$
• Rectangular Solid Volume: $V = \ell wh$ (l = length, w = width, h = height)
• Cylinder Volume: $V = \pi r^2 h$
• Cylinder Surface Area: $S = 2 \pi r h + 2 \pi r^2$
• Sphere Volume: $V = \frac{4}{3} \pi r^3$
• Sphere Surface Area: $S = 4 \pi r^2$
• Right Triangle: $a^2 + b^2 = c^2$
  Trigonometric definitions:
• $\sin \theta = \frac{a}{c}$
• $\cos \theta = \frac{b}{c}$
• $\tan \theta = \frac{a}{b}$

Calculus

• $\frac{df}{dx} \frac{d}{dx} (ax^n) = nax^{n-1}$
• $\frac{d}{dx} e^x = e^x$
• $\frac{d}{dx} \ln x = \frac{1}{x}$
• $\frac{d}{dx} \sin x = \cos x$
• $\frac{d}{dx} \cos x = -\sin x$
• $\int x^n dx = \frac{1}{n+1} x^{n+1}, n \neq -1$
• $\int e^x dx = e^x$
• $\int \frac{dx}{x} = \ln |x|$
• $\int \cos x dx = \sin x$
• $\int \sin x dx = -\cos x$