AP Physics Comprehensive Study Guide and Exam Review

AP Physics Exam Breakdown and Kinematics

AP Physics Full Review Past Exam Breakdown: - Energy: $25\%$ (Most prominent topic) - Dynamics/Newton's Law: $20\%$ - Kinematics: $17\%$ - Rotational Motion: $16\%$ - Momentum: $14\%$ - Circular Motion/Gravitation: $5\%$ - Simple Harmonic Motion: $3\%$
Motion Graphs: - Position Graph: The velocity is the slope of the line. - Velocity Graph: The acceleration is the slope of the line, and the area under the line represents the displacement. - Acceleration Graph: The area under the line represents the velocity.
Basic Kinematic Formulas: - Acceleration: $a = \frac{\Delta v}{\Delta t}$ - Velocity: $v = \frac{d}{t}$ - Average Velocity: $v_{avg} = \frac{v_1 + v_2}{2}$
Projectile Motion Basics: - Horizontal motion variables: Velocity, displacement, and time. - Vertical motion variables: Initial velocity, final velocity, displacement, time, and acceleration. - Key Principle: Horizontal time is equal to vertical time ( $t_{horiz} = t_{vert}$ ). - Methodology: Apply kinematic equations to determine vertical quantities first, then use those results to find horizontal quantities. - Highest Point Calculation: Time to reach the peak can be found using $t = \frac{-v_{0y}}{-g}$ .
Angled Projectile Motion: - Break the motion into horizontal and vertical components. - If the vector is directed up and to the right, use sine for vertical ( $v_y$ ) and cosine for horizontal ( $v_x$ ). - Use average velocity equations ( $v = \frac{d}{t}$ ) to find missing horizontal values. - Horizontal Range Formula: $Range = \frac{v_{0y}^2 \times \sin(2\theta)}{g}$ .

Forces and Newton’s Laws

Newton's Three Laws of Motion: 1. Law of Inertia: If a center of mass is at rest, it will remain at rest unless acted upon by a net force. A moving center of mass will maintain a constant velocity unless acted upon by a net force. 2. Law of Acceleration: The acceleration ( $a$ ) of an object or system is equal to the net force ( $F_{net}$ ) acting upon it divided by its mass ( $m$ ): $a = \frac{F_{net}}{m}$ . 3. Law of Action-Reaction: Every action has an equal and opposite reaction. This requires two forces from two different objects acting specifically on each other.
Equilibrium States: - Static Equilibrium: Occurs when the net force on a motionless object or system is $0$ . - Dynamic Equilibrium: Occurs when the net force on a moving object or system is $0$ . This requires the object to maintain a constant velocity with no acceleration. - General Rule: If an object is not moving or is moving at a constant rate, all forces are balanced.
Normal Force and Friction: - Normal Force ( $F_n$ ): Acts perpendicular to the surface applying it ( $F_g = mg$ ). - Friction Force: Acts parallel to the surface applying it. - Static Friction ( $F_s$ ): Acts on a motionless object. Its magnitude and direction will always be whatever is necessary to keep the object from moving ( $F_s < M_s \times F_n$ ). It opposes the applied force until it reaches a maximum threshold, then switches to kinetic friction. - Kinetic Friction ( $F_k$ ): Acts on moving objects once the static friction threshold is passed. The direction is always opposite the direction of motion ( $F_k = M_k \times F_n$ ), where $M$ is the coefficient of friction. - Ramp Components: "Sine makes it slide, cosine keeps it close" (Cosine acts into the ramp, sine acts down the ramp).
Atwood Machines: - Consists of two masses hanging from a massless string with a massless, frictionless pulley. - Tension is consistent throughout the string and accelerations of the masses are equal. - The system is accelerated by gravity: $a = \frac{F_{net}}{m}$ . - Tension Calculation: $F_{net} = ma = T_g - T$ , meaning $T = mg - ma$ . - Equilibrium occurs if $m_1 + m_2 = m$ ; otherwise, the system accelerates. - Special Velocity Case: For an object sliding down a ramp of height ( $h$ ), the velocity at the bottom is $v = \sqrt{2gh}$ . - Frictionless Ramp Relation: $M_k = \frac{h}{x}$ .

Mechanical Energy, Work, and Power

Work ( $W$ ): - Mechanical transfer of energy to or from a system by pushing or pulling when a force displaces the system. - Scalar Quantity. - Formula: $W = Fd\cos(\theta)$ or $W = KE_f - KE_i$ . - Positive Work: Adds mechanical energy; occurs when force and displacement are in the same direction ( $\theta$ is between $0$ and $89$ degrees). - Negative Work: Removes mechanical energy; occurs when force and displacement are in different directions ( $\theta$ is between $91$ and $180$ degrees). - Zero Work: Occurs when work is perpendicular to the direction of motion. - Graphical Analysis: Work is the area under a Force vs. Position graph.
Energy Types: - Kinetic Energy ( $K$ ): Energy of motion measured in Joules ( $J$ ). Formula: $K = \frac{1}{2}mv^2$ . If KE is doubled, velocity increases by a factor of $\sqrt{2}$ . - Gravitational Potential Energy ( $U_g$ ): Based on location in a gravitational field: $U_g = mgh$ . In the absence of friction, $\frac{1}{2}v^2 = gh$ because mass cancels. - Elastic Potential Energy ( $U_s$ ): Based on contact with a stretched/compressed spring: $U_s = \frac{1}{2}kx^2$ .
Conservation and Dynamics: - Conservation of Energy: The total energy of an isolated system is constant. Mechanical energy is conserved in the absence of friction. - Blocks on Ramps: Steeper ramps result in faster descent (less time), but height alone determines final velocity ( $v$ ) and displacement ( $\Delta x$ ) if the block becomes a projectile. - Coefficient of Kinetic Friction: $M_k = \frac{F_f}{F_n}$ .
Power ( $P$ ): - Rate at which energy changes state. - Scalar Quantity measured in Watts ( $W$ ), where $1\,W = 1\,J/s$ . - Formulas: $P = F \times v$ , $P = \frac{W}{t}$ , $P = \frac{Fd}{t}$ , or $P = Fv\cos(\theta)$ .

Momentum and Collisions

Impulse ( $J$ ): - Impulse is the change in momentum ( $\Delta p$ ). - Graphically, it is the area under a Force vs. Time graph. - Formulas: $I = F\Delta t$ or $I = \Delta(mv)$ . Therefore, $F \times t = m \times \Delta v$ .
Collision Types: - Elastic Collisions: Conserve both momentum and kinetic energy ( $p_{1i} + p_{2i} = p_{1f} + p_{2f}$ ); objects bounce off each other. - Inelastic Collisions: Conserve momentum but NOT kinetic energy. Objects frequently stick together (perfectly inelastic). - Explosions: Conserve momentum but NOT kinetic energy; one object separates into multiple parts. - Law of Conservation of Momentum: The total momentum of an isolated system remains constant.
Center of Mass: - The point where an object or system balances on a fulcrum. - Calculation: Establish a zero point ( $x = 0$ ), multiply each part's mass by its distance from the zero point, sum these values, and divide by the total mass.
Open vs. Closed Systems: - Closed System: Conserves energy. - Open System: Energy is changed by something outside the objects in question. Energy loss to surroundings (e.g., through friction) indicates an open system.

Circular Motion and Gravitation

Uniform Circular Motion: - Motion along a circular path at constant speed. - Tangential speed is constant, but tangential velocity is not (changes direction). - Tangential velocity is always tangent to the path.
Centripetal Acceleration and Force: - Centripetal Acceleration ( $a_c$ ): Directed toward the center. Formula: $a_c = \frac{v^2}{r}$ . Measured in $m/s^2$ . - Centripetal Force ( $F_c$ ): The net force directed toward the center ( $F_c = m \times \frac{v^2}{r}$ or $F_c = m\omega^2r$ ). It is perpendicular to motion and does zero work. - Orbit Mechanics: For a planet in orbit, gravitational force is the centripetal force: $\frac{GMm}{R^2} = \frac{mv^2}{R}$ .
Universal Gravitation: - Newton’s Law: $F_g = \frac{Gm_1m_2}{r^2}$ . If distance doubles, force decreases by a factor of $4$ . - Gravitational Constant ( $G$ ): $6.67 \times 10^{-11}\,N \cdot m^2 / kg^2$ . - Field Strength ( $g$ ): $g = \frac{GM}{r^2}$ . On Earth, this is approximately $9.81\,m/s^2$ . - Potential Energy ( $U_g$ ): Between two systems: $U_g = \frac{Gm_1m_2}{r}$ . - Earth Constants: Radius $= 6.37 \times 10^6\,m$ ; Mass $= 5.973 \times 10^{24}\,kg$ . - Orbital Velocity & Acceleration: For one revolution: $v = \frac{2\pi r}{T}$ ; $a_c = \frac{4\pi^2r}{T^2}$ ; $F_c = \frac{4\pi^2mr}{T^2}$ .

Rotational Motion

Types of Motion: - Translational: Object moves across a trajectory. - Angular: Rotation where points move in a circle. - Combination: Both rotation and translation (e.g., rolling).
Angular Quantities: - Angular Velocity ( $\omega$ ): Radians rotated per unit time. $\omega = \frac{\Delta\theta}{t}$ . Area under an angular velocity vs. time graph represents angular displacement. - Angular Acceleration ( $\alpha$ ): Rate of change of angular velocity. $\alpha = \frac{\Delta\omega}{t}$ or $\alpha = \frac{\tau}{I}$ . Slope of the angular velocity vs. time graph.
Torque ( $\tau$ ): - Rotational analog to force, required to change angular velocity. - Unit: Newton-meter ( $N\cdot m$ ). - Formulas: $\tau = Fr\sin(\theta)$ and $\tau = I\alpha$ . Max value at $90^{\circ}$ . - Direction: Negative for clockwise; positive for counter-clockwise. - Net Torque: Slope of an angular momentum vs. time graph. If net torque is $0$ , net force is $0$ .
Rotational Inertia and Momentum: - Moment of Inertia ( $I$ ): Measure of how easily an object rotates, based on mass distribution. Unit: $kg\cdot m^2$ . Slope of torque vs. angular acceleration graph. - Newton’s 2nd Law for Rotation: $\alpha = \frac{\tau_{net}}{I}$ . Smaller inertia leads to greater acceleration. - Angular Momentum ( $L$ ): $L = I\omega$ . Change in momentum: $\Delta L = \tau t$ . Conserved if no external torque is applied. - Rotational Kinetic Energy ( $RKE$ ): $K_{rot} = \frac{1}{2}I\omega^2$ or $K_{rot} = \frac{1}{2}I(\frac{v}{r})^2$ . Total KE for rolling objects includes $\frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ .

Simple Harmonic Motion (SHM)

Basics: - Periodic motion where restoring force is proportional to distance from equilibrium. - Amplitude ( $A$ ): Initial displacement. - Period ( $T$ ): Time for one revolution ( $2\pi r$ ). - Frequency ( $f$ ): Cycles per second ( $Hz$ ). $f = \frac{1}{T}$ .
SHM Key Locations: - At Amplitude: Elastic potential energy is Max; Spring force is Max; Acceleration is Max; Kinetic energy is Min; Speed is Min. - At Equilibrium: Elastic potential energy is Min; Spring force is Min; Acceleration is Min; Kinetic energy is Max; Speed is Max.
Hooke’s Law: - $F_s = kx$ . Restoring force pulls toward equilibrium. - Spring Constant ( $k$ ): Measured in $N/m$ . Slope of $F_s$ vs. $x$ graph. - Spring Energy: $U_s = \frac{1}{2}kx^2$ or $PE = \frac{1}{2}kA^2$ .
Periods of Systems: - Mass-Spring: $T_s = 2\pi\sqrt{\frac{m}{k}}$ . Frequency $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ . - Pendulum: $T_p = 2\pi\sqrt{\frac{L}{g}}$ . Frequency $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$ . - Variable Impact: Doubling $k$ increases $f$ by $\sqrt{2}$ . Halving $L$ divides $T$ by $\sqrt{2}$ . Period is not affected by displacement ( $x$ ).

Waves and Sound

Transverse Wave Parts: - Crest: Max vertical displacement above horizontal. - Trough: Max vertical displacement below horizontal. - Wavelength (\lambda): Distance between adjacent crests/troughs.
Wave Speed Formulas: - $v = f\lambda$ . - On a Stretched String: $v = \sqrt{\frac{F_T}{\mu}}$ , where $\mu$ is linear mass density ( $mass/length$ ) and $F_T$ is tension force. - Rule #1: Speed is determined by the medium, not frequency. - Rule #2: Entering a new medium changes speed, but frequency remains constant.
Interference (Superposition): - Constructive: Overlapping displacements add up to a greater magnitude. - Destructive: Opposite displacements meet and magnitude decreases. - In Phase: Crest meets crest, trough meets trough. - Out of Phase: Crest meets trough.
Standing Waves: - Combination of waves in opposite directions; includes nodes and antinodes (spaced by $\frac{1}{2}\lambda$ ). - Condition: $L = n(\frac{1}{2}\lambda)$ . - Wavelength of harmonics: $\lambda_n = \frac{2L}{n}$ . - Harmonic Frequencies: $f_n = \frac{nv}{2L}$ .
Sound Waves: - Longitudinal waves consisting of compressions (high pressure) and rarefactions (low pressure). - Speed: $v = \sqrt{\frac{B}{\rho}}$ , where $B$ is Bulk Modulus and $\rho$ is density. - Beats: $f_{beat} = |f_1 - f_2|$ . - Resonance: - Closed Tube: $f_n = \frac{nv}{4L}$ for odd integers $n$ . - Open Tube: $f_n = \frac{nv}{2L}$ for any integer $n$ . - Doppler Effect: Frequency increases as source and detector move toward each other.

Electrostatics and DC Circuits

Electric Charge ( $q$ ): - Protons ( $+$ , in nucleus) and Electrons ( $r$ , orbiting). - Elementary charge ( $e$ ): Smallest unit of charge. - Quantization: $q = n(\pm e)$ , where $n$ is an integer. - Charge is conserved; net charge cannot be created or destroyed.
Coulomb’s Law: - $F_E = k\frac{q_1q_2}{r^2}$ . Negative $F_E = attraction$ ; Positive $F_E = repulsion$ . - Coulomb Constant ( $k_0$ ): $9 \times 10^9\,N \cdot m^2 / C^2$ . - Permittivity of Free Space (\epsilon_0): $8.85 \times 10^{-12}\,C^2 / N\cdot m^2$ . - Relation: $k_0 = \frac{1}{4\pi\epsilon_0}$ .
Electric Field ( $E$ ): - Field Strength: $E = \frac{F}{q}$ . Vectors point away from positive source and toward negative source. - Electric Dipole: Pair of equal but opposite charges. - Force in a field: $F = qE$ .
Electric Current ( $I$ ): - Rate of flow of charge: $I_{avg} = \frac{\Delta Q}{\Delta t}$ . Unit: Ampere ( $1\,A = 1\,C/s$ ). - Direction is defined by the flow of positive charge carriers.
Resistance ( $R$ ): - Ohm's Law: $R = \frac{V}{I}$ (Unit: Ohm $\Omega$ ). - Resistivity Formula: $R = \rho\frac{L}{A}$ , where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area.
Energy and Power in Circuits: - Power: $P = IV$ , $P = I^2R$ , or $P = \frac{V^2}{R}$ . - Voltage ( $V$ ) provides Electromotive Force (emf) to drive the flow.
Circuit Analysis and Kirchhoff’s Rules: - Resistors in Series: $R_s = \sum R_i$ . - Resistors in Parallel: $\frac{1}{R_p} = \sum \frac{1}{R_i}$ . - Junction Rule: Sum of currents entering a junction equals sum of currents leaving. - Loop Rule: Sum of potential differences in a closed loop is zero. Potential drops by $IR$ across resistors and changes across terminals of emf sources based on direction.