Physics 163 Final Exam Study Guide: Key Concepts and Equations

Kinematics and Vectors

Position, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. It involves understanding the meaning and units of position, velocity, and acceleration.

• Position: $x$ (meters, m)

• Velocity: $v = \frac{dx}{dt}$ (meters per second, m/s)

• Acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ (meters per second squared, m/s2)

• Constant Acceleration Equations:

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v = v_0 + a t$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Graphical Representation: Be able to interpret and create graphs of position, velocity, and acceleration.

Conservation of Energy

• Definition: The total energy in a closed system remains constant.

• Equation: $E = K + U$ where $K$ is kinetic energy and $U$ is potential energy.

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy (gravitational): $U = mgh$

Free-Body Diagrams (FBDs)

• FBDs are used to visualize all the forces acting on an object.

• Identify all forces, their directions, and points of application.

Vectors and Cross Products

• Vector Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

• Cross Product: $\vec{A} \times \vec{B} = \vec{C}$, magnitude $|\vec{A} \times \vec{B}| = AB \sin \theta$

• Use the right-hand rule to determine direction.

• Dot product to determine angle: $\vec{A} \cdot \vec{B} = AB \cos \theta$

Rigid Bodies and Rotational Motion

Angular Kinematics

Angular kinematics describes the rotational analogs of position, velocity, and acceleration.

• Angular Position: $\theta$ (radians)

• Angular Velocity: $\omega = \frac{d\theta}{dt}$ (rad/s)

• Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ (rad/s2)

• Constant Angular Acceleration Equations:

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

  • $\omega = \omega_0 + \alpha t$

  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Torque

• Definition: Torque is the rotational equivalent of force.

• Equation: $\tau = r F \sin \phi$

• Direction determined by right-hand rule.

Moment of Inertia

• Definition: Moment of inertia quantifies an object's resistance to angular acceleration.

• Point Particles: $I = \sum m_i r_i^2$

• Extended Rigid Body: $I = \int r^2 dm$

• Parallel axis theorem: $I = I_{cm} + Md^2$

Newton's Second Law for Rotation

• Equation: $\tau = I \alpha$

• Analogous to $F = ma$ for linear motion.

Center of Mass and Gravitational Torque

• Center of Mass: $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

• Gravitational Torque: $\tau_{cm} = r_{cm} F_{grav}$

Oscillations

Simple Harmonic Motion (SHM)

SHM describes periodic motion where the restoring force is proportional to displacement.

• Position: $x(t) = A \cos(\omega t + \phi)$

• Velocity: $v(t) = -A \omega \sin(\omega t + \phi)$

• Acceleration: $a(t) = -A \omega^2 \cos(\omega t + \phi)$

• Period: $T = \frac{2\pi}{\omega}$

• Frequency: $f = \frac{1}{T}$

Energy in SHM

• Total Energy: $E = \frac{1}{2} m \omega^2 A^2$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy: $U = \frac{1}{2} k x^2$

Damped and Driven Oscillations

• Damped Oscillation Equation: $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0$

• Solution: $x(t) = A e^{-\gamma t} \cos(\omega' t + \phi)$

• Driven Oscillation: $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \cos(\omega t)$

Waves

Wave Basics

Waves transfer energy through a medium without transferring matter.

• Wave Speed: $v = f \lambda$

• Transverse vs. Longitudinal Waves:

  • Transverse: Medium moves perpendicular to wave direction.

  • Longitudinal: Medium moves parallel to wave direction.

• Graphical Representation: Be able to interpret wave graphs and identify amplitude, wavelength, and frequency.

Spherical Waves and Doppler Effect

• Intensity: $I = \frac{P}{4\pi r^2}$

• Doppler Effect: For a moving source, $f' = f \frac{v}{v \pm v_s}$

Superposition and Standing Waves

Principle of Superposition

When two or more waves overlap, the resulting displacement is the sum of the individual displacements.

• Standing Waves: $y(x, t) = A \sin(kx) \cos(\omega t)$

• Nodes and antinodes are points of zero and maximum amplitude, respectively.

Interference

• Constructive Interference: $\Delta r = m \lambda$

• Destructive Interference: $\Delta r = (m + \frac{1}{2}) \lambda$

Wave Optics

Double-Slit Interference

• Path Difference: $d \sin \theta = m \lambda$

• Bright spots occur where path difference is a multiple of wavelength.

Diffraction Grating

• Similar principle to double-slit, but with many slits.

• Sharp maxima occur at $d \sin \theta = m \lambda$

Single-Slit Diffraction

• Dark Spots: $a \sin \theta = m \lambda$

Ray Optics

Reflection and Refraction

• Law of Reflection: $\theta_i = \theta_r$

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Total Internal Reflection: Occurs when light travels from a medium with higher to lower refractive index and $\theta_1 > \theta_c$.

Thin Lenses and Mirrors

• Lens Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Ray tracing diagrams are used to locate images formed by lenses and mirrors.

Summary Table: Key Equations

Topic	Equation	Description
Kinematics	$v = \frac{dx}{dt}$	Velocity as rate of change of position
Kinematics	$a = \frac{dv}{dt}$	Acceleration as rate of change of velocity
Energy	$K = \frac{1}{2} m v^2$	Kinetic energy
Energy	$U = mgh$	Gravitational potential energy
Rotation	$\tau = I \alpha$	Rotational analog of Newton's 2nd law
SHM	$x(t) = A \cos(\omega t + \phi)$	Position in simple harmonic motion
Waves	$v = f \lambda$	Wave speed
Optics	$n_1 \sin \theta_1 = n_2 \sin \theta_2$	Snell's law for refraction
Optics	$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$	Thin lens equation

Additional info: These notes are expanded from a final exam study guide, covering major topics in introductory college physics including kinematics, energy, rotation, oscillations, waves, and optics. All equations are presented in LaTeX format for clarity and academic rigor.