Scalars: Quantities described by a single number (e.g., mass, age).

Vectors: Quantities described by both magnitude and direction.

• Vector Addition/Subtraction: Combining or removing vectors to result in a new vector.
• Scalar Multiplication: Multiplying a vector by a scalar, affecting magnitude but not direction.
• Dot Product (Scalar Product): \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \)
• Cross Product: \( \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \), resulting in a vector perpendicular to both \( \vec{A} \) and \( \vec{B} \).

Displacement: Change in position, \( \Delta x = x_2 - x_1 \).

Velocity: Rate of change of displacement. Velocity is a vector, while speed is its magnitude.

• Average Velocity: \( \vec{v}_{avg} = \frac{\Delta x}{\Delta t} \)
• Instantaneous Velocity: The derivative of position with respect to time.

Acceleration: Rate of change of velocity.

• Instantaneous Acceleration: The derivative of velocity with respect to time.

Equations of Motion:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

Second Law (Force): The net force acting on an object is equal to the mass of the object multiplied by its acceleration, \( \vec{F} = m\vec{a} \).

Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.

Weight: Gravitational force, \( W = mg \).

Tension: Force transmitted through a string or rope when it is pulled tight by forces acting from opposite ends.

Friction: The force opposing relative motion.

• Static Friction: Prevents motion, \( f_s \leq \mu_s N \).
• Kinetic Friction: Opposes motion, \( f_k = \mu_k N \).

Normal Force: The perpendicular contact force exerted by a surface on an object.

Definition: Work is done when a force causes a displacement, \( W = \vec{F} \cdot \vec{d} \cos(\theta) \).

Work-Energy Theorem: The total work done by all forces on a particle equals the change in its kinetic energy, \( W_{tot} = \Delta KE \).

Kinetic Energy: Energy of motion, \( KE = \frac{1}{2}mv^2 \).

Potential Energy: Energy stored due to position.

• Gravitational PE: \( U_g = mgh \).
• Elastic PE (Spring): \( U_s = \frac{1}{2}kx^2 \).

Definition: The rate at which work is done, \( P = \frac{dW}{dt} \).

Units: Watts (W), where \( 1 \text{W} = 1 \text{J/s} \).

Angular Displacement: The angle through which a point or line has been rotated in a specified sense about a specified axis, \( \theta \) (radians).

Angular Velocity: The rate of change of angular displacement, \( \omega = \frac{d\theta}{dt} \) (radians per second).

Angular Acceleration: The rate of change of angular velocity, \( \alpha = \frac{d\omega}{dt} \) (radians per second squared).

Definition: The resistance of a body to change its rotational motion, \( I = \sum m_ir_i^2 \).

Common Moments of Inertia:

• Solid Cylinder: \( I = \frac{1}{2}MR^2 \)
• Solid Sphere: \( I = \frac{2}{5}MR^2 \)
• Thin Rod (about center): \( I = \frac{1}{12}ML^2 \)

Definition: A measure of the force that can cause an object to rotate about an axis, \( \tau = rF\sin(\theta) \).

Relationship with Angular Acceleration: \( \tau = I\alpha \).

Definition: The kinetic energy due to the rotation of an object, \( KE_{rot} = \frac{1}{2}I\omega^2 \).

Angular Momentum: The quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity, \( L = I\omega \).

Conservation Law: In the absence of external torques, the total angular momentum of a system remains constant, \( L_{initial} = L_{final} \).

Definition: Motion that repeats itself in equal intervals of time, such as the oscillation of a mass on a spring or a pendulum.

Equation of Motion: \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

Angular Frequency: \( \omega = \sqrt{\frac{k}{m}} \) for a mass-spring system, \( \omega = \sqrt{\frac{g}{l}} \) for a pendulum.

Kinetic Energy: \( KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2 - x^2) \).

Potential Energy: \( PE = \frac{1}{2}kx^2 \).

Total Energy: \( E = KE + PE = \frac{1}{2}kA^2 \).

Definition: A disturbance that transfers energy through matter or space.

Types of Waves:

• Transverse Waves: Particles of the medium move perpendicular to the direction of wave propagation (e.g., light waves).
• Longitudinal Waves: Particles of the medium move parallel to the direction of wave propagation (e.g., sound waves).

Wavelength (\(\lambda\)): The distance between consecutive crests or troughs of a wave.

Frequency (f): The number of waves that pass a point in one second, \( f = \frac{1}{T} \).

Wave Speed (v): The speed at which a wave propagates, \( v = f\lambda \).

Definition: The increase in internal energy of a system is equal to the heat added to the system minus the work done by the system, \( \Delta U = Q - W \).

Definition: The entropy of an isolated system always increases over time, and energy transformations are not 100% efficient.

Entropy (\(S\)): A measure of the disorder or randomness of a system.

Definition: A device that converts heat energy into mechanical work, often operating in a cyclic process.

Efficiency: The efficiency of a heat engine is given by \( \eta = \frac{W_{out}}{Q_{in}} \), where \( W_{out} \) is the work output and \( Q_{in} \) is the heat input.

Conduction: Transfer of heat through a material by direct contact.

Convection: Transfer of heat by the movement of fluids (liquids or gases).

Radiation: Transfer of heat through electromagnetic waves without the need for a medium.

Definition: A property of particles that causes them to experience a force when placed in an electric field. Charges can be positive or negative, with like charges repelling and opposite charges attracting.

Units: Coulombs (C).

Conservation of Charge: The total charge in an isolated system remains constant.

Definition: The force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them, \( F = k_e \frac{|q_1 q_2|}{r^2} \).

Constant: \( k_e = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).

Definition: A field that surrounds electric charges and exerts a force on other charges within the field. The electric field \( \vec{E} \) is defined as the force \( \vec{F} \) per unit charge \( q \), \( \vec{E} = \frac{\vec{F}}{q} \).

Units: Newtons per Coulomb (N/C) or Volts per meter (V/m).

Field of a Point Charge: \( \vec{E} = k_e \frac{q}{r^2} \hat{r} \).

Definition: The work done per unit charge in bringing a charge from infinity to a point in space, \( V = \frac{W}{q} \).

Units: Volts (V), where 1 V = 1 J/C.

Potential Due to a Point Charge: \( V = k_e \frac{q}{r} \).

Definition: The ability of a system to store electric charge, \( C = \frac{Q}{V} \).

Units: Farads (F), where 1 F = 1 C/V.

Capacitance of a Parallel Plate Capacitor: \( C = \frac{\epsilon_0 A}{d} \), where \( \epsilon_0 \) is the permittivity of free space, \( A \) is the area of the plates, and \( d \) is the separation between the plates.

Definition: The electric flux through a closed surface is proportional to the charge enclosed by the surface, \( \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \).

Electric Flux: \( \Phi_E = \oint \vec{E} \cdot d\vec{A} \).

Definition: The flow of electric charge, \( I = \frac{dQ}{dt} \).

Units: Amperes (A), where 1 A = 1 C/s.

Direction: Conventional current flows from positive to negative, opposite to the flow of electrons.

Definition: The current flowing through a conductor between two points is directly proportional to the voltage across the two points, \( V = IR \).

Resistance: \( R = \frac{V}{I} \), measured in Ohms (Ω).

Series: \( R_{eq} = R_1 + R_2 + R_3 + \dots \).

Parallel: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \).

Kirchhoff's Current Law (KCL): The total current entering a junction equals the total current leaving the junction.

Kirchhoff's Voltage Law (KVL): The sum of all voltages around a closed loop equals zero.

Series: \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots \).

Parallel: \( C_{eq} = C_1 + C_2 + C_3 + \dots \).

RC Circuits: The time-dependent behavior of resistors and capacitors in circuits is characterized by the time constant \( \tau = RC \).

Power Delivered by a Resistor: \( P = I^2R = IV = \frac{V^2}{R} \).

Units: Watts (W), where 1 W = 1 J/s.

Definition: A field produced by moving electric charges or by changing electric fields, \( \vec{B} \).

Units: Tesla (T), where 1 T = 1 N/(A·m).

Magnetic Force on a Moving Charge: \( \vec{F}_B = q\vec{v} \times \vec{B} \).

Definition: A current-carrying wire in a magnetic field experiences a force, \( \vec{F} = I\vec{L} \times \vec{B} \).

Right-Hand Rule: Point your thumb in the direction of the current, and your fingers in the direction of the magnetic field; your palm points in the direction of the force.

Definition: Describes the magnetic field generated by a steady current, \( d\vec{B} = \frac{\mu_0 I d\vec{L} \times \hat{r}}{4\pi r^2} \).

Permeability of Free Space: \( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \).

Definition: The integral of the magnetic field \( \vec{B} \) around a closed loop is proportional to the total current passing through the loop, \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \).

Definition: A changing magnetic field induces an electromotive force (EMF) in a circuit, \( \mathcal{E} = -\frac{d\Phi_B}{dt} \).

Magnetic Flux: \( \Phi_B = \oint \vec{B} \cdot d\vec{A} \).

Definition: The direction of the induced EMF is such that it opposes the change in magnetic flux that caused it.

Law of Reflection: The angle of incidence \( \theta_i \) is equal to the angle of reflection \( \theta_r \), \( \theta_i = \theta_r \).

Specular Reflection: Reflection from a smooth surface, where parallel rays remain parallel.

Diffuse Reflection: Reflection from a rough surface, where parallel rays scatter in different directions.

Snell's Law: The relationship between the angles of incidence and refraction when a wave passes from one medium to another, \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \).

Index of Refraction: \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the medium.

Definition: When a wave strikes the boundary between two media at an angle greater than the critical angle, it reflects entirely back into the original medium.

Critical Angle: The angle of incidence at which the refracted ray is parallel to the boundary, \( \sin \theta_c = \frac{n_2}{n_1} \) (for \( n_1 > n_2 \)).

Thin Lens Equation: Relates the focal length \( f \) of a lens to the object distance \( d_o \) and the image distance \( d_i \), \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \).

Magnification: The ratio of the height of the image \( h_i \) to the height of the object \( h_o \), \( M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \).

Converging (Convex) Lens: Brings parallel rays to a focus, focal length is positive.

Diverging (Concave) Lens: Causes parallel rays to diverge as if they originated from a point, focal length is negative.

Mirror Equation: Similar to the lens equation, \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length.

Concave Mirror: A converging mirror, where parallel rays converge to a focal point.

Convex Mirror: A diverging mirror, where parallel rays diverge, making them appear to come from a focal point behind the mirror.

Constructive Interference: When two waves meet in phase, their amplitudes add up, producing a wave of greater amplitude.

Destructive Interference: When two waves meet out of phase, their amplitudes subtract, potentially canceling each other out.

Double-Slit Experiment: Demonstrates the interference pattern created by light passing through two closely spaced slits, producing a pattern of bright and dark fringes.

Definition: The bending of waves around obstacles and openings.

Single-Slit Diffraction: Produces a central bright fringe with alternating dark and bright fringes on either side, intensity decreases with distance from the center.

Definition: The emission of electrons from a material when light of sufficient frequency shines on it.

Einstein's Equation: \( E_k = hf - \phi \), where \( E_k \) is the kinetic energy of the emitted electrons, \( hf \) is the energy of the photons, and \( \phi \) is the work function of the material.

Work Function: The minimum energy required to remove an electron from the surface of a material.

Definition: The concept that particles such as electrons exhibit both wave-like and particle-like properties.

De Broglie Wavelength: \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

Bohr Model: Describes electrons orbiting the nucleus in quantized energy levels. Electrons emit or absorb energy when transitioning between levels.

Quantum Mechanical Model: Describes electrons as existing in probability clouds (orbitals) around the nucleus, with their exact position and momentum described by the Heisenberg Uncertainty Principle.

Special Relativity: Describes the physics of objects moving at constant speeds, especially those close to the speed of light.

Key Equations:

• Time Dilation: \( \Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}} \)
• Length Contraction: \( L' = L\sqrt{1 - \frac{v^2}{c^2}} \)
• Mass-Energy Equivalence: \( E = mc^2 \)

Radioactive Decay: The process by which an unstable atomic nucleus loses energy by emitting radiation. Types include alpha, beta, and gamma decay.

Half-Life: The time required for half of the radioactive nuclei in a sample to decay.

Nuclear Fission: The process of splitting a large atomic nucleus into smaller particles, releasing a significant amount of energy.

Nuclear Fusion: The process of combining two light atomic nuclei to form a heavier nucleus, releasing energy in the process.