Physics and Math

Kinematics and Dynamics

Vectors and Scalars

Vectors: Physical quantities that have both magnitude and direction. Examples: displacement, velocity, acceleration, and force

Scalars: Quantities without direction. Scalar quantities may be the magnitude of vectors, like speed, or may be dimensionless, like coefficients of friction

Vector Addition: Tip-to-tail method, or you can break the vector into its component parts and use Pythagorean theorem

Vector Subtraction: Change the direction of the subtracted vector and then do a tip-to-tail addition

Vector Multiplication:
By scalar: Changes the magnitude and may reverse the direction.
Dot Product: A • B = |A||B| cos (θ), results in a scalar quantity
Cross Product: A × B = |A||B| sin (θ), results in a new vector. Direction of the new vector can be found using the right-hand rule

Forces and Acceleration

Force: Any push or pull that has the potential to result in an acceleration

Gravity: The attractive force between two objects as a result of their masses

Friction: A force that opposes motion as a function of electrostatic interactions at the surfaces between two objects
Static Friction: Stationary object
Kinetic Friction: Sliding object
f = uN

Mass: A measure of the inertia of an object – its amount of material

Weight: The force experienced by a given mass due to the gravitational attraction to the Earth

Acceleration: The vector representation of the change in velocity over time.

Torque: A twisting force that causes rotation
τ = r F sin(θ)
POS = counterclockwise
NEG = clockwise

Mechanical Equilibrium

Free Body Diagrams: Representations of the forces acting on an object.

Translational Equilibrium: Occurs in the absence of any net forces acting on an object

Rotational Equilibrium: Occurs in the absence of any net torques acting on an object. The center of mass is the most commonly used pivot point.

Newton’s Laws

First Law: An object will remain at rest or move with a constant velocity if there is no net force on the object
Fnet = m a = 0 if at rest or constant velocity

Second Law: Any acceleration is the result a net force > 0
Fnet = m a

Third Law: Any two objects interacting with one another experience equal and opposite forces as a result of their interaction
FAB = -FBA

Displacement and Velocity

Displacement: The vector representation of a change in position. Path independent

Distance: A scalar quantity that reflects the path traveled

Velocity: The vector representation of the change in DISPLACEMENT with respect to time

Avg Velocity = ^{Total displacement} / _{Total time}
Avg Speed = ^{Total distance traveled} / _{Total time}

Instantaneous Velocity: The change in displacement over time as the time approaches 0
Instantaneous Speed: The magnitude of the instantaneous velocity vector

Motion with Constant Acceleration

Linear Motion: Includes free fall and motion in which the velocity and acceleration vectors are parallel or antiparallel

Kinematics Equations for Linear Motion
vf = v0 + a Δt      Δx = v0 Δt + ½ a (Δt)²
vf² = v0² + 2 a Δx      Δx = v̄ Δt (if a = 0)

Projectile Motion: Contains both an x- and y-component. Assuming negligible air resistance, the only force acting on the object is gravity. X velocity is constant throughout.

Inclined Planes: Force components:
Parallel to the ramp use sinθ. “Sin is sliding ↓ the slide”.
Perpendicular to the ramp use cosθ.

Circular Motion: Best thought of as having radial and tangential dimensions. Centripetal force vector points radially inward, the instantaneous velocity vector points tangentially.
Centripetal force: Fc = (mv²)/r

Work and Energy

Energy

Structural Proteins: The property of a system that enables it to do something or make something happen, including the capacity to do work. SI units are joules (J).
J = (kg • m²) / s²

Kinetic Energy: Energy associated with the mvmt of objects. It depends on mass and speed squared.
KE = ½ m v²

Potential Energy: Energy stored within a system.

Gravitational Potential Energy: Related to the mass of an object and its height above a zero point.
U = m g h

Elastic Potential Energy: Related to the spring constant and the degree of stretch or compression of a spring squared.
U = ½ k x²

Electrical Potential Energy: The energy between two charged particles.

Chemical Potential Energy: The energy stored in the bonds of compounds.

Conservative Forces: Path independent and do not dissipate the mechanical energy of a system. Examples: Gravity and electrostatic forces.

Nonconservative Forces: Path dependent and cause dissipation of mechanical energy from a system. Examples: Friction, air resistance, and viscous drag.

Work

Work: The process by which energy is transferred from one system to another. Can be expressed as the dot product of force and displacement:
W = F d = F d cos(θ)

Power: The rate at which work is done or energy is transferred. SI unit is watt (W).
W = J / s = N m / s = kg m² / s³

Work-Energy Theorem: When net work is done on or by a system, the system’s kinetic energy will change by the same amount.
Wnet = ΔK = Kf – Ki

Mechanical Advantage

Mechanical Advantage: The factor by which a simple machine multiplies the input force to accomplish work. The input force necessary to accomplish the work is reduced and the distance through which the reduced input force must be applied is increased by the same factor.

MA of an Inclined Plane:
MA = (Length of incline) / (Height of incline)

Simple Machines: Inclined plane, wedge, wheel and axle, lever, pulley, and screw.

Efficiency: The ratio of the machine’s work output to work input when nonconservative forces are taken into account.
Mechanical Advantage = Fout / Fin

Thermodynamics

0^th Law of Thermodynamics

Thermal Equilibrium: When systems have the same average KE and thus the same temperature. No heat transfer.

Temperature: The average kinetic energy of the particles that make up a substance.
°F = (9/5) °C + 32
°C = (5/9) (°F – 32)
K = °C + 273

Thermal Expansion: Describes how a substance changes in length or volume as a function of the change in temperature.
ΔL = α L ΔT
ΔV = β V ΔT

1^st Law of Thermodynamics

A statement of conservation of energy: The total energy in the universe can never decrease or increase. For an individual system:
ΔU = Q – W

• ΔU = change in system’s internal energy
• Q = energy transferred into the system as heat
• W = work done by the system

Heat: The process by which energy transfer between two objects at different temperatures that occurs until the two objects come into thermal equilibrium (reach the same temperature).
q = m c ΔT

Specific Heat: The amount of energy necessary to raise one gram of a substance by 1° C or 1 K.
Specific heat of H₂O = 1 cal/g•K = 4.184 J/g•K

Heat of Transformation: The energy required for a phase change of a substance (temperature does not change during the transformation).
q = m L     L = heat of transformation

Processes with Constant Variable:
Isobaric: Pressure is constant, ΔP = 0
Isothermal: Temp is constant, ΔU = 0
Adiabatic: No heat is exchanged, Q = 0
Isovolumetric (isochoric): Volume is constant, ΔV = 0, so Work = 0

Work of a Gas:
W = –P ΔV

Systems

Isolated System: Do not exchange matter or energy with surroundings.

Closed System: Exchange energy but not matter with their surroundings.

Open System: Exchange both energy and matter with their surroundings.

State Functions: Pathway independent and are not themselves defined by a process. Include: Pressure, density, temp, volume, enthalpy, internal energy, Gibbs free energy, and entropy.

Process Functions: Describe the pathway from one equilibrium state to another. Include: work and heat.

Note: An isolated system does not exchange energy or matter with surroundings.

2^nd Law of Thermodynamics

In a closed system, up to and including the universe, energy will spontaneously and irreversibly go from being localized to being spread out.

Entropy: A measure of how much energy has spread out or how spread out energy has become.

Low Entropy → High Entropy

Fluids

Characteristics of Fluids and Solids

Fluids: Substances that flow and conform to the shape of their containers, includes liquids and gases. They can exert perpendicular forces but not shear forces.

Solids: Do not flow. They maintain their shape regardless of their container.

Density: Mass per unit volume of substance.
ρ = m / V

Pressure: A measure of force per unit area; it is exerted by a fluid on the walls of its container and on objects placed in the fluid. Scalar quantity. The pressure exerted by a gas on its container will always be perpendicular to the container walls.
P = F / A

Absolute Pressure: The sum of all pressures at a certain point within a fluid; it is equal to the pressure at the surface of the fluid plus the pressure due to the fluid itself.
Ptotal = P0 + ρ g h
In water, every additional 10m of depth adds ≈ 1 atm to Ptotal

Gauge Pressure: The difference between absolute pressure and atmospheric pressure. In liquids, gauge pressure is caused by the weight of the liquid above the point of measurement.
Pgauge = P – Patm = (P0 + ρ g z) – Patm

Hydrostatics

Pascal’s Principle: A pressure applied to an incompressible fluid will be distributed undiminished throughout the entire volume of the fluid.

Hydraulic Machines: Operate based on the application of Pascal’s principle to generate mechanical advantage.

Archimedes’ Principle: When an object is placed in a fluid, the fluid generates a buoyant force against the object that is equal to the weight of the fluid displaced by the object.
FB = ρ V g
Also, m = ρ V and F = P A

(Densityobject / Densitydisplaced fluid) = (Weightobject in air / (Weightobject in air – Weightobject in water))

If the max buoyant force is larger than the force of gravity on the object, the object will float. If the max buoyant force is smaller than the force of gravity on the object, the object will sink.

If FB > m_object g, then the object floats.
If FB < m_object g, then the object sinks.

Specific Gravity: Ratio of density of an object to density of water.
Specific gravity = ρobject / ρwater

Cohesive vs. Adhesive: Fluids experience cohesive forces with other molecules of the same fluid and adhesive forces with other materials.

Surface Tension: Cohesive forces give rise to surface tension.

Hydraulic Lift:
P = F1 / A1 = F2 / A2
F2 = F1 (A2 / A1)

Fluid Dynamics

Viscosity: A measure of a fluid’s internal friction. Viscous Drag is a nonconservative force generated by viscosity.

Laminar Flow: Smooth and orderly.

Turbulent Flow: Rough and disorderly.

Poiseuille’s Law: Determines the rate of laminar flow.
Q = (π r⁴ ΔP) / (8 η L)
The relationship between radius and pressure gradient is inverse exponential to the fourth power.

Flow Rate: Q = (Vol / time) = A v
A = cross sectional area, v = velocity

Continuity Equation: Fluids will flow more quickly through narrow passages and more slowly through wider ones.
Q = v1 A1 = v2 A2

Bernoulli’s Equation: The sum of the static pressure and the dynamic pressure will be constant between any two points in a closed system.
P1 + ½ ρ v1² + ρ g h1 = P2 + ½ ρ v2² + ρ g h2

Venturi Effect: The velocity of a fluid passing through a constricted area will INCREASE and its static pressure will DECREASE.

Venturi Tube: The average height of the horizontal tube remains constant, so ρ g h remains constant at points 1 and 2. As cross-sectional area decreases from point 1 to point 2, the linear speed must increase. As the dynamic pressure increases, the absolute pressure must decrease at point 2, causing the column of fluid sticking up from the Venturi tube to be lower at point 2.

Fluids in Physiology

Circulatory System: The circulatory system behaves as a closed system with nonconstant flow. The nonconstant flow = our pulse.
v = Q / A = (cardiac output) / (cross–sectional area)
ΔP = Q × R = cardiac output × resistance
ΔP = v A R

Pressure is directly related to velocity, area, and resistance. Area is inversely related to resistance and velocity. Cross–sectional area↑ → Resistance↓ and/or velocity↓

Breathing: Inspiration and expiration create a pressure gradient not only for the respiration system, but for the circulatory system too.

Alveoli: Air at the alveoli has essentially zero speed.

Electrostatics and Magnetism

Charges

Coulomb: The SI unit of charge

Protons & Electrons: Protons have a positive charge and electrons have a negative charge. Both protons and electrons possess the fundamental unit of charge (q = 1.60 × 10⁻¹⁹ C). Protons and electrons have different masses.

Attraction & Repulsion: Opposite charges exert attractive forces, and like charges exert repulsive forces.

Conductors: Allow the free and uniform passage of electrons when charged.

Insulators: Resist the movement of charge and will have localized areas of charge that do not distribute over the surface of the material.

Coulomb’s Law

Coulomb’s Law: Gives the magnitude of the electrostatic force vector between two charges. The force vector points along the line connecting the centers of the two charges.
F = k |q₁ q₂| / r²

Electric Field: Every charge generates an electric field, which can exert forces on other charges.
E = (Force exerted on a test charge / magnitude of that charge) = F / q = kQ / r²

Field Lines: Used to represent the electric field vectors for a charge. They show the activity of a positive test charge, which would move away from a positive charge and move toward a negative charge (north to south). The field is stronger where the field lines are closer together.

Special Cases in Electrostatics

Equipotential Lines: A line on which the potential at every point is the same. Equipotential lines are always perpendicular to electrical field lines. Work will be done when a charge is moved from one equipotential line to another. No work is done when a charge moves from a point on an equipotential line to another point on the same line.

Electric Dipole: Generated by two charges of opposite sign separated by a fixed distance d. In an external electric field, an electric dipole will experience a net torque until it is aligned with the electric field vector. An electric field will not induce any translational motion in the dipole regardless of its orientation with respect to the electric field vector.
V = (k q d / r²) cos(θ)

Net Torque: τ = p E sin(θ)

Dipole Moment: The product of charge and separation distance
p = q d

Essential Equations for Test Day

Fe = k q₁ q₂ / r²     U = k Q q / r E = k Q / r²     V = k Q / r

Electrical Potential Energy

Electrical potential energy is the amount of work required to bring the test charge from infinitely far away to a given position in the vicinity of a source charge.

Increases: Like charges move toward each other. Opp charges move apart.
Decreases: Opp charges move toward each other. Like charges move apart.

Electrical Potential Energy: U = k Q q / r

Electrical Potential

Electrical potential is the electrical potential energy per unit charge. Different points in the space of an electric field surrounding a source charge will have different electrical potential values.

Electrical Potential: V = U / q    1 volt = 1 J/C

Voltage: Potential difference. The change in electrical potential that accompanies the mvmt of a test charge from one position to another.
ΔV = Vb – Va = Wab / q

Test Charges: Will move spontaneously in whichever direction results in a decrease in their electrical potential energy.
POS Test Charges: High potential → Low potential
NEG Test Charges: Low potential → High potential

Magnetism

Magnetic Field: Created by magnets and moving charges. SI unit is the tesla (T). 1 T = 10,000 gauss.
Straight Wire: B = μ₀ I / (2πr)
Loop of Wire: B = μ₀ I / (2r)

Diamagnetic Materials: Possess NO unpaired electrons and are slightly REPELLED by a magnet.

Paramagnetic Materials: Possess SOME unpaired electrons and become WEAKLY MAGNETIC in an external magnetic field.

Ferromagnetic Materials: Possess SOME unpaired electrons and become STRONGLY MAGNETIC in an external magnetic field.

Characteristics of Magnetic Fields: Current-carrying wires create magnetic fields that are concentric circles surrounding the wire. External magnetic fields exert forces on charges moving in any direction except parallel or antiparallel to the field. Point charges moving through uniform circular motion in a uniform magnetic field will experience the centripetal force as the magnetic force acting on the point charge. Determine direction using the right-hand rule.

Moving Point Charge: Fb = q v B sin(θ)

Current-Carrying Wire: Fb = I L B sin(θ)

Lorentz Force: Sum of the electrostatic and magnetic forces acting on a body

Circuits

Charges

Current: The movement of charge that occurs between two points that have different electrical potentials. By convention, current is defined as the mvmt of positive charge from the high-potential end of a voltage source to the low-potential end. In reality, it is negatively-charged particles (electrons) that move in a circuit, from low potential to high potential.
I = Q / Δt

Conductive Materials:
Metallic Conduction: The flow of current due to movement of electrons.
Electrolytic Conduction: The movement of free ions under electric field.
Insulators: Materials that do not conduct a current.

Kirchhoff’s Laws:
Junction Rule: The sum of the currents flowing into a junction is equal to the sum of the currents flowing out of that junction.
Loop Rule: In a closed loop, the sum of voltage sources is always equal to the sum of voltage drops.
Vsource = Vdrop

Resistance

Resistance: The opposition that a substance offers to the flow of e⁻.

Resistors: Conductive materials with a moderate amount of resistance that slow down electrons without stopping them.
R = ρ L / A    where ρ = resistivity, L = length of resistor, A = cross sectional area

Ohm’s Law: For a given resistance, the magnitude of the current through a resistor is proportional to the voltage drop across the resistor.
V = I R

Resistors in Series: Additive. Sum together to create the total resistance of a circuit.
Rₛ = R₁ + R₂ + R₃ + … + Rn

Resistors in Parallel: ↓Equivalent resistance of a circuit. To get the total resistance, add the reciprocals of the resistances of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance.
1 / Rp = 1 / R₁ + 1 / R₂ + 1 / R₃ + … + 1 / Rn

Capacitance and Capacitors

Capacitors: Have the ability to store and discharge electrical potential energy.

Capacitance: In parallel plate capacitors, it is determined by the area of the plates and the distance between the plates.
C = Q / V
Capacitance based on parallel plate geometry: C = ε₀ (A/d)

Electric field in a capacitor: E = V/d

Potential energy of a capacitor: U = (1/2) C V²

Series: ↓Equivalent capacitance of a circuit

Parallel: Sum together to create a large equivalent capacitance

Dielectric Materials: Insulators placed between the plates of a capacitor that increase capacitance by a factor equal to the material’s dielectric constant, κ.

Meters

Ammeter: Inserted in SERIES in a circuit to measure current; they have negligible resistance.

Voltmeter: Inserted in PARALLEL in a circuit to measure a voltage drop; they have very large resistances.

Ohmmeter: Inserted around a resistive element to measure resistance; they are self-powered and have negligible resistance.

Capacitors in Series: The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances. Total capacitance will always be less than the value of the smallest capacitor.

Capacitors in Parallel: Total capacitance is equal to the sum of all the individual capacitances.

Waves and Sound

General Wave Characteristics

Transverse Waves: Have oscillations of wave particles perpendicular to the direction of wave propagation. LIGHT

Longitudinal Waves: Have oscillations of wave particles parallel to the direction of wave propagation. SOUND

v = fλ     v = wave speed     f = frequency     λ = wavelength

v = √(B/ρ)     B = bulk modulus (increases from gas to liquid to solid)     ρ = density

Displacement (x): Refers to how far a point is from the equilibrium position, expressed as a vector quantity.

Amplitude (A): The magnitude of its maximal displacement. The maximum point is called a crest. The minimum point is called a trough.

Wavelength (λ): The distance between two crests or two troughs.

Frequency (f): The number of cycles it makes per second. Expressed in Hz.

Angular Frequency (ω): Also known as radial or circular frequency, measures angular displacement per unit time. Expressed in radians per second. ω = 2πf = 2π/T

Period (T): The number of seconds it takes to complete a cycle. It is the inverse of frequency. T = 1/f

Interference: Describes the ways in which waves interact in space to form a resultant wave.

• Constructive Interference: Occurs when waves are exactly in phase with each other. The amplitude of the resultant wave is equal to the sum of the amplitudes of the two interfering waves.
• Destructive Interference: Occurs when waves are exactly out of phase with each other. The amplitude of the resultant wave is equal to the difference in amplitude between the two interfering waves.
• Partially Constructive / Destructive Interference: Occurs when two waves are not quite perfectly in or out of phase with each other. The displacement of the resultant wave is equal to the sum of the displacements of the two interfering waves.

Traveling Waves: Have continuously shifting points of maximum and minimum displacement.

Standing Waves: Produced by the constructive and destructive interference of two waves of the same frequency traveling in opposite directions in the same space.

Antinodes: Points of maximum oscillation.

Nodes: Points where there is no oscillation.

Resonance: The increase in amplitude that occurs when a periodic force is applied at the natural (resonant) frequency.

Damping: A decrease in amplitude caused by an applied or nonconservative force.

Sound

Sound: Produced by mechanical disturbance of a material that creates an oscillation of the molecules in the material.

Propagation: Sound propagates through all forms of matter but not through a vacuum. Fastest through solids, followed by liquids, and slowest through gases. Within a medium, as density increases, speed of sound decreases.

Pitch: Our perception of frequency.

Doppler Effect: A shift in the perceived frequency of a sound compared to the actual frequency of the emitted sound when the source of the sound and its detector are moving relative to one another.

• The apparent frequency will be higher than the emitted frequency when the source and detector are moving toward each other.
• The apparent frequency will be lower than the emitted frequency when the source and detector are moving apart from each other.
• The apparent frequency can be higher, lower, or equal to the emitted frequency when the two objects are moving in the same direction, depending on their relative speeds.

f′ = f ( (v ± v_D) / (v ∓ v_S) )
f′ = perceived freq     f = emitted freq
Use the top sign for “toward”, bottom sign for “away”

Intensity: Intensity is related to a wave’s amplitude. Intensity decreases over distance and some energy is lost to attenuation from frictional forces.

I = P/A     P = power     A = area

Strings and Open Pipes: Support standing waves and the length of the string or pipe is equal to some multiple of half-wavelengths.
L = nλ/2 (n = 1, 2, …)

Closed Pipes: Closed at one end. Support standing waves, and the length of the pipe is equal to some odd multiple of quarter-wavelengths.
L = nλ/4 (n = 1, 3, …)

Ultrasound: Uses high frequency sound waves to compare the relative densities of tissues in the body. Doppler Ultrasound is used to determine the flow of blood within the body.

1st, 2nd, and 3rd Harmonics of a String: N = node, A = antinode. As a shortcut, for strings attached at both ends, the number of antinodes present will tell you which harmonic it is.

Light and Optics

Electromagnetic Spectrum

Electromagnetic Waves: Transverse waves that consist of an oscillating electric field and an oscillating magnetic field. The two fields are perpendicular to each other and to the direction of propagation of the wave.

Electromagnetic Spectrum: The range of frequencies and wavelengths found in EM waves.

EM Spectrum:

Small λ → high energy
High λ → low energy

Note: Gamma, X-ray, and higher UV are ionizing. They can liberate electrons from nearby atoms and create free radicals.

Visible Spectrum:

Wavelength (nm)

Hydrogen Spectral Series:

• Lyman: Ultraviolet, n = 1
• Balmer: Visible, n = 2
• Paschen: Infrared, n = 3

Acrostic: “Loves Beer Pong”, then n = 1, n = 2, n = 3

Rydberg Formula: hf = R (1/n_final² – 1/n_initial²)

Diffraction

Diffraction: The bending and spreading out of light waves as they pass through a narrow slit. Diffraction may produce a large central light fringe surrounded by alternating light and dark fringes with the addition of a lens.

Interference: When waves interact with each other, the displacements add together in a process called interference.

Young’s Double-Slit Experiment: Shows the constructive and destructive interference of waves that occur as light passes through parallel slits, resulting in minima (dark fringes) and maxima (bright fringes) of intensity.

Polarization

Plane-Polarized Light: A polarizing filter only lets light through if the E field of the wave aligns with the openings in the filter. The E fields of the exiting light oscillate along the same axis.

Circular Polarized Light: All of the light rays have electric fields with equal intensity but constantly rotating direction. Circularly polarized light is created by exposing unpolarized light to special pigments or filters.

Geometric Optics

Reflection: Rebounding of incident light waves at a medium’s boundary

Law of Reflection: θ₁ = θ₂

Spherical Mirrors:

Mirror	Image Produced	Position	Cause
Concave	Real	Inverted	Object’s position is greater than the focal length
	Virtual	Upright	Object’s position is less than the focal length
Convex	Virtual	Upright & smaller
Plane	Virtual	Upright & same size	Can think of these as spherical mirrors with infinite radii of curvature

Refraction: The bending of light as it passes from one medium to another. The speed of light changes depending on index of refraction of the medium. This speed change causes refraction. The amount of refraction depends on the wavelengths involved.

Index of refraction: n = c/v
c = speed of light in vacuum, v = speed of light in the medium

Dispersion: When various wavelengths of light separate from each other.

Snell’s Law: n₁ sin θ₁ = n₂ sin θ₂

Total Internal Reflection: When light cannot be refracted out of a medium and is instead reflected back inside the medium. Occurs when light moves from a medium with a HIGHER index of refraction to a medium with a LOWER index of refraction with a high incident θ.

Critical Angle: The minimum incident angle at which total reflection occurs.
θ_c = sin⁻¹(n₂/n₁)

Lenses:

Refract light to form images of objects. Thin symmetrical lenses have focal points on each side.

Lens	Image Produced	Position	System
Convex	Real	Inverted	Converging system
	Virtual	Upright	Converging system
Concave	Virtual	Upright	Diverging system

Lenses Formula: f = focal length, F = focus, 1/f = 1/o + 1/i

Lensmaker’s Equation: 1/f = (n – 1)(1/r₁ – 1/r₂)

Atomic and Nuclear Phenomena

The Photoelectric Effect

The ejection of an electron from the surface of a metal in response to light

Energy of a photon of light: E = hf

To calculate λ from f use: c = fλ,    c = speed of light = 3 × 10⁸ m/s

Maximum kinetic energy in the photoelectric effect: K_max = hf – W

Threshold Frequency: The minimum light frequency necessary to eject an electron from a given metal.

Work Function: The minimum energy necessary to eject an electron from a given metal.

W = hf_T    h = Planck’s constant = 6.626 × 10^-34 J·s

Absorption and Emission of Light

Bohr Model: States that electron energy levels are stable and discrete, corresponding to specific orbits.

Absorption: An electron can jump from a lower-energy to a higher-energy orbit by absorbing a photon of light of the same frequency as the energy difference between the orbits.

Emission: When an electron falls from a higher-energy to a lower-energy orbit, it emits a photon of light of the same frequency as the energy difference between the orbits.

Absorption Spectra: May be impacted by small changes in molecular structure.

Fluorescence: Occurs when a species absorbs high-frequency light and then returns to its ground state in multiple steps. Each step has less energy than the absorbed light and is within the visible range of the electromagnetic spectrum.

Nuclear Binding Energy and Mass Defect

Nuclear Binding Energy: Is the amount of energy that is released when nucleons (protons and neutrons) bind together.

4 Fundamental Forces of Nature: Strong and weak nuclear force, electrostatic forces, gravitation.

Mass Defect: The difference between the mass of the unbonded nucleons and the mass of the bonded nucleons within the nucleus. The unbonded constituents have more energy and, therefore, more mass than the bonded constituents. The mass defect is the amount of mass converted to energy during nuclear fusion.

Nuclear Reactions

Fusion: Occurs when small nuclei combine into larger nuclei.

Fission: Occurs when a large nucleus splits into smaller nuclei.

Energy is released in both fusion and fission because the nuclei formed in both processes are more stable than the starting nuclei.

Radioactive Decay:
The loss of small particles from the nucleus.

Half-Life: The amount of time required for half of a sample of radioactive nuclei to decay. Or, the time it takes to reduce the radioactivity of a substance by half.

Exponential Decay: The rate at which radioactive nuclei decay is proportional to the number of nuclei that remain.

n = n₀ e^-λt
n = # of undecayed nuclei
n₀ = # of undecayed nuclei at t = 0
λ = known decay constant

Note: If the problem just says “beta”, they mean “beta negative”. Beta-negative is the default.

Mathematics

Arithmetic and Sig Figs

Scientific Notation: Improves the ease of calculation. It is usually helpful to convert a number to scientific notation

(3 × 10³) (9 × 10²) = (3 × 9)(10³ × 10²) = (3 × 9)(10³⁺²) = 2.1 × 10³

(1.5 × 10³)(3 × 10²) = 4.5 × 10⁵ – Add exponents

^{8 × 10-2} / _{2 × 10^-2} = 4 × 10⁰ – Subtract exponents

(2 × 10^-2)³ = 8 × 10^-6 – Multiply exponents

√(9 × 10⁸) = (9 × 10⁸)^1/2 = 3 × 10⁴ – Divide the exponent by 2

LARS mnemonic when moving the decimal within scientific notation.
Left → Add, Right → Subtract

481.2 × 10⁷ = 4.812 × 10⁹ – Left Add
0.00314 × 10^-3 = 3.13 × 10^-6 – Right Subtract

Significant Figures: Include all nonzero digits and any trailing zeroes in a number with a decimal point.

Estimation:
Multiplication: If one number is rounded up, the other should be rounded down in proportion.
Division: If one number is rounded up, the other should also be rounded up in proportion.

Exponents, Log and Ln

Estimating: To calculate the square root of any number less than 400, you can approximate its value by determining which two perfect squares it falls between. For example, √180 is between 13 and 14.

√180 = √4 × √45 = 2 × 3 × √5 = 6√5
√5 ≈ 2.2 so √65 ≈ 13.2

Square Roots: approximate values:

Common: 1² = 1, 6² = 36, 11² = 121, 16² = 256

Squares: 2² = 4, 7² = 49, 12² = 144, 17² = 289

3² = 9, 8² = 64, 13² = 169, 18² = 324

4² = 16, 9² = 81, 14² = 196, 19² = 361

5² = 25, 10² = 100, 15² = 225, 20² = 400

Log and Ln: log₁₀(A) = B     ln(A) = B
10^B = A     e^B = A     e = 2.7

log_A(1) = 0
log_A(A) = 1
log_A (greater than 1) = Positive
log_A (less than 1) = Negative

log(A × B) = log(A) + log(B)
log(A / B) = log(A) – log(B)
log(A^B) = B log(A)
log(1/A) = –log(A)

Estimating Log: log(A × 10^B) = log(A) + log(10^B) = log(A) + B
log(A × 10^B) ≈ B + 0.4

Trigonometry

SOH CAH TOA:
sin(θ) = O/H, cos(θ) = A/H, tan(θ) = O/A = sin(θ)/cos(θ)

Common Values:

θ	cos(θ)	sin(θ)	tan(θ)
0°	1	0	0
30°	√3/2	1/2	√3/3
45°	√2/2	√2/2	1
60°	1/2	√3/2	√3
90°	0	1	undefined
180°	-1	0	0

45-45-90 triangle

30-60-90 triangle

The Unit Circle:

x = cos(θ), y = sin(θ)
tan(θ) = y/x = sin(θ)/cos(θ)

Reasoning About the Design and Execution of Research

The Scientific Method

Initial steps: Focus on formulating a hypothesis.
Intermediate steps: Focus on testing that hypothesis.
Final steps: Provide results for further testing of the hypothesis.

FINER Method: Assesses the value of a research question on the basis of whether or not it is feasible, interesting, novel, ethical, and relevant.

Ethics

Medical Ethics:

4 tenets: beneficence, nonmaleficence, respect for patient autonomy, and justice

Research Ethics:

Respect for persons, justice, beneficence.
Must have equipoise – a lack of knowledge about which arm of research study is better for the subject.

Research in the Real World

Populations: All of the individuals who share a set of characteristics. Population data are called parameters.

Samples: A subset of a population that are used to estimate population data. Sample data are called statistics.

Internal Validity: If the outcome of the research is that the DV has been affected as a result of manipulating the IV. Any confounding variables have been controlled for.

External Validity: Refers to the ability of a study to be generalized to the population that it describes.

Within-Subject Design: Controls for individual variations in a measurement by comparing the scores of a subject in one condition to the scores of the same subject in other conditions. So the subject serves as its own control.

Statistical Significance: Refers to the low likelihood of the experimental findings being due to chance.

Basic Science Research

Occurs in the lab, not in human subjects. Basic science research is often the best type for demonstrating causality because the experimenter has the highest degree of control over the experimental conditions.

Variables:
Independent Variable: Manipulated
Dependent Variable: Observe for change.

Controls:
Positive Controls: Ensure that a change in the dependent variable occurs when expected.
Negative Controls: Ensure that no change in the dependent variable occurs when none is expected.

Accuracy (Validity): The quality of approximating the true value.
Precision (Reliability): The quality of being consistent in approximations.

Human Subject Research

Human subjects research is subject to ethical constraints that are generally absent in basic science research. Causal conclusions are harder to determine because circumstances are harder to control. Much of human subject research is observational.

Cohort Studies: Record exposures throughout time and then assess the rate of a certain outcome.

Cross-sectional Studies: Assess both exposure and outcome at the same point in time.

Case-Control Studies: Assess outcome status and then assess for exposure history.

Hill’s Criteria: Used to determine if causality can be supported. The criteria include temporality (necessary for causality), strength, dose-response, relationships, consistency, plausibility etc.

Bias:
Selection Bias: The sample differs from the population.
Detection Bias: Arises from educated professionals using their knowledge in an inconsistent way by searching for an outcome disproportionately in certain populations.
Hawthorne Effect: Behavior of subjects is altered simply by knowing that they are being studied.
Social Desirability Bias: A type of response bias that is the tendency of survey respondents to answer questions in a manner that will be viewed favorably by others.

Placebo Effect: Results are influenced by the fact that the subjects are aware they are or are not in the control group.

Confounding Variable: An extraneous variable that relates to BOTH the dependent and independent variables.

Mediating Variable: The means by which the IV affects the DV. It is the variable “middleman” between the IV and DV.

Moderating Variable: Influences the already established relationship between the IV and DV. Moderators affect the strength of the relationship between the two variables.

Data-Based and Statistical Reasoning

Measures of Central Tendency

Provide a single value representation for the middle of the data set.

Mean: The average.

Median: The value that lies in the middle of the data set. Tends to be least susceptible to outliers, but may not be useful for data sets with large ranges.

Mode: The data point that appears most often.

Distributions

Normal Distribution: Symmetrical and the mean, median, and mode are equal.

Standard Distribution: A normal distribution with a mean of 0 and a standard deviation of 1. It is used for most calculations.

Skewed Distribution: Have differences in their mean, median, and mode. Skew direction is the direction of the tail.

Bimodal Distribution: Multiple peaks, although not necessarily multiple modes.

Measures of Distribution

Range: Difference between largest and smallest value.

Interquartile Range: The difference between the value of the third quartile and first quartile. Can be used to determine outliers.

Standard Deviation (σ): A measurement of variability about the mean. Can be used to determine outliers.

Outliers: In general, any value that lies more than 3 standard deviations from the mean.

Probability

Independent Events: The probability of independent events does not change based on the outcomes of other events.

Dependent Events: The probability of a dependent event changes depending on the outcomes of other events.

Terminology: Mutually Exclusive: Cannot occur simultaneously.

When a set of outcomes is exhaustive, there are no other possible outcomes.

Statistical Testing

Hypothesis Tests: Use a known distribution to determine whether the null hypothesis can be rejected.

p-value: Whether or not a finding is statistically significant is determined by the comparison of a p-value to the selected significance level (α). A significance level of 0.05 is commonly used.

Confidence Intervals: Are a range of values about a sample mean that are used to estimate the population mean. A wider interval is associated with a higher confidence level (95% is common).

Hypothesis Testing Chart with Type 1 and Type 2 Errors

Charts, Graphs, and Tables

Pie and Bar Charts: Used to compare categorical data.

Histograms and Box Plots: Used to compare numerical data.

Linear, Semilog, and Log-log Plots: Can be distinguished by the axes.

Slope: rise / run

Exponential Relationship

Logarithmic Relationship

Quadratic Relationship

Curvilinear Relationship