MCAT Physics - Mandatory Assignment PDF

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Mandatory Assignment...

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MCAT Physics & Math 1. Kinematics, Dynamics 1.1. Units ● SI units: meters (m), kilograms (kg), seconds (s), etc.

1.2. Vectors, Scalars ● ● ● ●

Vector addition: add tip to tail Scalar vector multiplication Dot product: a • b = ||a|| ||b|| cos θ = axbx + ayby + … Cross product: a × b = ||a|| ||b|| sin θ n = det [[î, ĵ, k ], [ax, ay, az], [bx, by, bz]] for 3D ○ Right-hand rule

1.3. Displacement, Velocity ● Distance (x, scalar), displacement (x, vector) ● Speed (v, scalar), velocity (v, vector) ○ vavg = Δx/Δt

1.4. Forces, Acceleration ● Force (F): push/pull, in newtons (N = kg·m/s2) ○ Gravity: attractive force felt, exerted by all objects ■ F = GmM/r2, where G = universal gravitational constant ○ Friction: opposes motion ■ Static friction (fs): stationary object, depends on force being applied ● 0 ≤ fs ≤ μsN, where μ = coefficient of friction, N = normal force ● Object moves when force exceeds max static friction: fs, max = μsN ● Rolling w/o slipping: rolling object experiences instantaneous static, not kinetic, friction ■ Kinetic friction (fk): sliding object ● fk = μkN ● Usually μs ≥ μk ● Mass (m, object’s inertia, scalar), weight (w, gravity on object, vector) ○ w = mg, where g = acceleration due to gravity ■ gEarth = GMEarth/rEarth2 ○ Weight is applied to center of gravity: xCM = Σ mixi / Σ mi, etc. ● Acceleration (a): rate of change of v ○ aavg = Δv/Δt

1.5. Newton’s Laws of Motion ● 1st Law: inertia, Σ F = 0 if a = 0 ● 2nd Law: Σ F = ma ● 3rd Law: action and reaction, FA→B = –FB→A

1.6. Motion w/ Constant Acceleration ● Linear motion ○ v = v0 + at ○ x = x0 + vt + ½ at2 ○ v2 = v02 + 2a Δx ○ Δx = ½ (v + v0)t ● Free fall: object falls w/ constant acceleration (g), no air resistance ○ Terminal velocity: when gravitational force = drag force ● Projectile motion: free fall w/ ax = 0, constant ay = g ● Inclined plane: split forces into components perpendicular, parallel to plane ○ Fg, ⊥ = mg cos θ = –N, where θ = angle of incline ○ Fg, || = mg sin θ ● Circular motion ○ Uniform circular motion: v is always tangent to path, centripetal force always points inward ○ Centripetal acceleration: ac = v2/r ■ Centripetal force: Fc = mac = mv2/r

1.7. Mechanical Equilibrium ● Equilibrium: Σ F = 0, Σ τ = 0 ● Torque (τ): in N·m (= kg·m2/s2) ○ τ = r × F, where r = length from fulcrum to point of force ○ CCW τ > 0, CW τ < 0

2. Work, Energy 2.1. Energy ● Kinetic energy (K): energy of motion, in joules (J = N·m = kg·m2/s2) ○ K = ½ mv2 ● Potential energy (U): energy of position/etc. ○ Gravitational: Ug = mgh, where h = height ■ Generally, Ug = –GmM/r ○ Elastic: Uel = ½ kx2, where k = spring constant, x = extension of spring

● Mechanical energy (E): total energy, conserved (1st Law of Thermodynamics) ○ ΔE = ΔK + ΔU ○ Energy can still be “lost” as heat, etc. ● Conservation ○ Conservative forces: path-independent, conserves E (e.g., gravitational, electrostatic) ■ ΔE = 0 ○ Nonconservative forces: path-dependent, dissipates E as thermal/chemical energy (e.g., friction, etc.) ■ ΔE = WNC

2.2. Work ● Work (W): transfer of mechanical energy, in J ○ W=F•d ■ Only forces anti/parallel to displacement do work ○ Wnet = Kf – Ki ○ Gas compression: (+) work done on gas, (–) work done by gas ○ Gas expansion: (–) work done on gas, (+) work done by gas ● Pressure (P), volume (V) ○ P–V graphs: pressure vs. volume ■ Work = integral of curve ○ W = P ΔV when process is isobaric (ΔP = 0) ● Power (Ƥ): rate of work, in watts (W = J/s) ○ Ƥavg = ΔW/Δt = ΔE/Δt

2.3. Mechanical Advantage ● Simple machines: inclined plane, wedge (2 inclined planes), wheel + axle, lever, pulley, screw (rotating inclined plane) ○ Apply less force over more distance → work stays the same ■ e.g., to lift load by load distance using 3 pulleys (w = 3T), you need ⅓ as much effort, 3x as much effort distance ● Mechanical advantage = Fout/Fin ○ Factor by which a simple machine multiplies input force ● Efficiency = Wout/Win = (load)(load distance) / (effort)(effort distance) ○ Some work is wasted on nonconservative forces (e.g., friction)

3. Thermodynamics 3.1. Zeroth Law ● 0th Law of TD: thermal equilibrium between systems is transitive ● Temperature (T): avg. KE of particles, in Fahrenheit (°F)/Celsius (°C)/Kelvin (K)

○ K = °C + 273.15 ○ Absolute zero (0 K): no thermal energy ● Heat (Q): transfer of thermal energy from higher to lower temp ○ Thermal equilibrium: no net heat flow, equal temps ● Thermal expansion: for most objects, temp ↑ → size ↑ ○ Solids: ΔL = αL0ΔT, where L = length, α = coefficient of linear expansion ■ “When temp changes, length changes a lot (αL0ΔT)” ○ Liquids: ΔV = βV0ΔT, where V = volume, β = coefficient of volumetric expansion ■ For a given material, β = 3α (expands in 3 dimensions)

3.2. Systems ● Types ○ Isolated: no exchange w/ surroundings ■ Δ internal energy = 0 ○ Closed: closed to matter, exchanges only energy w/ surroundings ○ Open: exchanges both energy, matter w/ surroundings ● State functions: path-independent (e.g., “PD TV HUGS”: P, ρ, T, V, H, U, G, S) ● Process functions: path-dependent (e.g., W, Q)

3.3. First Law ● 1st Law of TD: ΔU = Q – W, where U = internal energy ○ System gains U from gaining heat, loses U from doing work ● Heat: energy transfers from higher- to lower-temp object, until thermal equilibrium (2nd Law of TD) ○ Heat transfer ■ Conduction: direct transfer thru molecular collisions (physical contact) ● Metals are best (e– sea = fast energy transfer), gases are worst (particles are far apart) ■ Convection: transfer by liquid/gas flow over object ■ Radiation: transfer thru EM waves (only type that can transfer thru a vacuum) ○ Specific heat (c): heat needed to raise 1 g substance by 1 K ■ cwater = 1 cal/(g·K)= 4.184 J/(g·K) ■ q = mc ΔT ○ Heat of transformation (L)/latent heat ■ Phase change: heat transfer doesn’t change temp until all of substance is converted ● Add heat to substance at phase-transition temp → # microstates (degrees of freedom, S) ↑, but avg. KE (temp) remains same ■ q = mL, where L = heat of transformation ● Heat of fusion (ΔHfus): freezing/solidification, melting/fusion at melting point ● Heat of vaporization (ΔHvap): boiling/evaporation/vaporization, condensation at boiling point ● Thermodynamic processes

○ Isothermal: constant temp, ΔU = 0 ■ 1st Law: Q = W ○ Adiabatic: no heat exchange, Q = 0 ■ 1st Law: ΔU = –W ○ Isovolumetric/isometric/isochoric: constant volume, ΔV = 0 ■ 1st Law: ΔU = Q ■ No work possible: W = 0 ○ Isobaric: constant pressure, ΔP = 0 ■ W = P ΔV

3.4. Second Law, Entropy ● 2nd Law of TD: heat transfers from higher- to lower-temp object, until thermal equilibrium ○ i.e., energy spontaneously disperses ● Entropy (S): spontaneous energy dispersal at some temp ○ ΔS = Qrev/T, where Qrev = heat transferred in reversible process ■ Reversible process: takes infinite time, system is always in equilibrium, no energy is dissipated ○ Concentrating energy requires work ○ Entropy of universe is always increasing: ΔSuniv = ΔSsys + ΔSsurr > 0

4. Fluids 4.1. Characteristics ● Fluids: flow, conform to shapes of their containers ● Solids: rigid, withstand shear (tangential) forces ● Density: ρ = m/V ○ ρwater = 1 g/mL = 1 g/cm3 = 1,000 kg/m3 ○ w = ρVg ○ Specific gravity = ρ/ρwater ● Pressure (P): in pascals (Pa = N/m2) ○ P = F/A, where F = magnitude of normal force ○ 1 atm = 760 mmHg = 760 torr = 101.3 kPa ○ Absolute/hydrostatic pressure: total pressure exerted on object submerged in fluid ■ P = P0 + ρgy, where P0 = incident/ambient pressure (at surface), ρ = density of fluid, y = depth ● Gauge pressure: ρgy ○ Gauge pressure = (abs. pressure) – (atm. pressure)

4.2. Hydrostatics ● Pascal’s Principle: in incompressible fluids, pressure is applied equally to all parts of fluid and its container ○ P = F1/A1 = F2/A2 ■ Area ↑ → force ↑ (not a simple machine) ○ W = F1d1 = F2d2 ■ Force ↑ → distance ↓ ● Archimedes’ Principle: buoyant force on object immersed in fluid = weight of displaced fluid ○ Fbuoy = ρfluidVdisplacedg ○ Object floats if avg. ρobject ≤ avg. ρfluid, sinks if avg. ρobject > avg. ρfluid ○ Objects sink only until Fbuoy = wobject ○ % submerged = ρobject/ρfluid ● Molecular forces ○ Cohesion: intramolecular attraction in liquid ■ Surface tension: thin, strong layer on surface ○ Adhesion: molecular attraction toward another substance ■ Meniscus: curved surface up side of container ● Concave (adhesion > cohesion), convex (cohesion > adhesion)

4.3. Fluid Dynamics ● Viscosity (η): fluid’s resistance to flow, measured in Pa·s ○ Ideal fluids: inviscid, incompressible ● Laminar flow: smooth, parallel ○ Poiseuille’s Law: Q = (πr4 ΔP) / (8ηL), where Q = flow rate, r = tube radius, L = tube length, ΔP = pressure gradient between 2 ends of tube ● Turbulent flow: rough, eddies (swirls usually downstream of object) ○ Critical speed: vc = (NRη) / (ρd), where NR = Reynolds number, d = tube diameter ■ When vfluid ≥ vc, laminar flow → turbulent flow, except at boundary layer (thin layer next to tube wall) ● Flow rate (Q): constant for closed system, independent of cross-sectional area ○ Continuity equation: Q = v1A1 = v2A2, where A = cross-sectional area ■ Fluids flow faster thru small openings, slower thru large openings ■ Q and v are different concepts! ○ Streamlines: pathways by moving fluid particles ● Bernoulli’s Equation: P + ½ ρv2 + ρgh is constant, where P = absolute pressure, h = height of fluid ○ Dynamic pressure: ½ ρv2 = K/V ○ Static pressure: P + ρgh = P + U ○ Energy for fluid movement ↑ → energy for static pressure ↓ ○ Pitot tube: measures static pressure to find v ○ Venturi effect: A ↓ → P ↓ ■ Continuity eq.: A ↓ → v ↑ → dyn. pressure (½ ρv2) ↑

■ Bernoullli’s eq.: dyn. pressure ↑ → static pressure (P + ρgh) ↓ ■ Avg. height stays same (ρgh does not change): static pressure ↓ → abs. pressure ( P) ↓ ■ Abs. pressure (P) ↓ → height of liquid from tube ↓

4.4. Physiology ● Circulatory system ○ Closed loop, non-constant flow rate (pulses), volume ↓ (oncotic/hydrostatic) ○ Farther from heart → resistance ↑ in each vessel, total resistance ↓ (parallel) ○ Skeletal muscles increase P in veins ● Respiratory system ○ Inspire (ΔP < 0), expire (ΔP > 0) ○ In alveoli, v ≈ 0

5. Electrostatics, Magnetism 5.1. Charges ● Protons (p+), electrons (e–) ○ Opposite charges attract, like charges repel ● Charge: in coulombs (C) ○ Charge of e– = 1.6 × 10–19 C ● Insulators: charge is localized, can’t transfer charge ● Conductors: charge distributed throughout surface, can transfer charge

5.2. Coulomb’s Law ● Coulomb’s Law: F = k qq0/r2, where F = electrostatic force, q = charge, r = distance ○ Coulomb’s constant: k = 1/(4πε 0) ≈ 9 × 109 N·m2/C2, where ε0 = vacuum permittivity ● Electric field (E): produced by source charge (q), exerts electrostatic force on test charge (q0) ○ E = F/q0 = k q/r2 ○ Field lines: from (+) to (–), never cross ■ Field lines closer together = stronger E

5.3. Electric Potential Energy ● Electric PE: energy of charge’s relative position to another ○ U = k qq0/r ■ U > 0 if repel, U < 0 if attract ○ W = ΔU ■ Work needed to bring q0 from ∞ to some distance in electric field

○ Charges spontaneously move from high to low electrical PE

5.4. Electric Potential ● Electric potential: V = U/q0 = k q/r, in volts (V = J/C) ● Potential diff./voltage: ΔV = Vb – Va = Wa→b/q0, where Wa→b = work needed to move q0 from a to b ○ W = q ΔV ■ Derivation: V = Ed = Fd/q = W/q ○ (+) charges spontaneously move from high to low V (ΔV < 0, Wab < 0) ○ (–) charges spontaneously move from low to high V (ΔV > 0, Wab < 0)

5.5. Special Cases ● Equipotential lines: every point on equipotential line has same potential ○ Electric field lines ⊥ equipotential lines ● Electric dipoles: 2 equal opposite charges, separated by a distance ○ Dipole moment: p = qd ■ Points (–) to (+) in physics ○ Potential at point far from dipole: V = U/q = k (p cos θ)/r2, where p = dipole moment, r = dist. from dipole to point ○ Perpendicular bisector of dipole: equipotential line bisects bipole ■ E = k p/r3, V = 0 ○ In electric field: τ = p × E, U = –p • E ■ At θ = 0, τ = 0, min U (stable equilibrium) ■ At θ = π/2, max +τ, U = 0

■ At θ = π, τ = 0, max U (unstable equilibrium) ■ At θ = 3π/2, max –τ, U = 0

5.6. Magnetism ● Magnetic materials ○ Diamagnetic: no unpaired e–, no net magnetic field, weakly antimagnetic ○ Paramagnetic: unpaired e– (parallel spins), randomly oriented magnetic dipoles ■ Weakly magnetized in ext. B, but thermal energy makes dipoles reorient randomly ○ Ferromagnetic: unpaired e–, strongly magnetized in ext. B/high temps ● Magnetic field (B): formed by moving charge, in teslas (T = (N·s)/(C·m) = 10,000 gauss) ○ By a long, straight wire: B = (μ0I)/(2πr), where I = current, r = distance, μ0 = 4π × 10–7 T·m/A = vacuum permeability ○ By a circular loop: B = (μ0I)/(2r), where r = radius ● Magnetic forces ○ Force by B on moving charge: F = qv × B ■ Lorentz force: F = FE + FB = qE + (qv × B) ○ Force by B on current-carrying wire: F = IL × B, where L = length of wire in direction of current

6. Circuits 6.1. Current ● Conductance = 1/R, in siemens (S = 1/Ω) ○ Metallic: free flow of e– across metallic bonds ○ Electrolytic: flow of solutes, depends on [electrolytes] ● Current (I): flow of (+) charge, in amperes (A = C/s) ○ I = Q/t ○ Direct (DC), alternating (AC) current ○ Electromotive force (emf, ℰ): voltage between 2 terminals of galvanic cell ● Kirchhoff’s Laws ○ Junction rule: Σ Ijunction = 0 ○ Loop rule: Σ Vloop = 0

6.2. Resistance ● Resistance (R): material opposes current, in ohms (Ω) ○ R = ρL/A, where ρ = resistivity, L = resistor length, A = resistor cross-sectional area ■ L ↑ → e–’s need to travel longer thru resistor → R ↑ ■ A ↑ → more conduction pathways thru resistor → R ↓ ● Ohm’s Law: V = IR

○ Internal resistance (r): intrinsic to non-superconducting batteries, V = ℰ – Ir ○ Batteries are galvanic cells when discharging, electrolytic cells when recharging ● Power (Ƥ): rate of energy expenditure/work ○ Ƥ = W/t = VQ/t = VI ○ Rate of energy dissipation by resistor: Ƥ = VI = I2R = V2/R (substitute Ohm’s Law) ● Resistors ○ Series: Req = R1 + R2 ■ V drops, I is constant ■ Req > R1, R2 ○ Parallel: 1/Req = 1/R1 + 1/R2 ■ I drops, V is constant ■ Req < R1, R2

6.3. Capacitance ● Capacitance (C): charges build up when 2 separated materials are connected to emf source, in farads (F = C/V) ○ Q = CV ○ For parallel-plate capacitors: C = ε0A/d ■ Uniform electric field between plates: E = V/d ○ PE stored in capacitors: U = ½ CV2 ● Dielectric: insulator between capacitor plates ○ C = κC0, where C0 = capacitance w/o dielectric ■ Dielectric constant (κ): 1 for vacuum, > 1 for all else ○ In isolated capacitors, Q is constant, so dielectric → V ↓ → C ↑ ○ In circuit capacitors, V is constant, so dielectric → Q ↑ → C ↑ ● Capacitors: hold charge at some voltage ○ Series: 1/Ceq = 1/C1 + 1/C2 ■ V drops, Q is constant ○ Parallel: Ceq = C1 + C2 ■ Q drops, V is constant

6.4. Meters ● Ammeter: measures I (R ≈ 0) ● Voltmeter: measures V (R ≈ ∞) ● Ohmmeter: measures R (circuit should be off)

7. Waves, Sound 7.1. Characteristics ● Types ○ Transverse: direction of oscillation ⊥ propagation (e.g., EM waves) ○ Longitudinal: direction of oscillation || propagation (e.g., sound waves) ● Waves ○ Wavelength (λ): distance between crests ○ Frequency (f): # cycles per second, in hertz (Hz = 1/s) ■ Propagation speed: v = λf ■ Angular frequency: ω = 2πf, in rad/s ○ Period (T): # seconds per cycle, T = 1/f ○ Amplitude (A): max height of displacement from equilibrium position ● Superposition: A = A1 + A2 ○ Constructive interference: waves in phase, A’s add ○ Partially constructive/destructive ○ Destructive interference: waves out of phase, A’s cancel out ● Resonance: vibrates at multiple natural frequencies (fundamental pitch + overtones) ○ Forced oscillation: periodically varying force applied to object ■ Force frequency = natural frequency → A ↑ ○ Damping/attenuation: A ↓ due to nonconservative force

7.2. Sound ● Sound: longitudinal wave thru oscillation of particles ○ Speed of sound: v = √(B/ρ), where B = bulk modulus (resistance to compression), ρ = density ■ Bsolid ≫ Bliquid ≫ Bgas ■ Fastest in low-density solid, slowest in dense gas ● Pitch: frequency of sound ○ Range of human hearing: 20–20,000 Hz ■ Infrasonic (< 20 Hz), ultrasonic (> 20 kHz) ○ Doppler effect: sound waves in front of moving object are compressed, behind object are stretched ■ f′ = f (v ± vo)/(v ∓ vs), where vo = speed of observer, vs = speed of source ● Use top sign when observer/source is moving toward the other ● Use bottom sign when observer/source is moving away from the other ■ Shock waves: wave fronts build up in front of object moving at speed of sound ● Sonic boom: observer perceives very high pressure, then very low pressure ● If faster than speed of sound, wave fronts trail behind object, destructively interfere ● Loudness/volume: intensity (I) of sound, in W/m2

○ I = Ƥ/SA, where Ƥ = power, SA = surface area of sound front ■ I ∝ 1/r2 (distance from source), since SA ∝ r2 ○ I = 2π2f2A2ρv, where f = frequency, A = amplitude, ρ = density of medium, v = speed of sound ○ Sound level (β): audible intensities, in decibels (dB) ■ β + 10 dB ↔ I × 10 ● Threshold of hearing: 0 dB = 10–12 W/m2 ● Threshold of pain: 130 dB = 10 W/m2 ● Perforate eardrum: 160 dB = 10,000 W/m2 ■ β = 10 log I/I0, where I0 = 0 dB = threshold of hearing ● Changing intensity: βf = βi + 10 log If/Ii ● Beat frequency: fbeat = |f1 – f2| ● Standing waves: incident/reflected waves interfere → stationary waveform ○ Nodes (A = 0), antinodes (max A) ○ Displacement (open ends), pressure (closed ends) nodes ■ Displacement nodes = pressure antinodes ○ Strings/open pipes: closed/open at both ends ■ λ = 2L/n, f = nv / 2L, where n = harmonic number ■ For strings, n = # displacement antinodes ■ For open pipes, n = # displacement nodes ○ Closed pipes: closed at 1 end ■ λ = 4L/n, f = nv / 4L, where n = odd harmonic number

8. Light, Optics 8.1. Electromagnetic Spectrum ● Electromagnetic (EM) waves ○ Propagates in direction E × B ○ EM spectrum: γ rays, X-rays, ultraviolet (UV), visible (400 nm/violet to 700 nm/red), infrared (IR), microwaves, radio waves ● Speed of light: c = λf ≈ 3 × 108 m/s in vacuum

8.2. Geometric Optics ● Rectilinear propagation: light travels linearly thru homogeneous medium ● Sign conventions ○ Radius (R), focal length (f): same (+), opposite side as outgoing light (–) ○ Distance (s): same (+), opposite side as outgoing light (–) ○ Magnification: virtual/upright (+), real/inverted (–)

● Reflection: θr = θa ○ Real image (s′ > 0): rays converge at image, inverted (m < 0) ○ Virtual image (s′ < 0): rays only appear to converge at image, upright (m > 0) ○ Plane mirrors: virtual upright image behind mirror ■ Distances: 1/s + 1/s′ = 0 ■ Lateral magnification: m = 1 ○ Spherical mirrors ■ Types ● Concave/converging: center of curvature is in front of mirror (R > 0) ● Convex/diverging: center of curvature is behind mirror (R < 0) ■ Focal point: where parallel rays intersect after reflection ● Focal length: f = R/2 ● Object at focal point (s = f): all reflected rays are parallel, s′ = ∞ (no image) ● Object at center of curvature (s = 2f = R): real image at same point (s′ = s) ■ Distances: 1/f = 1/s + 1/s′ = 2/R ■ Lateral magnification: m = –s′/s ○ Principal rays ■ Thru focal point → reflected parallel to axis ■ Parallel to axis → reflected thru focal point ■ Thru center of curvature → reflected normally back along incident path ■ To vertex → reflected symmetrically (θr = θa) ● Refraction: na sin θa = nb sin θb ○ Index of refraction: n = c/v, where c = speed of light in vacuum ■ nvacuum = 1, nair ≈ 1 ○ Total internal reflection: no refraction when θa ≥ θc ■ Critical angle: sin θc = nb/na ■ Implies na > nb ■ When θa = θc, refracted ray is 90° w/ normal ● Lenses: twice refracted, assumed to be negligibly thin ○ Types ■ Convex/converging: thickest in middle, treat hyperopia (R > 0) ■ Concave/diverging: thickest at end...