Notes for Physics and Math

Physics and Math 1: Kinematics and Dynamics Vectors and Scalars Forces and Acceleration 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 Change the direction of the subtracted vector and then Subtraction: do a tip-to-tail addition Vector By scalar: Changes the magnitude and may reverse the Multiplication: direction. Dot Product: !"¥"# $ %!%%#%"&'(")q* , results in a scalar quantity Cross Product: !" + "# $ % !%%#%"(,-")q* , results in a new vector. Direction of the new vector can be found using the right-hand rule 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 = $ >"? 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 t $ @"A (,-)q* BCD" $ &'E-FGH&I'&JK,(G LMN" $ &I'&JK,(G Newton’s Laws Mechanical Equilibrium Free Body Representations of the forces acting on an object. Diagrams: Translational Occurs in the absence of any net forces acting on an object Equilibrium: Rotational Occurs in the absence of any net torques acting on an Equilibrium: object. The center of mass is the most commonly used pivot point. 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 = Avg Speed = ./012"3456217898:0 ./012"0498 ./012"34501:78"0;1<8283 ./012"0498 Instantaneous Velocity: The change in displacement over time as the time approaches 0 Instantaneous Speed: The magnitude of the instantaneous velocity vector 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 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 W OP $ OQ R S"TU TV $ OQ "TU R" X "S")TU*X X OP TV $ " OZ"TU $ OQX R Y"S"TV (if a = 0) Projectile Contains both an x- and y-component. Assuming Motion: 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 sinq. “Sin is sliding ¯ the slide”. Perpendicular to the ramp use cosq. Circular Best thought of as having radial and tangential Motion: dimensions. Centripetal force vector points radially inward, the instantaneous velocity vector points tangentially. Centripetal force: A7 $ " 9"<[ ; Physics and Math 2: Work and Energy Energy Work 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= !"#¥#$% &% Work: The process by which energy is transferred from one system to another. Can be expressed as the dot product of force and displacement: 0 * 1##2 * 1#2#345#6 q7 Power: The rate at which work is done or energy is transferred. SI Kinetic Energy: Energy associated with the mvmt of objects. It ' depends on mass and speed squared. KE = m v2 ( Potential Energy: Energy stored within a system. Gravitational Related to the mass of an object and its height Potential Energy: above a zero point. ) * +#,#Elastic Potential Related to the spring constant and the degree of Energy: stretch or compression of a spring squared. ' ) * # #.#/ ( ( Electrical Potential The energy between two charged particles. Energy: Chemical Potential The energy stored in the bonds of compounds. Energy: Conservative Forces: Path independent and do not dissipate the mechanical energy of a system. Examples: Gravity and electrostatic forces. Nonconservative Path dependent and cause dissipation of mechanical Forces: energy from a system. Examples: Friction, air resistance, and viscous drag. 8 unit is watt (W). W = 9 = :##; 9 = <=##;% 9> Work-Energy When net work is done on or by a system, the system’s Theorem: kinetic energy will change by the same amount. 0?@A * #DB * BC D BE Mechanical Advantage Mechanical The factor by which a simple machine multiplies the Advantage: 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 FG * # H@?"AI#JC#E?KLE?@ M@E"IA#JC#E?KLE?@ Plane: 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. X FN3OPQR3PS#GTUPQVPWN * # XYZ[ \] Physics and Math 3: Thermodynamics 0th 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. % ! " #$& $'$( ) *+ & ' " % $#! , *+( - " $' ) +.* Thermal Expansion: Describes how a substance changes in length or volume as a function of the change in temperature. D/ " $a$/$D0 D1 " $b$1$D0 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. 1st Law of Thermodynamics A statement of conservation of energy: The total energy in the universe can never decrease or increase. For an individual system: 23 " 4 , 5 23 " 6789:;$<9$=>=?;@A =$<9?;B98C$;9;B:> 4 " ;9;B:>$?B89=D;BB;E$<9?F$?7;$=>=?;@$8=$7;8? 5 " GFBH$EF9;$I>$?7;$=>=?;@ Heat: The process by which energy transfer between two objects at different temperatures that occurs until the two objects come into thermal equilibrium (reach the same temperature). J " K$L$D0 Specific Heat: The amount of energy necessary to raise one gram of a substance by 1¡ C or 1 K. R MNO Specific heat of H2O = 1 P$¥$Q = 4.184 P$¥$Q Heat of The energy required for a phase change of a substance Transformation: (temperature does not change during the transformation). J " K$/ L = heat of transformation Processes with Isobaric: Pressure is constant, DP = 0 Constant Variable: Isothermal: Temp is constant, DU = 0 Adiabatic: No heat is exchanged, Q = 0 Isovolumetric (isochoric): Volume is constant, DV = 0, so Work = 0 Work of a Gas: 5 " ,S$21 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 Describe the pathway from one equilibrium state to Functions: another. Include: work and heat. 2nd 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. Note: An isolated system does not exchange energy or matter with surroundings Entropy: A measure of how much energy has spread out or how spread out energy has become. Physics and Math 4: Fluids Characteristics of Fluids and Solids Fluid Dynamics 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 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. p!9 : ! D ; 8 "! Density: Mass per unit volume of substance. !r " # $ 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. % " ! ' ? @A#B " C!D C " EFGHH!HIEJKGLMN!MFIM D " OINGEKJP Absolute The sum of all pressures at a certain point within a fluid; it is Pressure: equal to the pressure at the surface of the fluid plus the pressure due to the fluid itself. %()(*+ " %, - ! r !.!/ In water, every additional 10m of depth adds » 1 atm to %()(*+ Continuity Fluids will flow more quickly through narrow passages Equation: and more slowly through wider ones. 8 " DQ !CQ " DR !CR Gauge The difference between absolute pressure and atmospheric Pressure: pressure. In liquids, gauge pressure is caused by the weight of the liquid above the point of measurement. %01*02 " % 3 %*(4 " 5%, - !r !.!67 3 %*(4 Bernoulli’s The sum of the static pressure and the dynamic pressure Equation: will be constant between any two points in a closed system. Q Q % Q - R !r !DQR - !r!.!/Q " %R - R !r!DRR - ! r !.!/R Venturi Effect: The velocity of a fluid passing through a constricted area will INCREASE and its static pressure will DECREASE Hydrostatics Pascal’s A pressure applied to an incompressible fluid will be Principle: distributed undiminished throughout the entire volume of the fluid. % "! &S 'S "! &T 'T Hydraulic Operate based on the application of Pascal’s principle to Machines: generate mechanical advantage. Archimedes’ When an object is placed in a fluid, the fluid generates a Principle: buoyant force against the object that is equal to the weight of the fluid displaced by the object. UV " !r!W!. Also, X " !r !W and U " %!C. Y2Z[\(]^_`abc Y2Z[\(]defghibad!jhked " l2\0m(^_`abc!en!ieo l2\0m(^_`abc!en!ieo!p!l2\0m(^_`abc!en!qicao If the max buoyant force is larger than the force of gravity on the object, the object will float. If the max buoyant force is smaller than the force of gravity on the object, the object will sink. If UV r X)st2u(!., then the object floats. If UV v X)st2u(!., then the object sinks. Specific Gravity: Ratio of density of an object to density of water. r^_`abc wxIEKyKE!zFMOKJP " r qicao Cohesive vs. Fluids experience cohesive forces with other molecules of Adhesive: the same fluid and adhesive forces with other materials. Surface Tension: Cohesive forces give rise to surface tension. Venturi Tube: The average height of the horizontal tube remains constant, so r./ remains constant at points 1 and 2. As cross-sectional area decreases from point 1 to point 2, the linear speed must increase. As the dynamic pressure increases, the absolute pressure must decrease at point 2, causing the column of fluid sticking up from the Venturi tube be to be lower at point 2. Fluids in Physiology Circulatory The circulatory system behaves as a closed system with System: nonconstant flow. The nonconstant flow = our pulse. { D "! ' " u*|}\*u!)1(~1( u|)[[p[2u(\)Z*+!*|2* 8 " D!C •% " 8! € !• " EMF‚KME!GƒJxƒJ! € !FIHKHJMLEI •% " D!C!• Pressure is directly related to velocity, area, and resistance. Area is inversely related to resistance and velocity. Cross-sectional area& Þ Resistance¯ and/or velocity¯ Hydraulic Lift: % "! &S 'S "! &T 'T 'T UR " UQ 5' 7 S 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. Physics and Math 5: Electrostatics and Magnetism Essential Equations for Test Day Charges .: " /'012 0'013 0 43 Coulomb: The SI unit of charge J "' /'I Protons & Protons have a positive charge and electrons have a negative Electrons: charge. Both protons and electrons possess the fundamental unit of charge (! " #$%&' ('#&)*+',-. Protons and electrons have different masses. Attraction & Opposite charges exert attractive forces, and like charges Repulsion: exert repulsive forces 5 " ' 43 /'I'1 K "' 4 /'I 4 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. 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 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: J " ' /'I'1 4 Coulomb’s Law Electrical Potential 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. /'01 0'01 0 . " 23 3 4 Electric Field: Every charge generates an electric field, which can exert forces on other charges G /'I 6789:':;:8<:='7>'?'<:@<'9A?8B: " ' H " ' 43 5 "' C?B>D "# @ r " ABCDCEDFDEG, H " IBJKEL#MN#ABCDCEMA , O " PAMCC#CBPEDMJQI#QABQ @ Capacitance based on parallel plate geometry: R " # eT #U W V Electric field in a capacitor: X " # S V Z Potential energy of a capacitor: Y " # #R#9[ [ Series / Series: ¯equivalent capacitance of a circuit Parallel: 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, k Meters Ammeters: Resistance: The opposition that a substance offers to the flow of e-. Have the ability to store and discharge electrical potential energy. Inserted in SERIES in a circuit to measure current; they have negligible resistance Voltmeters: Inserted in PARALLEL in a circuit to measure a voltage drop; they have very large resistances Ohmmeters: Inserted around a resistive element to measure resistance; they are self-powered and have negligible resistance Ohm’s Law: For a given resistance, the magnitude of the current through a resistor is proportional to the voltage drop across the resistor. 9 " !#> Resistors in Series: Additive. Sum together to create the total resistance of a circuit. Resistors in Parallel: ¯equivalent resistance of a circuit. Resistors in Series: Total resistance is equal to the sum of all the individual resistors. >: " >Z ] >[ ] >^ ] _ ] >- 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. Z Z Z Z Z "# #]# ]# ] _]# fg Resistors in Parallel: 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. Z Z Z Z Z "# #]# ]# ] _]# `a `b `c `d `e fb fc fd fe Capacitors in Parallel: Total capacitance is equal to the sum of all the individual capacitances. R= " RZ ] R[ ] R^ ] _ ] R- Physics and Math 7: Waves and Sound 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. 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 ! " #$# l 7 ! "# r !# " %&'(#)*((+ $# " ,-(./(012 l# " %&'(3(0456 8# " #9/3:#;<+/3/) (increases from gas to liquid to solid) r# " +(0)=52 Displacement (x): Refers to how far a point is from the equilibrium position, expressed as a vector quantity. Amplitude (A): The magnitude of its maximal displacement. The maximum point is called a crest. The minimum point is called a trough. Wavelength (l ): The distance between two crests or two troughs. Frequency (f): The number of cycles it makes per second. Expressed in Hz. Pitch: Our perception of frequency. Doppler Effect: A shift in the perceived frequency of a sound compared to the actual frequency of the emitted sound when the source of the sound and its detector are moving relative to one another. The apparent frequency will be higher than the emitted frequency when the source and detector are moving toward each other. The apparent frequency will be lower than the emitted frequency when the source and detector are moving away from each other. The apparent frequency can be higher, lower, or equal to the emitted frequency when the two objects are moving in the same direction, depending on their relative speeds. F#GF Angular Frequency (w ): Also known as radial or circular frequency, measures angular displacement per unit time. ?#p Expressed in radians per second. w# " ># p#$ " @ Period (T): The number of seconds it takes to complete a B cycle. It is the inverse of frequency. #A# " # $ D " #$#E F#IFH K Constructive Occurs when waves are exactly in phase with each Interference: other. The amplitude of the resultant wave is equal to the sum of the amplitudes of the two interfering waves. Destructive Occurs when waves are exactly out of phase with Interference: each other. The amplitude of the resultant wave is equal to the difference in amplitude between the two interfering waves. Partially Constructive / Occurs when two waves are not quite perfectly in Destructive Interference: or out of phase with each other. The displacement $ " #(;=55(+#,-(. Intensity: Intensity is related to a wave’s amplitude. Intensity decreases over distance and some energy is lost to attenuation from frictional forces. C Interference: Describes the ways in which waves interact in space to form a resultant wave. $ D " #*(-1=('(+#,-(. J Use the Top sign for “toward”, bottom sign for “away” L" M N O# " #*<%(- P " #&-(& Strings and Support standing waves and the length of the string or Open Pipes: pipe is equal to some multiple of half-wavelengths. R# l Q " # ##E S " TU >U V K ? Closed Pipes: Closed at one end. Support standing waves, and the length of the pipe is equal to some odd multiple of quarterl wavelengths. Q " # R# ##E S " TU XU V K W 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. 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. 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 Physics and Math 8: Light and Optics Electromagnetic Spectrum Electromagnetic Transverse waves that consist of an oscillating electric Waves: field and an oscillating magnetic field. The two fields are perpendicular to each other and to the direction of propagation of the wave. Geometric Optics Reflection: Rebounding of incident light waves at a medium’s boundary Law of q ' $ "q 2 Reflection: Electromagnetic The range of frequencies and wavelengths found in EM Spectrum: waves. EM Spectrum: small l ¬® high l high energy ¬® low energy Mirror Image Produced Position Cause Concave Real Inverted Object’s position is greater than the focal length Virtual Upright Convex Virtual Upright & smaller Plane Virtual Upright & same size Spherical Mirrors: Note: Gamma, X-ray, and higher UV are ionizing. They can liberate electrons from nearby atoms and create free radicals. Visible Spectrum: Object’s position is less than the focal length Can think of these as spherical mirrors with infinite radii of curvature Wavelength (nm) Hydrogen Lyman: Ultraviolet, n = 1 Spectral Series: Balmer: Visible, n = 2 Paschen: Infrared, n = 3 Acrostic “Loves Beer Pong”, then n = 1, n = 2, n = 3 Rydberg Formula: !"# $ %"& ' (. )*+,- / (. ' 1 *+*0*,- 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. 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- Shows the constructive and destructive interference of waves Slit Experiment: that occur as light passes through parallel slits, resulting in minima (dark fringes) and maxima (bright fringes) of intensity. Polarization Plane-Polarized A polarizing filter only lets light through if the E field of Light: the wave aligns with the openings in the filter. The E fields of the exiting light oscillate along the same axis. Circular All of the light rays have electric fields with equal Polarized Light: intensity but constantly rotating direction. Circularly polarized light is created by exposing unpolarized light to special pigments or filters. Plane-Polarized Light Index of refraction: 3 $ " 4 5 6 $ 7899:";<"=>?@A">B"CD6EEF G $ 7899:";<"=>?@A">B"A@9"F9:>EF Dispersion: When various wavelengths of light separate from each other. Snell’s Law: The law of refraction. There is an inverse relationship between the index of refraction and the sine of the angle of refraction (measured from the normal) 3' 7>B& q' 1 $ 32 7>B&q2 1 Total Internal When light cannot be refracted out of a medium and is instead Reflection: 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 q. Critical The minimum incident angle at which total reflection occurs. Angle: q 4 $ 7>BH'&(.1 ( I Lenses: Refract light to form images of objects. Thin symmetrical lenses have focal points on each side. Lens Image Produced Position Convex Real Inverted Converging system Virtual Upright Converging system Virtual Upright Diverging system Concave System Circular-Polarized Light F = focus f = focal length ' "# $ " "J 2 Lensmaker’s Lenses with non-negligible thickness require the lensmaker’s eq. Equation: ' $ &3 / L1& ' / " ' 1 K MI M. Physics and Math 9: 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: ! " #$% 7 & " '())*$+,$-./01 " 2$ 3$456 To calculate l from % use: & " %$l 8 Maximum kinetic energy in the photoelectric effect: 97:; " #$% < = Threshold The minimum light frequency necessary to eject an electron Frequency: from a given metal. Work The minimum energy necessary to eject an electron from a Function: given metal. $$# " ?-@ABCD 'B+A'1@A1 " EFEGE$ 3$ 45HIJ$K$' = " #$%> $ 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 higherenergy 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 lowerenergy orbit, it emits a photon of light of the same frequency as the energy difference between the orbits. Absorption May be impacted by small changes in molecular structure. Spectra: 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 Is the amount of energy that is released when nucleons Energy: (protons and neutrons) bind together. 4 Fundamental Strong and weak nuclear force, electrostatic forces, Forces of Nature: 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 The loss of small particles from the nucleus. Decay: Mass N H J a N L b b LM LO HO Charge Ng N Alpha ( a) The emission of an alpha particle (a, JM aP MJ Q)), which is a Decay: helium nucleus. R T U $ RHJ V W$ J a S SHM LM Beta-negative The decay of a neutron into a proton, with emission of an (b -) Decay: electron (e-, b-) and an antineutrino (XY ). R R N H T U SLO V W HO b S Beta-positive “Positron emission”, the decay of a proton into a neutron, (b+) Decay: with emission of a positron (e+, b+) and a neutrino ( X). R R N L T U SHO V W LO b S Gamma (g ) The emission of a gamma ray, made up of photons, which Decay: converts a high-energy nucleus into a more stable nucleus. R Z T U SR T W NN g S Electron Is the absorption of an electron from the inner shell that Capture: combines with a proton in the nucleus to form a neutron. R R T W )H U SHO V S 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 The rate at which radioactive nuclei decay is proportional to Decay: the number of nuclei that remain. [ " [N $\H l$] [ " ^$+,$_A*)B@`)*$A_B-). [N " ^$+,$_A*)B@`)*$A_B-).$@1$a " 5 l $ " $CA+bA$*)B@`$B+A'1@A1 Note: If the problem just says “beta”, they mean “beta negative”. Beta-negative is the default. Physics and Math 10: Mathematics Arithmetic and Sig Figs Exponents, Log and Ln Scientific Notation: Improves the ease of calculation. It is usually helpful to convert a number to scientific notation !"# $#%&' ( ) !*# $# %&+ ( , !"# $# %&' ( ) !&-*# $ #%&' ( , #.-%# $# %&' !%-/# $# %&' (!"# $# %&+ ( , #0-/# $# %&1 2#$#3456 +#$#347 , 0# $#%&81 !.# $#%& ;*# 8+ (' $#%&2 , 9# $#%& Common %+ , % + Squares: .+ , 0 , !*# $# %& - Multiply exponents 2 (3<+ , "# $# %&= falls between. For example, ;%9& is between 13 and 14. ;%9& , # ;0 # $ # ;* # $ # ;/ , .# $ #"# $# ;/ , @;/ ;/ # A .-. so @;/ # A %"-.. - Add exponents - Subtract exponents 8: Estimating To calculate the square root of any number less than 400, you can Square Roots: approximate its value by determining which two perfect squares it - Divide the exponent by 2 LARS mnemonic when moving the decimal within scientific notation. Left Þ Add, Right Þ Subtract - Left Add 09%-.# $ #%&> , 0-9%. $ %&? - Right Subtract &-&&"%0# $ %&8' , "-%" $ %&8: 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. " ,* 0+ , %@ /+ , ./ Log and Ln: CDE!F ( , #G HIG , F @+ , "@ B+ , 0* 9+ , @0 *+ , 9% %&+ , %&& %%+ %.+ %"+ %0+ %/+ CJ!F( , #G KG , F %@+ %B+ %9+ %*+ .&+ , ./@ , .9* , ".0 , "@% , 0&& L , M- N OPQ R !%( , & OPQ R !QTUVWUT#WXVY#%( , ZP[\W\]U OPQ R !S( , % OPQ R !OU[[#WXVY#%( , ^UQVW\]U OPQ !S# $ #_( , OPQ#!S( ` OPQ#!_( S OPQ a b , #OPQ#!S( ) OPQ#!_( _ OPQ!Sc ( , _#OPQ#!S( % OPQ#! ( , # )OPQ#!S( S Division: If one number is rounded up, the other should also be rounded up in proportion. Estimating Log: OPQ !S# $#%&c ( , OPQ! S(` OPQ!%&c ( , # OPQ!S( ` _ Trigonometry SOH CAH TOA: [\Y!d ( , # e f R gP[!d( , # f e WVY!d( , # R , # hij#!d( OPQ !S# $#%&c ( A _ ` &- S klh#!d( The Unit Circle Common Values: , %.% , %00 , %@* , %*@ , ../ q cos( q) sin(q ) 0¡ 1 0 0 30¡ ;" . % . ;" " 45¡ 60¡ ;. . ;. . . . % tan(q ) 1 ;" ;" 90¡ 0 1 undefined 180¡ -1 0 0 m , gP[!q( WVY!q( 45-45-90 triangle 30-60-90 triangle n , [\Y!q( o ,#p ,# hij#! q( klh!q( Physics and Math 11: 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 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 The quality of approximating the true value. (Validity): Precision The quality of being consistent in approximations. (Reliability): Human Subject Research 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 Controls for individual variations in a measurement by Design: 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 Refers to the low likelihood of the experimental findings Significance: being due to chance. Clinical Refers to the usefulness or importance of experimental Significance: findings to patient care or patient outcomes. 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 Assess both exposure and outcome at the same point in Studies: time. Case-Control Assess outcome status and then assess for exposure Studies: 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 An extraneous variable that relates to BOTH the Variable: dependent and independent variables. Mediating The means by which the IV affects the DV. It is the Variable: “middleman” between the IV and DV. Moderating Influences the already established relationship between Variable: the IV and DV. Moderators affect the strength of the relationship between the two variables. Physics and Math 12: Data-Based and Statistical Reasoning Measures of Central Tendency Probability 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. Independent The probability of independent events does not change Events: based on the outcomes of other events. Dependent The probability of a dependent event changes depending on Events: the outcomes of other events. Terminology: Mutually Exclusive: Cannot occur simultaneously. When a set of outcomes is exhaustive, there are no other possible outcomes. Distributions Statistical Testing Normal Symmetrical and the mean, median, and mode are equal. Distribution: Hypothesis Use a known distribution to determine whether the null Tests: 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 (a). A significance level of 0.05 is commonly used. Confidence Are a range of values about a sample mean that are used Intervals: to estimate the population mean. A wider interval is associated with a higher confidence level (95% is common). Standard A normal distribution with a mean of 0 and a standard Distribution: deviation of 1. It is used for most calculations. Hypothesis Testing Chart with Type 1 and Type 2 Errors Skewed Have differences in their mean, median, and mode. Skew Distribution: direction is the direction of the tail. Mode Median Mode Median Mean LEFT skewed (negative skewedness) Mean RIGHT Skewed (positive skewedness) Bimodal Multiple peaks, although not necessarily multiple modes. Distribution: Measures of Distribution Range: Difference between largest and smallest value. Interquartile The difference between the value of the third quartile and Range: first quartile. Can be used to determine outliers. Standard A measurement of variability about the mean. Can be used Deviation ( s): to determine outliers. 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. Outliers: In general, any value that lies more than 3 standard deviations from the mean. Exponential Relationship Logarithmic Relationship Slope: Quadratic Relationship !"#$ !%& Curvilinear Relationship