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WHAT IS PHYSICS ALL ABOUT? Physics seeks to understand the natural phenomena that occur in our universe; a description of a natural phenomenon uses many specific terms definitions and mathematical equations Solving Problems in Physics In physics, we use the SI units (International System) for data and calculations Base Quantity Length Mass Temperature Time Electric Current ;;;;;;S=;;=;;;=~T~he position ofa motion with position, velocity and acceleration as variables; mass is the measure of the amount of matter; the standard unit for mass is kg I kg = 1000 g.; inertia is a property of matter, and as such, it occupies space I Motion along a straight line is called rectilinear; the equation of motion describes the position of the particle and velocity for elapsed time t a Velocity (v): The mte of change of the displacement h' () s WIt tIme ( t):v cis = Tt Ll s = rlt b Acceleration (a): The rate of change of the · h dv Ll v ve IoClty WIt tnne: a = dt = Tt a & v'are vectors , with magnitude and direction c Speed is the absolute value of the velocity; scalar with the same units as velocity Equations of Motion for One Dimension (I-D) Equations of motion describe the future position (x) and velocity (v) of a body in terms of the initial velocity (Vi), position (XII) and acceleration (a) a For constant acceleration the position is related to the time and acceleration by the following x = vi, I ·1 ; a , t' y = vi, ( + ;a, (' For a rotating body, use polar coordinates, an angle variable, , and r a radial distance from the rotational center C 'lotion in TlJI'(~e Dimensions (3-D) I Cartesian System: Equations of motion with x y and z components Spherical Coordinates: Equations of motion based on two angles ((} and 'P) and r the radial distance from the origin m,M T t I position and velocity: F = m a OR ~ F = m a Newton's 3rd Law: Every action is countered by an opposing action F: ~ pe\ of Forcc\ I A body force acts on the entire body, with the force acting at the center of mass a A gravitational force Fg pulls an object toward the center of the Earth: Fg = mg b Weight = Fg; gravitational force c Mass is a measure of the quantity of material , independent of g and other forces Surface forces act on the body's surface a Friction Fe is proportional to the force normal to the part of the body in contact with a sUlface, =" Fn·: Fr Fn i Static friction resists the move-ment of a body ii Dynamic friction slows the motion of a body For an object on a horizontal plane: F f = Il Fn = ll m g Net force = FI - F f Polar: (r, 9) x = r cos9, y = r sin9, r- Circular Motion (} %, Radian Radian/second a Radian/second s Meter W = F d cos (8) = F F D • r F_ _ r D Maximum work r p = LlWork = LlWork Ll t f P(t)dt The Sl unit for power is the Watt (W): I W = I Joule/second = I J/s Work for a constant output of power: W = P Lit Energ~ & Enrrg~ Consenation I The total energy of a body, E, is the sum of kinetic r The angle between rand the (x) axis te angular velocity The angular acceleration The circular motion arc s = r8 (8 in rad) Tangential acceleration & velocity: v , = rw; a , = r a ; v and a along the path of the K, & potential energy U: E = K +:Eu Potential energy arises from the interaction with a potential from an external force Potential energy is energy of position: U(r); the form of U depends on the force generating the potential: Gravitation: U(h) = mgh q,q, Electrostatic: U (r,,) = '"'"F;; If there are no other forces acting on the system, E is constant and the system is called conservative I Collisions & Linear 'loml'lItulll Collisions I Types of Collisions a Elastic: conserve energy b.lnelastic: energy is lost as heat or deformation Relative Motion & Frames of Reference: A body moves with vc:locity v in frame S; in frame S' the velocity is v' ; ifYs' is the velocity of frame S' relative to S, therefore: v = V,' + v' Elastic Collision Conserve Kinetic Energy: t m v ,' = t m vI L: motion arc Centripetal acceleration: a , = v' r; a is directed toward the rotational center a TIle centripetal force keeps the body in circular motion with a tangential acceleration and velocity No work Power (P) is energy expended per unit time: II Potrnthll distance from the rotation center (center of mass) z Newton's Laws are the core x = r simp cos9, principles for describing the motion y = r sinep sin9, z = r coslj), of classical objects in response to r2=x2 + y2+z2 forces The SI ullit of force is the Newton, N: IN=lkg m/s2 ; the cgs unit is the dyne: dyne = I g cm/s2 ~ 1,- I The I Meter r J Work = Key Varia_b_le_s_: _ , _ _ _._ _ _ - I energy is the energy of motion ; mass m and velocity v: K = t mv' The SI energy unit is the Joule (J): \J = I kg m 2/s 2 Momentum, p: Momentum is a property of motion, defined as the product of mass and velocity: p = m v Work (W): Work is a force acting on a body moving a distance; for a general force F, and a body moving a path , s: W = F ds For a constant force work is the scalar product of the two vectors: force F and path r: Ll time Fn polar coordinates: (r,8) r2=x +y2 x Dynamk Friction F Circular 'lotion I Motion along a circular path uses ~, Spherical Other physical quantities are derived from these basic units: Prefixes denote fractions or multiples of units; many variable symbols are Greek letters Math Skills: Many physical concepts are only understood with the use of algebra, statistics, trigonometry and calculus Unit Meter - m Kilogram - kg Kelvin - K Second - s Ampere - A (C/s) /, x Newton's 1st Law: A body remains motion unless influenced by a force Newton's 2nd Law: Force and acceleration determine the motion of a body and predict future equation of motion: x (I) = X u + V i ( + t ar b For constant acceleration the velocity vs time is given by the following: v r(t) = V i + at c.lf the acceleration is a function of time, the equation must be solved using a = aCt) R 'Iotioll ill 1\\0 Dimcnsiolls (2-0) I For bodies moving along a y Polar straight line derive x- and y equations of motion Symbol Conserve Momentum: Lm Vi = L: L rn Vr ~ Impulse is a force acting over time Impulse = F Ll t or f F (t) dt Impulse is also the momentum change: z Pfin - Pini! .1 Rul:ltiulI 411 a Rigid Bud~ I Center of Mass: The "average" position in the body, accounting for the object's mass distribution Moment ofInertia, 1: The moment ofinertia is a measure of the distribution of the mass about the rotational axis: ~ rn, r,' rio is the radial distance from mj to the rotational axis Sample I for bodies of mass m: rotating cylinder (radius R): +rn R' M O,cillatur~ Motion I Simple Harmonic Motion a Force: F = - k x (Hooke's Law) b Potential Energy: Uk = + k x' c Frequency of the oscillation: T Rotating Bodies Law Spring T = 21l'jI nr, b Frequency of oscillation: f= L Simple Pendulum !K 21l'V T \ L = Iw = r • P = f r • v dm Torque is also the change in L with time: T=r'F=~7 h: Static E(llIilihrium & Angular Momentum Elasticit~ t Equilibrium is achieved when: ~f = O , 0~ ~T = O ~ The body has no linear or angular acceleration Deformation of a solid body a Elasticity: A material returns to its original shape after the force acting on it is removed b Stress & Strain i Strain is the deformation of the body ii Stress is the force per unit area on the body c Hooke's Law: The stress IS linearly proportional to the strain; stress = elastic modulus x strain: i Linear Stress: Young's Modulus, symbolized Y ii Shape Stress: Shear Modulus, symbolized S iii Volume Stress: Bulk Modulus, symbolized B L lIniH'rsal (;nl\ itatiull _ _._ r _ , M Universal Gravitation M2 Gravitational Force & Energy a I energy: U,= -GM,M, GravltatlOna -r- GM,M, I rlorce: F, = ~ b Gravltatlona Fg is a vector, along r, connecting M J and M2 c Acceleration due to Gravity, g: For an object on the Earth's surface, Fg can be viewed as Fg =m g; g is the acceleration due to gravity on the Earth's surface: g = 9.8 m1s Hooke's Simple Pendulum a Period of oscillation: rotating sphere (radius R): trn r' T = la = r • f (angular acceleration force) Angular momentum is · the momentum associated with rotational motion: u mll f= L Ik 21l'V m twirling thin rod (length L): ,', rnL' Rotational Kinetic Energy = +LQ' The rotational energy varies with the rotational velocity and moment of inertia I Angular force is defined as torque, T: ~ Ful'l'~s in Solids & Fluids I p , the density of a solid, gas or liquid: p = mass/ volume = M/V Pressure, P, is the force divided by the area of the forces acted upon: P = forcelarea The SI unit of pressure is the Pascal, Pa: I Pa = N I m a Pascal's Law: For a Pascal's Law fluid enclosed in a vessel, the pressure is equal at all points in the vessel b Pressure Variation with Depth Pf The pressure below the surface of a liquid: P, = PI + pgh h is the depth, beneath the surface p is the density of the water PI is the pressure at the surface Pressure Variation ,-,.-.,., ,.,.,.,.,,, .==1 , ,, , , ,.'"., Surface Liquid P Pl h c Archimedes' Principle: An object of volume V immersed in liquid with density p, feels a buoyant force that tends to force the object out of the water: g, = p V g Earth's Archimedes' Principle ,,\.\I\i{fWV""hXXhhH omAA"""" Surface Liquid Examine Fluid Motion & Fluid Dynamics a Properties of an Ideal Fluid i Nonviscous - minimal interactions ii Incompressible - the density is constant iii Steady flow - no turbulence iv At any point in the flow, the product of area and velocity is constant: AI VI = Al VI b Variable Fluid Density If the density changes, the following equation described properties of the fluid: p,A,v, = p,A,v, Variable Fluid Density c Bernoulli's Equation is a more general b Weight is the gravitational description of fluid flow force exerted on a body by the i For any point y in the fluid tlow: P + +p v' + p g y constant Earth: Weight = Fg = mg Weight is ill!! the same as mass = Gravitational Potential Energy 01 \\a,cs •Transverse ·Traveling • Harmonic • Longitudinal • Standing • Quantum mechanical General fonn for a transverse OR traveling wave: y = fix - vt) (to the right) OR y = f(x + vt) (to the leli) General form for a harmonic wave: y A sin (kx - w t) OR Standing '\ constructive and destructive interference a Constructive Interference: Thc wave amplitudes add up to produce a I ) wave with a larger amplitude than either of the two waves Harmonic ~ave b Destructive Interference: The wave amplitudes add up to produce a wave with a smaller amplitude than either of the two waves B lIarnwnk \\:I'l' Propertil's Wavelength A (m) Period T (sec) Frequency f(Hz) Angular Frequency w (rad/s) Wave Amplitude A Speed Distance between cycles Cycles per ,eeond: f - IT r:: = 21l'/ T = 2m Height of wave I v (m/s) Linear velocity v = Af C Sound \\ aH', Wave Nature of Sound: Sound is a compression wave that displaces the medium carrying the wave; sound cannot tra, el through a vacuum General Speed of Sound: v = ~ b p is the density For a Gas: v = gravitational potential => Ug = mgh T~fI~S A Esampfl's 01 a B is the bulk modulus, the volume compressibilit of the solid, liquid or gas Gravitational Potential Energy, Ug a The WAVE MOTION ii.For a fluid at rest (special case): P,-P,=pgh J,r RT M r = Cp/C,· (the ratio of heat capacities) Loudness - Intensity & Relative Intensit~· Loudness (sound i11lensity) is the power carried by a sound wave a Relative Loudness - Decibel Scale (dB): P(dB) = 10 log (f.) i The decibel scale is delined relative to the threshold of hearing, I,,: P(I,,) = dB ii A change in 10 dB, represents a lOx increase in sound intensity, I b Doppler Effect The sound frequency shifts (f'/t) due to relative motion of the source of the sound and the observer or listener: Vo speed of the observer; vs Doppler Effect speed of the source;v speed of sound O => - A maximum force) RlPt-llaDd iii The "right hand rule" Rule defines the force direction b Force on a conducting segment: For a current I passing through a conductor of length I in a magnetic field B, the force is given by: F=II· B i For a general current path s: F=ljds.B E B = c b The speed of light, c, correlates the magnetic constant, 11", and the electric constant, _ _ 1_ Cu.c- IlloE!) speed oflight, c: c = fA d X-rays have short wavelength, compared with radio waves e Visible light is a very small part of the spectrum F = q v·B = qvB sin (B followmg equatIOn: wavelength, t ,and frequency, f, travels at the B: a B is the angle between vectors v and i For v parallel to B; F = I Electromagnetic wa ves are formed by transverse and E fields a The relative field strengths arc defined by the c In a vacuum, an electromagnetic wave with Magnetic Force: F mag on charge, q, moving at III A Q)"" EMF = fEds AND EMF = -~tl/)m U (magnetic): Magnetic potential energy arises from c The CGS unit is the Gauss, G: T = 104 G d For a bar magnet, the field is generated from the ferromagnetic properties of the metal forming the magnet i The poles of the magnet are denoted North/South The field lines are show in the figure below The EMF induced in a circuit is directly proportional to the time rate of change of the Torque OD a Loop the total of the magnetic flux, B dS, must be consistent with the current, I: f B • dS = Po I Magnetic Flux, 1/)", Summarize the general behavior of electrical and magnetic fields in free space I Gauss's Law for Electrostatics: fE dA = ~ Gauss's Law for Magnetism: a The magnetic flux, 1/)"" associated with an area, dA, of an arbitrary surface is given by the following equation: I/)m = j B • dA; dA is vector perpendicular to the area dA b Special Case - Planar area A and uniform B at angle I with dA: I/)m = B A cos B fB'dA=O Ampere-Maxwell Law: f B • ds = p"I + p"e" ~~" Faraday's Law: f E • dS = - ~~ I Light exhibits a duality, having both wave and particle properties Key Variables a Speed of light in a vacuum, c b Index of refraction, n: The index of refraction, symbolized n, is the ratio of the speed of light in a vacuum divided by the speed of light in the material: c (vacuum) n= c (material) c View light as a wave Iocus on wave properties: wavelength and frequency i For light as an electromagnetic wave : Af =c ii Light is characterized by its wavelength ("color"), or by its frcquency, f d View light as a particle in order to o - ~ ~'- N Images & Objects frequcncy, f, with the proportionality constant h, Planck's Constant: E (photon) = h f Reflection & Refraction of Light Renection of Light Incident Ray \"2 Lenses and mirrors are characterized by a number of optical paramcters: u The radius of curvature, R, defines the shape of the lens or milTor; R is two times the foca l lengt h, f: R = f i+ Sign Parameters ::j '"'"g;", t., f foca l length I s obj ct distance - Y diverging lens convex mirror virtual image erect inverted h' image size erect inverted = I speed bends the light ray as it passes from n I to 11 i The angles of the incident and relracted rays are governed by Snell's Law: n, sin 8, n , s in 8,; n l, n2: indices of retraction of two materials n ·, c Internal Reflectance: SID " n; ; Light = = passing fromlllaterial of higher n to a lower nmay be trapped in the material if the angle of incidence is too large Polarized Light: The E tield of th.: electromagnetic wave is not spherically symmetric (EX: plane (linear) polarized light, circularly polarized light) a One way to generate a polarized wave is by retlecting a beam on a surface at a preci se angle , called B, b The angle depends on the relative indices of refraction and is defined by Brewster's n·, Law: tan B, = n b The optic axis: Line from base of object through center of lens or mi rror c Magnification: The magnifying power of a * lens is given by M, the ratio of image si ze to object size: M = d Laws of Geometric O ptics i The m irror equ ation: The focal length, image distance and object distance are described by the following relationship: 1 ~
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