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# physicsfinal.pdf

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Date Created: 03/14/16

Physics 1A Final Study Guide mathematical and visual examples are in the documents attached Introduction A Physics a deals with the behavior and structure of matter and energy B Terms a Model useful idea to explain what we observe b Theory more broadly applicabledetaileduseful must make predictions that are testablefalsi able c Law concise general statement about how nature behaves d Uncertainty quanti es how well we can make a measurement and what affects the certainty of our measurement i If not speci ed it is assumed to be 1 or a few units in the last digit speci ed e Signi cant gures the number of reliably known digits i Adding or subtracting round to the least signi cant place ii Multiplication or division keep the last digit of signi cant gures iii Leading zeroes don39t count iv Keep 1 digit in the intermediate steps and round at the end f Scienti c notation write in powers of ten i Tetra T 1012 ii Giga G 109 iii Mega M 106 iv Kilo K 103 v Centi c 10392 vi Milli m 10393 vii Micro u 10396 viii Nano n 10399 g Measurements are made relative to a particular standard which we call a unit we use Systeme International for our units Sl Math Preliminaries A Kinematics and Dynamics a Kinematics description of how objects move b Dynamics forces and why objects move the way they do B Coordinates a Space and time b Properties of coordinates speci es a location and time are quanti able allows discussion of relative positions the point where all coordinates are 0 is the origin c Example C Scalars and Vectors a Scalars completely speci ed by numerical value magnitude b Vectors need both a magnitude and direction to completely specify represented by arrows in diagrams c Tensors type of vector d Speed and distance are scalars because they are just magnitudes e Velocity and displacement are vectors because in order to nd them we must take direction into account D Reference Frame freedom to specify your coordinate system a Inertial reference frame pick a reference frame with a constant velocity b The laws of mechanics cannot distinguish between constant velocity reference frames standing on a train platform watching it go by vs being on the train E Average and Instantaneous Quantities a Average velocity vector can be calculated using displacement and time elapsed b VD Xf39Xitf39ti Velocity quanti es how position changes over time Average speed is a scalar using distance traveled and time elapsed found by determining the slope between two points on a graph of x vs t essentially the secant line an e Instantaneous velocity speci c instant in time use derivative f VDdedt g Graphically you would nd the slope at the given point tangent line h I i Acceleration how the slope changes over time derivative of velocity j Average acceleration vfvitfti k Instantaneous acceleration dvDdt Constant Acceleration a Velocity changes linearly over time b Position changes quadratically over time Kinematic Equations used to quantify motion with constant acceleration in one dimension a vtvoat b xtxovot12at2 c v2v022axfxi Application Free Fall a A case of constant acceleration due to gravity g b Mass does not matter and gravity is constant Vectors Example 3 Trigonometry Review 1 FP P FP N problems will often include velocities or positions set at certain angles we use trig functions to split these angles into their x and y components sinoppositehypotenuse cosadjacenthypotenuse tanoppositeadjacent Gsin391oppositehypotenuse Gcos1adjacenthypotenuse etan391oppositeadjacent 8 given 55ms at angle of 53 degrees a velocity in the x direction 55cos53 b velocity in the y direction 555in53 Projectile Motion 1 break down the motion into x and y components 2 de ne your coordinate system 3 x direction equations a vxvox b change in xvoxt y direction equations a vyvoygt b ytyovoyt12gt2 b Dynamics why things move the way they do A Forces physical quantities that measure the strength of interactions between 2 objects a push or a pull a Is a vector quanitity because it has a magnitude and a direction b SI units are Newtons N or kgms2 B Newton s Laws a First law of motion every object continues in it s state of rest or of uniform velocity in a straight line as long as no net force acts upon it law of inertia i lnertia tendency of an object to maintain its state b Second law of motion acceleration of an object is directly proportional to the net force acting n it and inversely proportional to its mass i Direction of acceleration direction of force ii Fund ZFDm iii Mass becomes an objects resistance to accelerate c Third law of motion whenever one object exerts a force on a second object the second object exerts an equal and opposite force on the rst object C Weight and the Force of Gravity a Central force acts towards the center of the earth b Fgmg c The magnitude of the force of gravity related to object s weight d Gravity long range force objects do not have to be physically touching the earth s surface to experience an interaction D Normal Force a Contact forces perpendicular to the surface of contact b ND or FM E Types of Forces a Tension ropestring b Friction opposing motion between 2 surfaces in contact c Electric and magnetic forces F Strategy Sketch the problem De ne your system Draw a Free Body diagram with forces labeled Choose a coordinate system Break vectors into components Apply Newton s second law Solve wmeom LO Summary Thus Far A Summary of Newton s Laws a Fnet0 b Fnetma C IAonBFBonA d Larger massmore inertia B Types of Forces a Gravitational mg acts downwards b Tension acts in a spring or cable has the same magnitude all along the springcable FT or T c Normal perpendicular to surface FN or N d Friction C Friction a Force that acts between 2 surfaces in contact b Static friction keeps objects stationary when sliding would otherwise occur c Kinetic friction when 2 surfaces are sliding against each other d Force is always in the direction that opposes motion or potential motion e Ff is less than or equal to uFN where ucoefficient of friction f Is NOT a vector equation D Coefficient of Friction a Static friction varies in magnitude so the object stays stationary b Kinetic friction should be constant c For most cases uk will be less than us d Is always positive Uniform Circular Motion A lntro Something moving in a circle with a constant speed v Velocity is always tangential to the circle Achange in vchange int The change in velocity will always point towards the center of the circle B Centripetal Acceleration a Objects are kept on a circular path b Arv2r goom c Acceleration will always point towards the center of the circle d Velocity is always tangential to the circle VT e Velocity and acceleration are always perpendicular C Period and Frequency a PeriodT time it takes for one complete revolution around the circle b Frequencyf number of revolutions per second c Are always inverses of each other d T1f D Calculating Speed a Vdistancetimecircumferenceperiod2pirT E The Dynamics of Uniform Circular Motion a Newton s second law forces cause accelerations therefore there must be a centripetal force F Centripetal Force point towards center Fmamv2r a The net force in the radial direction G Misconceptions a There s no outside force or centrifugal force b Centrifugal force is just a myth c Is in an inertial reference frame as we ve constructed Newton s laws if we picked a non inertial reference frame from the point of view of the moving object there would be a quotCentrifugal forcequot in the equanns Gravity Continued A Gravity a Gravity typically is 98 but it changes as you get closerfurther from the earth higher elevationess gravity b Gravity depends on masses of both objects and the distance between them c FgGm1m2r2 d Guniversa constant667x103911Nm2kg2 e Rradius along the line connecting the center s of the two objects always attractive B Gravity Near the Earth s Surface a FgGmMERE2 b ME mass of the earth c REradius of the earth C Kepler s Laws laws of planetary motion a The path of each planet around the sun is an ellipse with the sun at one focus b Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal periods of time c Ration of the squares of the periods T of any 2 planets revolving around the sun is equal to the cubes of their mean distances from the sun d TlT2251SZ3 Energy A Noether s Theorem B Symmetries a Symmetries in the mathematical laws of physics lead to conserved quanUUes b Timetranslation invariance conservation of energy c Spatial translation invariance conservation of momentum d Rotation invariance conservation of angular momentum C Energy a Cannot be created nor destroyed b Can be transferred between objects c Can be converted between forms d Is a scalar quanitity e SI units are Joules J D Kinetic Energy a Energy associated with motion dependent on mass and speed b KE12mv2 c AKE12mAv212mv2iv2f E Potential Energy Energy associated with relative positions of particles Gravitational potential relative position of 2 masses PEgmgy APEgmgAymgyfyi Spring mass potential compression or expansion of spring relative to its equilibrium position Electric potential relative position of 2 charges g We re only focused on the changes in PE because PE has no physical meaning dependent on coordinate system h Kinetic energy doesn t depend on coordinate system so it does have physical meaning F Conservation of Energy EtotQW E total energy within a system Qenergy that enters system in the form of heat Wenergy that enters sytem in the form of work We ll work with mechanical energy so we can ignore Q If no work enters or leaves the system then AEO system is conserved DP00quot h rhme om Momentum A Review Momentum is a quantity of motion PDva Can be written as Newton s second law FDApDAt Conservation law for momentum 09069 f g h Apmtot InetexternalAt No external impulses momentum is conserved Momentum is useful in collision problems i Elastic energy is conserved ii Inelastic energy is not conserved iii Perfectly inelastic subcase of inelastic where objects stick together after collision iv lf momentum is conserved p tot0 or p totinitiap tot nal Change in direction change in momentum B 2D Momentum Problems a 990 we still only use one energy equation but the momentum equation can broken down into components Aptotx InetextxAt Apatchy InetextyAt M1V1ixm2V2ixmlvlfxm2v2fx M1V1iym2V2iymlvlfym2v2fy Example 16 C Center of Mass a an The motion of an extended object can be broken up i Translational motion of the center of mass of the object ii Motion around the center of mass rotation vibration XDcenter of mass maXamBXBmamB This can also be broken up into x and y components There will always be an equal amount of mass on either side of the center as well as above and below it Angular Momentum and Rotational Invariance gtwNi DD T39T39D39LQ h extended objectscenter of mass and roation aroud the center of mass translational motion x v and a rotational motion has matching ideas to x v and a Rotational Kinematics a b c De ne an axis of rotation also called a pivot point Often the axis perpendicular to the center of mass We can de ne angular position by how far it has rotated it is speci ed as an angle 6 and we must use radians Glr Average angular velocity how fast the object rotates around the axis of rotation on AGAt units are radiancssecond Instantaneous angular velocity to at a given time to dGdt Relationship between linear speed v and angular speed m V our to points along the axis of rotation positive directioncounter clockwise negative direction clockwise B Average Angular Acceleration a b How an object accelerates around axis of rotation dA 00 At c units radianss2 ol instantaneous angular acceleration how angular acceleration at a given time e ozd wdt LinearType Rotational Relation X displacement 6 Xr 6 Vtan velocity to Vr w Atan on Ar on acceleration C Constant Angular Acceleration Equations a to 000 at b a 60 wot12 ozt2 c 002 w022q G d Example 17 D Rotational Dynamics a In rotation the force equivalent is torque Tqu b Tnet torque c moment of inertia how resistant an object is to the rotation ol qangular acceleration E Torque a A rotational force b Momentlever arm a vector that goes from the axis of rotation to wherever the force is applied symbol rDdon t confuse with radius c Sometimes it will equal the radius but not always d TrDFTrDFsin e e This 6 is the angle between the rD and FD when placed tail to tail F Moment of Inertia a Rotational equivalent to mass b Is a sum of mass and it s position relative to the axis of rotation c for continuous objects this becomes an integral and will be given to you d miri2 G Conservation of Angular Momentum a Equivalent to the rotational form of Newton s second law b AL AtTnet c AtTnetangular impulse causes change in angular momentum d If net torqueO ALO so LiLf H Work from Torque a Torques do work when an object is displaced angularly b WT A e c PW At

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