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Chapter 2 - Motion

Part I - KINEMATICS physics how objects move why objects move how objects interact (environment) KINEMATICS - how objects move (not why). Chapter 2 - Motion. Description of motion - change in position cars sports: baseball, football, soccer, etc.

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Chapter 2 - Motion

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  1. Part I - KINEMATICS physics how objects move why objects move how objects interact (environment) KINEMATICS - how objects move (not why) Chapter 2 - Motion Description of motion - change in position cars sports: baseball, football, soccer, etc. world: rotates and revolves How to measure position? PHYSICAL QUANTITIES describe the physical universe two types of physical quantities: SCALARS - described by a magnitude or quantity how much, how far just describe amount mass, time, volume, length, temperature, density, speed

  2. Vectors: magnitude and direction! quantities for which direction is important i.e., where? (displacement - distance and direction) velocity, acceleration, force, momentum, magnetic &electric field displacement - like directions on map Difference between scalars and vectors 1 step = 2 ft 20 feet Distance= displacement =

  3. A C? B A C ~ How to describe vectors - distance and direction Vector variables - A A A A A ~ Geometric description - represent with a directed line scale factor - magnitude  length - ruler direction - points along arrow - protractor Scale1 inch = 1mile head magnitude = tail direction = VECTOR ALGEBRA - adding vectors + = head-to-tail method: head of first to tail of second result: from beginning to end magnitude : direction : displacement from adding displacements

  4. A Ax Ay A Ax Ay Resolution of Vectors - component description Break vectors into components up/down - left/right scale built in components give direction y m in miles x Looks like two perpendicular rulers = + x-component and y-component specifies vector easy component directions (perpendicular) like treasure map Vector - TWO SCALAR COMPONENTS Can treat each direction separately as a vector

  5. Rectilinear Kinematics MOTION- changes in position how objects move without regard to why one-dimensional motion Kinematic physical quantities-how you move position, how fast, speed up SCIENTIFIC MODEL straight line - start and end point particle- all mass and volume a point end start t to=0 d do=0 time (t) : use to keep track of the object at a particular instant - synchronize stop watches time interval or an instant in time start at t=0 to Position (d) : rectilinear - displacement, distance where the object is start-position at t=0 do stop watch at end time t, position d

  6. SPEED (scalar) – rate of change in position how far in a given time – how fast Speed and Velocity AVERAGE SPEED – OVER A TIME INTERVAL vAVG = = d/t can speed up or slow down during trip faster – cover more distance in a given time what constant speed to cover a distance in a given time distance time INSTANTANEOUS SPEED – AT A PARTICULAR TIME like looking at speedometer at an instant v measure: average speed in a short interval Speed of sound – uniform motion (constant speed) v = vAVG = 1100 ft/s Light faster than sound Hear thunder t=5s See lightning to=0s d How far away is the lightning strike?

  7. Velocity – how fast and what direction VECTOR! Magnitude – speed Direction - which way it’s going Rectilinear – speed is magnitude of velocity direction – left/right or up/down y + Direction using normal coordinate directions - x + - AVERAGE VELOCITY – time interval again vavg = = d / t displacement time INSTANTANEOUS VELOCITY at each instant during the interval -- short piece of time interval Speed and velocity t = 10 min. Not same for 2D 20 feet

  8. t to=0 d do=0 CHANGES IN VELOCITY } acceleration – rate of change in velocity can feel – force (cars, elevators,..) VECTOR speeding up slowing down changing direction a v = vo = Final instantaneous velocity initial instantaneous velocity rectilinear – change in speed Average and instantaneous – UNIFORM (CONSTANT) ACCELERATION a= = (v-vo) change in speed time t Two types of problems: Uniform motion Uniform acceleration

  9. a v = vo = t to=0 d do=0 DEFINITIONS : vavg = d / t a = (v-vo)/ t DERIVATIONS : not magic – use math to rearrange definitions Tools for 1D Uniform Acceleration - formulae UNIFORM ACCELERATION FORMULA d=vot + ½ at2 v=vo + at vavg= (v+vo)/2 vavg=d / t can find out how objects move Only works for constant acceleration. Identifies when acceleration begins Fill out line diagram and apply formula How to pick right formula -pick two with variable of interest (v, t, a, d) -find variable you don’t care about

  10. 1D uniform acceleration - examples • An airplane starts from rest and reaches the • final take-off velocity in 50 s. What is its acceleration? • 2. Space shuttle rocket accelerates 2 m/s2. a) What is • the velocity 90 s after lift-off? • b) How far did it go? Finish description • 3. A car travels 75 ft/s east and slams on brakes. a) If it • stops in a distance of 200 ft, then how long did it take • to stop? • b) What is the acceleration? • 4. Car accelerates on I-10 uniformly along ramp. The • speed is 30 ft/s at one instant and 5 s later it is • going 110 ft/s. What is the acceleration of the car?

  11. 2D Motion (planar) - projectiles supported by Church 2000 yrs Unnatural motion to Aristotle four elements natural state – rest (he said so) medium adds resistance to speed slow / fast – depends on object unnatural – medium pushes object DEMO – weight vs. crumpled paper ARISTOTLE RIGHT? Galileo – scientific method measure the motion - kinematics compare quantitatively inclined plane all objects fall the same rate: ay = - g = -10 m/s2 (ay = - g g = 10 m/s2) natural state – constant velocity everything we need to treat complex 2D motion projectile motion :objects projected

  12. , y x Planar Motion two dimensional motion in the plane of paper x and y graph describe vector position - vector velocity EXAMPLES Uniform Circular Motion: ball on a string Moves at a constant speed in circle At time t : position given by x and y Projectile Motion: cannonball, arrow vo Projectile: - projected (thrown) with initial velocity - falls in earth’s gravity

  13. to=0 to=0 t= t= yo=0 xo=0 y= x= vy= vx= voy= vox= Projectile motion thrown in the earth’s gravity separate into component directions(x,y) work with each component separately! VERTICAL DIRECTION Galileo: ALL objects fall the same ay= -g g =10 m/s2 approx. HORIZONTAL DIRECTION No acceleration if no force (friction) ax=0 (ignore air resistance) ay= - g =-10 m/s2 ax=0

  14. to=0 t= yo=0 y= vy= voy= Two Cases We Will Deal With: Vertical Projectile - Jump ball FREE FALL - freely falling in gravity thrown straight up-fall straight down rectilinear - y direction ay a) time to highest point? b) how high? EXAMPLES A baseball is thrown straight up with a speed of 35 m/s. a)How long does it take to reach the highest point? b) How high does the ball go? An egg is thrown straight downward with a speed of 12 m/s. What is its speed 3 seconds later?

  15. to=0 t= yo=0 y= vy= voy= Horizontal projectile - cannonball, soccer, gun Projected with a horizontal velocity -no initial velocity in vertical (y) direction -must work in two directions - separate ! -time connects the directions Separate components vertical free fall horizontal uniform motion two rectilinear problems vo=40 m/s 12 m range a) How much time to hit ground? what direction?

  16. to=0 t= xo=0 x= vx= vox= B) What is the range of the projectile? direction? time connects position for fall in both directions: how far does it go in x-direction in the time it takes to fall? NOTE: time of fall has nothing to do with x-direction! falls same independent of horizontal speed!!!!

  17. Motion - Part 2: Dynamics (MECHANICS)-Why objects move! not unnatural motion * force not required to keep object going * well-defined laws of motion Sir Isaac Newton - 1st theoretical physicist Great mathematician bubonic plague sent him home to orchard developed theories in 18 months- algebra, calculus, motion, gravitation fluid motion, optics (Principia, 1687) looked at results of others---Galileo, Keppler “on the shoulders of great men” Newton’s Laws explain events in everyday life! Newton’s First Law of Motion A body in uniform motion will remain in uniform motion unless acted on by an external force new natural state - uniform motion change motion with force - acceleration

  18. so objects travel in a straight line at a constant speed unless force (push or pull) acts – natural state inertia - tendency of an object to remain in uniform motion LAW OF INERTIA Decelleration – turning a curve PROJECTILE MOTION Galileo knew – but Newton published Newton’s Second Law of Motion(Force Law) The acceleration of a body is proportional to the force and inversely proportional to the mass a=F/mm proportionality constant (inertial)mass (kg) – resistance to a change in uniform motion-or force -ability to remain in uniform motion big mass – small acceleration small mass – accelerates easily train vs. bicycle

  19. a=F/m F=ma Example: How much force is required to accelerate a 1 kg block at 1 m/s2? MODEL a=1 m/s2 F=? m=1 kg F= ma = (1 kg)(1 m/s2) = 1 kg m/s2 SI force: kg m/s2 = 1 Newton = 1 N • F=ma not whole truth • F=ma vector equation – direction • 2N left gives 2 m/s2 left • Also talk about • net force – add all forces on object • Fnet=Fpush + (-Ffriction) friction opposes motion

  20. Second Law Examples: m=1000 kg 1. F=? a=1 m/s2 2. two people m=1000 kg F1=500 N a=? F2=800 N Fnet = 3. m=1000 kg F1=500 N a=? F2= 800 N Direction – be careful!!!

  21. Weight and mass Mass – intrinsic property of matter - doesn’t change -always resistance to force Weight – force of gravity on an object - depends on location -difficulty in lifting an object -weightless in space, but F=ma still no effort to lift! weight – force of gravity W= F = ma = mg on Earth surface g=10 m/s2 acc. due to gravity Surface gravity : depends on gsurface Moon gmoon=1/6 gearth =1.66 m/s2 M=100 kg Wearth= 1000 N Wmoon= 166 N Wspace= 0 g=0 weightless F=ma still

  22. Newton’s laws good for more than 200 years NO DISCREPANCIES Tribute to Newton 20th century- new observations measurement techniques improved failures in F=ma Jet planes, rockets – very very fast scale Einstein’s Theory of Special Relativity E-Microscope, scattering – very very small scale Quantum Mechanical Theory (Bohr, Scrodinger) Telescopy (BH, Neutron stars) – very very massive scale Einstein’s Theory of General Relativity ALL theories reduce to F=ma in the scale of everyday experience: - use Newton’s 2nd law for our purposes - F=ma valid for cars, buildings, etc. less complicated math!!!! - use other theories when needed….

  23. Forces of Nature: accelerate objects Gravitational – force between masses suns, planets, people Electromagnetic – force between charges opposites attract-likes repel “contact force” Strong Nuclear – nucleus of an atom keeps atom together “ likes repel” Weak Nuclear – nuclear decay gamma rays, beta-decay, nuclear reactions UNIFYING THEORY – binds all forces together at times beginning -all forces have same form HOLY GRAIL

  24. Newton’s Third Law of Motion(Action-Reaction) If one body exerts a force on a second, then the second exerts a force back on the first which is equal in magnitude and opposite in direction Easy statement: How objects push on one another - Forces exerted in pairs -Always exerts force back -Large mass, small acceleration pendulum toy leaning on wall walking Fhand Fwall

  25. Application of Newton’s Laws: Circular Motion Uniform Circular motion object traveling in a circle of constant radius at uniform speed Like planet motion Or a ball on a string 1st law – inertial movement constant speed in straight line - velocity R Velocity – tangent Acceleration - radial 2nd law – forced falls in toward center – direction change - acceleration Centripetal (center-seeking) acceleration - FORCE acceleration required to keep object on circle too fast, spirals in – too slow, spirals out ac=v2 / R depends on particular circle and speed

  26. Another Application: MOMENTUM momentum – difficulty in stopping an object p = mv linear momentum mass and velocity vector – direction! BASEBALL m=0.2 kg v=40 m/s p=mv=(0.2 kg)(40 m/s) = 8 kg m/s SI units kg m/s is almost N TRAIN m=100,000 kg v=1 m/s p=mv=(100,000 kg)(1 m/s) = 100,000 kg m/s Heavy or moving fast harder to stop! no motion—no momentum You can change the motion by changing momentum accelerate-fell the force from momentum changes fastball hurts more than slider

  27. DERIVATION: Impulse Second law definition:acceleration Fext=maa= (v-vo)/ t F=ma=m{(v-vo)/ t} IMPULSE-MOMENTUM THEOREM I = Ftc = mv-mvo Impulse- change in momentum tc contact time -during which force applied external force – accelerates as long as force applied Consequences: sports – tennis golf baseball } Hit ball as hard as possible and Follow-through (increase tc) SAME IMPULSE- increase tc safety (cars) -- metal dash padded dash airbags } Io=Ftc

  28. to=0 t=tc xo=0 x= vx= vox= IMPULSE – force applied for a time external force produces acceleration accelerates for time tc a Impulse momentum theorem includes acceleration I= F tc = mv-mvo • EXAMPLE: • A baseball is initially pitched toward the batter • at 40 m/s, and the batter hits it straight back to • the pitcher at 30 m/s. • What impulse is imparted to the ball? • What is the force on the bat? • The bat applies the external force which • changes the motion of the ball • external – connected to body • – connected to ground - etc

  29. Conservation of Momentum - COLLISIONS Momentum important in collisions COLLISION MODEL v2 Isolated System No external forces v1 BEFORE m2 m1 individual impulses cancel –equal & opposite F21 F12 DURING accelerated m1 internal forces 3rd lawaction-reaction pair m2 v2 v1 AFTER m2 m1 Momentum exchanged : I is change in momentum I12=-I21 one gains, other loses momentum

  30. Conservation laws conserved – same before as after constant if assumptions true Conservation of momentum ptot=m1v1+m2v2 if no external forces INTERACTING OBJECTS For collisions, conserved before and after collision =(m1v1+m2v2)after ptot= (m1v1+m2v2)before Internal forces transfer momentum ISOLATED FROM OUTSIDE FORCES No momentum lost transferred EXAMPLE: perfectly inelastic – stick together after colliding

  31. R Newton’s Law of Universal Gravitation Newton described how a gravitational force would act MOTIVATION: ASTRONOMY – circular motion inertial- - linear centripetal Falls toward center of earth MOON Moon falls like apple Earth What caused moon to fall? APPLES and the MOON fall due to same force -- gravity LAW OF UNIVERSAL GRAVITATION All objects with mass attract all other objects with mass -attractive force -smallest force in nature -universal (all objects the same) falling apples-orbiting planets-satellites EXPLAINED Heliocentric Model!

  32. Gravitation Model – picture to understand d point mass-centers m1 m2 F=G m1m2 / d2 m1, m2 masses (kg) d separation (m) G universal constant same for everything Gmust be measured Newton couldn’t do that, but he could: 1. explain motions of planets around Sun (satellites, comets) 2. explain the tides from moon 3. explain why g changes w/ altitude (distance from center of earth) 4. orbital perturbations – deviations from predicted path

  33. Cavendish experiment Established the universal gravitation constant G G = 6.67 x 10-11 N m2/kg2 Can do things like: - calculate forces between ordinary objects - ’weigh’ the earth - predict new planets (perturbations) - put man-made satellites into orbit centripetal force equals gravity force TV, mapping, weather, spy geosynchronous – same period as earth

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