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Lecture I: Basic Physics1

Velocity Velocity: Instantaneous change in position !⃗ %⃗' & Suppose object position () and constant velocity !⃗.After time step : () () !⃗ () () () !⃗ . !⃗ is never constant in practice A function of time !⃗ . Position is integrated in time: () () & 2 !⃗/ 1/. Velocity SI units: 3 456Lecture I: Basic Physics2

Acceleration Instantaneous change in velocity: !⃗ %' &. Constant acceleration: )⃗ !⃗ *. Otherwise, integrate: ) * V & 1 !/ 0/. Note vector quantities! Position: trajectory of a point. Velocity: tangent to trajectory curve. Speed: absolute value of velocity. Acceleration: the change in the tangent.Lecture I: Basic Physics3

Relative Quantities Using coordinates, our vector quantities arerelative to the chosen axis system (origin xyzdirection) They are viewpoint dependent. The derivation/integration relations are invariant!Lecture I: Basic Physics4

Forces Acceleration is induced by a force. Direction of force direction ofassociated acceleration. Net force (and net acceleration): thesum of all acting forces.Lecture I: Basic Physics5

Newton’s laws of motion In the late 17th century, SirIsaac Newton described threelaws that govern all motion onEarth. .ultimately, an approximation Small scale: quantum mechanics. Big scale: theories of relativity.Lecture I: Basic Physics6

1st Law of Motion Sum of forces on an object is null ó there is nochange in the motionIf !"# 0, there is no change in motion With zero force sum: An object at rest stays at rest. A moving object perpetuates in the same velocity. Behavior of objects in the outer space.Lecture I: Basic Physics7

2nd Law of Motion Each force induces a co-directional acceleration inlinear to the mass of the object:!⃗# % ' ( )⃗' is the mass and )⃗ the acceleration Consequently: More force ó faster speed-up. Same force ó lighter objects accelerate faster thanheavy objects.Lecture I: Basic Physics8

3rd Law of Motion Forces have consequences:When two objects come into contact, they exertequal and opposite forces upon each other. All forces are actually interactions between bodies!What happens here?Lecture I: Basic Physics9

Gravity Newton’s Law of Gravitation: the gravitation forcebetween two masses ! and " is:,' ,)#⃗% #⃗' ) #⃗) ' /'). : gravitational constant 6.673 10899 [, 89 ? 8. ].- @⃗' @⃗) : the distance between the objects./') A⃗B 8A⃗CD A⃗B 8A⃗C : the unit direction between them.Lecture I: Basic Physics10

Gravity on Earth By applying Newton’s 2nd law to an object withmass ! on the surface of the Earth, we obtain:"⃗ %& "⃗( ! ) *⃗,),-./01 23 -./01,-./01 23 *-./01!)*Mass of object is canceled out!* 4562&7 6.673 ous-thought-experiments-that-just.html5.98 10DED 9.81!/J6.377 10F DLecture I: Basic Physics11

Gravity on Other Planets On Earth at altitude ℎ: " %&'()* &'()* ,- . On the Moon /%001 7.35 1099 ; %001 1738 ;/ %001 1.62 //B 9 On Mars /%C D 6.42 109F ; %C D 3403 ;/ %C D 3.69 //B 9Lecture I: Basic Physics12

Weight Weight ó gravitational force! # %⃗ We weigh different on the moon (but have thesame mass ) Force units: ['%) * , - ]. Denoted as Newtons [0].Lecture I: Basic Physics13

Free-Body Diagram To get acceleration: sum forces anddivide by mass (D'Alembert's principle):!⃗# % ' !⃗( ) * ⃗ Forces add up linearly as vectors. Important: when all are represented in thesame axis system! The Free-Body Diagram includes: Object shape: center of mass, contactpoints. Applied forces: direction, magnitude, andpoint of application.Lecture I: Basic 69ad22bd5-46.html14

Normal force Force acting as a reaction to contact. Direction is normal to the surface of contact. Magnitude enough to cancel the weight so objectdoesn’t go through the plane. Here, !⃗# % cos(*) ,-⃗ cos(.) Related to collision handling (more later). Object slides down plane with remaining force:% sin(.).Lecture I: Basic Physics15

Friction Can the object stay in total equilibrium? An extra tangential friction force must cancel ! sin(&). Ability to resist movement. Static friction keeps an object on a surface from moving. Kinetic friction slows down an object in contact.Lecture I: Basic Physics16

Friction Static friction: a threshold force. object will not move unless tangential force is stronger. Kinetic friction: when the object is moving. Depends on the materials in contact. smoother ó less friction. Coefficient of friction ! determines friction forces: Static friction: "# !# "% Kinetic friction: "& !& "%Lecture I: Basic Physics17

Friction The kinetic coefficient of friction is always smallerthan the static friction. If the tangential force is larger than the staticfriction, the object moves. If the object moves while in contact, the kineticfriction is applied to the object.Lecture I: Basic Physics18

FrictionSurface FrictionStatic (!" )Kinetic (!# )Steel on steel (dry)0.60.4Steel on steel (greasy)0.10.050.0410.04Brake lining on cast iron0.40.3Rubber on concrete (dry)1.00.9Rubber on concrete (wet)0.300.25Metal on ice0.0220.02Steel on steel0.740.57Aluminum on steel0.610.47Copper on steel0.530.36Nickel on nickel1.10.53Glass on glass0.940.40Copper on glass0.680.53Teflon on steelLecture I: Basic Physics19

Fluid resistance An object moving in a fluid (air is a fluid) is sloweddown by this fluid. This is called fluid resistance, or drag, anddepends on several parameters, e.g.: High velocity ó larger resistance. More surface area ó larger resistance (“badaerodynamics”).Lecture I: Basic Physics20

Fluid resistance At high velocity, the drag force !" # %# is quadraticto the relative speed & of the object:!"()*( . / 0/ & . / 12 / 3 0 is the density of the fluid (1.204 for air at 20 ) 12 is the drag coefficient (depends on the shape ofthe object). 3 is the reference area (area of the projection ofthe exposed shape).Lecture I: Basic Physics21

Fluid resistance At low velocity, the drag force is approximatelylinearly proportional to the velocity!"# % ( ) *⃗where ( depends on the properties of the fluid andthe shape of the object. High/low velocity threshold is defined by ReynoldsNumber (Re).Lecture I: Basic Physics22

Buoyancy1.14 Develops when an object isimmersed in a fluid. A function of the volume of theobject ! and the density of thefluid ":# " & ' & ! Considers the difference ofpressure above and below theimmersed object. Directed straight up, counteractingthe weight.Lecture I: Basic Physics23

Springs React according to Hook’s Law on extension andcompression, i.e. on the relative displacement. The relative length ! to the rest length !"determines the applied force:# ' ! !" ' is the spring constant (in (/*). Scalar spring: two directions.Lecture I: Basic Physics24

Dampers Without interference, objects may oscillateinfinitely. Dampers slow down the oscillation betweenobjects ! and " connected by a spring. Opposite to the relative speed between the twoobjects:#⃗% ((*⃗ *⃗, ) ( is the damping coefficient. Resulting force applied on ! (opposite on "). Similar to friction or drag at low velocity!Lecture I: Basic Physics25

Free-Body Diagram To get acceleration: sum forces anddivide by mass (D'Alembert's principle):!⃗# % ' !⃗( ) * ⃗ Forces add up linearly as vectors. Important: when all are represented in thesame axis system! The Free-Body Diagram includes: Object shape: center of mass, contactpoints. Applied forces: direction, magnitude, andpoint of application.Lecture I: Basic 69ad22bd5-46.html26

Work A force !⃗ does work # (in %&'( * ,), if itachieves a displacement /⃗ in the direction of thedisplacement:# !⃗ /⃗ Note dot product between vectors. Scalar quantity.!3! cos 3Lecture I: Basic Physics27

Kinetic energy The kinetic energy !" is the energy of an object invelocity:1!" ' (⃗ *2 The faster the object is moving, the more energy it has. The energy is a scalar (relative to speed ( (⃗ ,regardless of direction). Unit is also Joule:4 ,('//01)* , 567 8 ' 9 ' :Lecture I: Basic Physics28

Work-Energy theorem The Work-Energy theorem: net work ó change inkinetic energy:! % % (' ') % (')i.e.1 ⃗ - .⃗ 1 2(' ')3 2(')32 Very similar to Newton’s second law.Lecture I: Basic Physics29

Potential energy (Gravitational) Potential energy is the energy‘stored’ in an object due to relative heightdifference. The amount of work that would be done if we were to setit free.!" % & % ℎ Simple product of the weight ( % & and height ℎ.* Also measured in Joules (as here )& % ,- . % ). Other potential energies exist (like a compressedspring).Lecture I: Basic Physics30

Conservation of mechanical energy Law of conservation: in a closed system, energycannot be created or destroyed. Energy may switch form. May transfer between objects. Classical example: falling trades potential and kineticenergies.!" ( ) !( ( ) !" ( ) !( ( )i.e.11.,-( ) ,/ℎ ,-( ). ,/ℎ 22Lecture I: Basic Physics31

Conservation: Example A roller-coaster cart at the top of the first hill Much potential energy, but only a little kinetic energy. Going down the drop: losing height, picking up speed. At the bottom: almost all potential energy switched tokinetic, cart is at its maximum speed.Lecture I: Basic Physics32

Conservation of Mechanical Energy External forces are usually applied: Friction and air resistance. Where does the “reduced” energy go? Converted into heat and air displacements (soundwaves, wind). We compensate by adding an extra term !" to theconservation equation:!# !' !" !# !' if !" 0, some energy is esdefault.jpgLecture I: Basic Physics33

Momentum The linear momentum !⃗: the mass of anobject multiplied by its velocity:!⃗ & ( Heavier object/higher velocity ó moremomentum (more difficult to stop). unit is [* , - ./0]. Vector quantity (velocity).Lecture I: Basic Physics34

Impulse A change of momentum:!⃗ % Compare: Impulse is change in momentum. Work is change in energy. Unit is also ['(*) ,-](like momentum). Impulse ó force integrated over time:44/⃗ 0 1⃗ 23 6 0 7⃗23 6Δ9⃗.55Lecture I: Basic Physics35

Conservation of Momentum Law of conservation: in a closed system (noexternal forces\impulses), momentum cannot becreated or destroyed. Compare: conservation of energy. Implied from 3rd law. Objects react with the same force exerted on them. Special case of Noether’s theorem: every physicalsystem (With a symmetric action) has aconservation law.Emmy NoetherLecture I: Basic Physics36

1. Lecture I: Basic Physics 2. Velocity: Instantaneouschange in position !⃗ %⃗. Suppose object position ()and constantvelocity !⃗. After time step : () () !⃗ () () () !⃗ . !⃗is never constant in practice. A function of time !⃗ . Position is integratedin time: () () !⃗ / 1/.File Size: 1MBPage Count: 36