Momentum & Energy conservation
Momentum
Newton’s 2nd law (shorthand version) F = ma change in v time a = change in v time F = m
Car truck collision Fc Ft Ft = mt change in vt time Fc = mc change in vc time Fc t = mc change in vc Ft t = mt change in vt Fc t + Ft t = mc change in vc + mt change in vt (Fc + Ft)t =change in mcvc + change in mtvt (Fc + Ft)t =change in (mcvc + mtvt)
0 = change in (mcvc + mtvt) mcvc + mtvt stays constant! Car truck collision Fc Ft (Fc + Ft)t =change in (mcvc + mtvt) Newton’s 3rd law: Fc = -Ft (Fc + Ft)t = 0 0 = change in (mcvc + mtvt) mcvc + mtvt stays constant!
Mtvt= momentum of thruck mcvc + mtvt = total momentum Momentum = mv mcvc = momentum of car this changes this changes Mtvt= momentum of thruck mcvc + mtvt = total momentum this stays constant Before = -40 After = -40 Momentum is conserved!
True for all collisions before =+20 after =+20 visit www.physicsclassroom.com/mmedia/index.html
Revisit the canoe at the dock Initial momentum canoe = 0 boy = 0 Total = 0 final momentum canoe = mcvc boy = mbvb Total = 0
Momentum is a vector: mv collision in 2 dimensions
eating Finding nemo
Billiard balls 2 before after ptot 2 ptot 1 1
Conservation of momentum on a sub-atomic level before p p ptot proton after p ptot p p- meson p- meson
Rocket travel before P0 after P0 + p exhaust p
Rifle recoil mV mV
Machine-gun granny
Work and Energy
Physicist’s definition of “work” dist∥ A scalar (not a vector) dist Work = F x dist∥
Atlas holds up the Earth But he doesn’t move, dist∥ = 0 Work= Fx dist∥ = 0 He doesn’t do any work!
Garcon does work when he picks up the tray but not while he carries it around the room dist is not zero, but dist∥ is 0
This scalar quantity is given a special name: kinetic energy Why this definition? A vector equation Newton’s 2nd law: F=m a Definition of work + a little calculus A scalar equation Work= change in ½mv2 This scalar quantity is given a special name: kinetic energy
Concept of Kinetic Energy Emilie du Châtelet (1706-1749) Brilliant mathematician One of Voltaire’s lovers
the Work-Energy Theorem Work = change in KE This is called: the Work-Energy Theorem
Units again… Kinetic Energy = ½mv2 m2 s2 kg work = F x dist∥ =1Joule m same! =1Joule m s2 N m =kg m
change in vertical height Work done by gravity end start dist dist∥ change in vertical height W=mg Work = F x dist∥ = -mg x change in height = -change in mg h
Gravitational Potential Energy Workgrav = -change in mgh This is called: “Gravitational Potential Energy” (or PEgrav) change in PEgrav = -Workgrav Workgrav = -change in PEgrav
If gravity is the only force doing work…. Work-energy theorem: -change in mgh = change in ½ mv2 0 = change in mgh + change in ½ mv2 change in (mgh + ½ mv2) = 0 mgh + ½ mv2 = constant
Conservation of energy mgh + ½ mv2 = constant Gravitational Potential energy Kinetic energy If gravity is the only force that does work: PE + KE = constant Energy is conserved
Free fall (reminder) height t = 0s 80m V0 = 0 75m t = 1s V1 = 10m/s
m=1kg free falls from 80m mgh ½ mv2 sum t = 0s V0 = 0 h0=80m 800J 0 750J 50J V1 = 10m/s; h1=75m 800J t = 2s V2 = 20m/s; h2=60m 600J 200J 800J t = 3s V3 = 30m/s; h3=35m 350J 450J 800J t = 4s V4 = 40m/s; h4=0 0 800J 800J
T is always ┴ to the motion pendulum T W=mg Two forces: T and W T is always ┴ to the motion (& does no work)
Pendulum conserves energy Etot=mghmax Etot=mghmax hmax Etot=1/2 m(vmax)2
Roller coaster
Work done by spring = - change in ½ kx2 Work done by a spring Relaxed Position F=0 x F I compress the spring (I do + work; spring does -work) Work done by spring = - change in ½ kx2
If spring is the only force doing work…. Work-energy theorem: -change in ½ kx2 = change in ½ mv2 0 = change in ½ kx2 + change in ½ mv2 change in ( ½ kx2 + ½ mv2) = 0 ½ kx2 + ½ mv2 = constant potential energy in the spring
Conservation of energy springs & gravity mgh + ½ kx2 + ½ mv2 = constant Gravitational potential energy spring potential energy Kinetic energy If elastic force & gravity are the only forces doing work: PEgrav + PEspring + KE = constant Energy is conserved
example grav PE KineticE Spring PE
Two types of forces: “Conservative” forces “Dissipative” forces forces that do + & – work Gravity Elastic (springs, etc) Electrical forces … “Dissipative” forces forces that only do – work Friction Viscosity …. -work heat (no potential energy.) -work change in PE
(-)Work done by frictionheat
Thermal atomic motion Heat energy= KE and PE associated with Air solid the random thermal motion of atoms
Work-energy theorem (all forces) Workfric = change in (PE+KE) Work done dissipative Forces (always -) potential energy From all Conservative forces Kinetic energy -Workfric = change in heat energy Workfric = -change in heat energy -change in Heat Energy = change in (PE+KE)
Work – Energy Theorem (all forces) 0 = change in Heat Energy + change in (PE+KE) 0 = change in (Heat Energy+PE+KE) Heat Energy + PE + KE = constant Law of Conservation of Energy
Energy conversion while skiing Potential energy Potential energykinetic energy Friction: energy gets converted to heat
1 calorie = heat energy required to raise the Units again Heat units: 1 calorie = heat energy required to raise the temp of 1 gram of H2O by 1o C Kg m2/s2 1 calorie= 4.18 Joules
Food Calories 1 Calorie = 1000 calories = 1Kcalorie The Calories you read on food labels 1 Calorie= 4.18x103 Joules 7 x 106 J 8 x 105 J 2 x 106 J
electrical energy each second to produce light Power amout of energy elapsed time Rate of using energy: Power = Joule second Units: 1 = 1 Watt A 100 W light bulb consumes 100 J of electrical energy each second to produce light
Other units 1 Horsepower = 750 Watts Over a full day, a work-horse can have an average work output of more than 750 Joules each second 1 Horsepower = 750 Watts
Kilowatt hours energy time Power = energy = power x time power unit x time unit = energy unit Kilowatts (103 W) hours (3600 s) Elec companies use: x 1 kilowatt-hour = 1kW-hr = 103 W x 3.6x103 s = 3.6x106 Ws J
In Hawaii electrical energy costs about 25cents /kW-hr about 300 won In Hawaii electrical energy costs about 25cents /kW-hr What is the cost in Seoul?