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-Conservation of angular momentum -Relation between conservation laws & symmetries. Lect 4. Rotation. Rotation. d 2. d 1. The ants moved different distances: d 1 is less than d 2. Rotation. q. q 2. q 1. Both ants moved the Same angle: q 1 = q 2 (= q ). - PowerPoint PPT Presentation
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-Conservation of angular momentum-Relation between conservation laws & symmetriesLect 4
Rotation
Rotationd1d2The ants moved differentdistances: d1 is less than d2
RotationBoth ants moved theSame angle: q1 = q2 (=q)qq1q2Angle is a simpler quantity than distance for describing rotational motion
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q velocity vchange in delapsed time=angular vel. wchange in qelapsed time=
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q acceleration achange in velapsed time=angular accel. achange in welapsed time=velocity vangular vel. w
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q acceleration aangular accel. avelocity vangular vel. wMoment of inertia = mass x (moment-arm)2mass mresistance to change in the state of (linear) motionMoment of Inertia I (= mr2)resistance to change in the state of angular motionMxmomentarm
Moment of inertialMMxrrI Mr2r = dist from axis of rotationI=smallI=large(same M)easy to turnharder to turn
Moment of inertia
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q acceleration aangular accel. avelocity vangular vel. wForce F (=ma) torque t (=I a)torque = force x moment-armSame force;bigger torqueSame force;even bigger torque mass mmoment of inertia I
Teeter-TotterFFbut Boys moment-arm is larger.. His weight produces a larger torqueForces are the same..
Torque = force x moment-armLine of action Moment Arm = dt = F x dF
Opening a doorFdifficultFeasydsmalldlarge
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q acceleration aangular accel. avelocity vangular vel. wForce F (=ma) torque t (=I a) mass mmoment of inertia Imomentum p (=mv) angular mom. L (=I w)Angular momentumis conserved: L=constIw = IwL= p x moment-arm = Iwxp
Conservation of angular momentumIw IwIw
High DiverIw IwIw
Conservation of angular momentumIwIw
Conservation of angular momentum
Angular momentum is a vectorRight-hand rule
Torque is also a vectorwrist bypivot pointFingers in F directionFThumb int directionanotherright-hand ruleFpivotpointt is out ofthe screenexample:
Conservation of angular momentumL has no verticalcomponentNo torques possible Around vertical axisvertical component of L= constGirl spins:net verticalcomponent of Lstill = 0
Turning bicycleLLThese compensate
Spinning wheelFtwheel precessesaway from viewer
Angular vs linear quantitiesLinear quantity symb. Angular quantity symb.distance d angle q acceleration aangular accel. avelocity vangular vel. wForce F (=ma) torque t (=I a) mass mmoment of inertia Imomentum p (=mv)kinetic energy mv2 angular mom. L (=I w) rotational k.e. I w2IwVKEtot = mV2 + Iw2
Hoop disk sphere race
Hoop disk sphere raceIIIhoopdisksphere
Hoop disk sphere raceIIIKE = mv2 + Iw2KE = mv2 + Iw2KE = mv2+ Iw2hoopdisksphere
Hoop disk sphere raceEvery sphere beats every disk
& every disk beats every hoop
Keplers 3 laws of planetary motionOrbits are elipses with Sun at a focus
Equal areas in equal time
Period2 r3Johannes Kepler1571-1630
Basis of Keplers lawsLaws 1 & 3 are consequences of the nature of the gravitational force
The 2nd law is a consequence of conservation of angular momentumA1=r1v1Tr1v1A2=r2v2Tr2v2L1=Mr1v1L2=Mr2v2L1=L2 v1r1 =v2r2
Symmetry and Conservation laws
Lect 4a
Hiroshige 1797-185836 views of FujiView 4View 14
Hokusai 1760-184924 views of FujiView 18View 20
Temple of heaven (Beijing)
Snowflakes600
Kaleidoscoperotateby 450Start with a random patternUse mirrorsto repeat itover & overThe attractionis all in thesymmetryInclude a reflection
Rotational symmetryNo matter which way I turn a perfect sphereIt looks identicalq1q2
Space translation symmetryMid-west corn field
Time-translation symmetry in musicrepeatrepeat again& again& again
Prior to Kepler, Galileo, etcGod is perfect, therefore nature must be perfectly symmetric:
Planetary orbits must be perfect circles
Celestial objects must be perfect spheres
Kepler: planetary orbits are ellipses; not perfect circles
Galileo:There are mountains on the Moon; it is not a perfect sphere!
Critique of Newtons LawsWhat is an inertial reference frame?: a frame where the law of inertia works.CircularLogic!!Law of Inertia (1st Law): only works in inertial reference frames.
Newtons 2nd LawF = m aBut what is F?whatever gives you thecorrect value for m aIs this a law of nature?or a definition of force??????
But Newtons laws led us to discover Conservation Laws!Conservation of Momentum
Conservation of Energy
Conservation of Angular MomentumThese are fundamental(At least we think so.)
Newtons laws implicitly assume that they are valid for all times in the past, present & futureProcesses that we see occurring in these distant Galaxies actually happened billions of years agoNewtons laws have time-translation symmetry
The Bible agrees that nature is time-translation symmetricThe thing that hath been, it is that which shall be; and that which is doneis that which shall be done: and there is no new thingunder the sun Ecclesiates 1.9
Newton believed that his laws apply equally well everywhere in the Universe Newton realized that the same laws that cause apples to fall from trees here on Earth, apply to planets billions of miles away from Earth.Newtons laws have space-translation symmetry
rotational symmetryFFaaF = m aSame rule forall directions
(no preferred directions in space.)Newtons laws have rotation symmetry
Symmetry recoveredSymmetry resides in the laws of nature, not necessarily in the solutions to these laws.
Emmy Noether1882 - 1935Conserved quantities: stay the same throughout aprocessSymmetry: something that stays the same throughout aprocessConservation laws are consequences of symmetries
Symmetries Conservation lawsSymmetryConservation lawRotationAngular momentumSpace translationMomentumTime translationEnergy
Noethers discovery:Conservation laws are a consequence of the simple and elegant properties of space and time!
Content of Newtons laws is in their symmetry properties