-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