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-Conservation of angular momentum -Relation between conservation laws & symmetries Lect 4 Rotation Rotation d1 d2 The ants moved different distances: d1 is less than d2 Rotation q q1 q2 Both ants moved the Same angle: q1 = q2 (=q) Angle is a simpler quantity than distance for describing rotational motion Angular vs “linear” quantities Linear quantity distance velocity = change in d elapsed time symb. d v Angular quantity symb. angle angular vel. change in q = elapsed time q w Angular vs “linear” quantities Linear quantity distance velocity acceleration = change in v elapsed time symb. d v a Angular quantity symb. angle q angular vel. w angular accel. a change in w = elapsed time Angular vs “linear” quantities Linear quantity symb. distance velocity acceleration mass d v a m resistance to change in the state of (linear) motion moment arm x Angular quantity symb. angle q angular vel. w angular accel. a Moment of Inertia I (= mr2) resistance to change in the state of angular motion M Moment of inertia = mass x (moment-arm)2 Moment of inertial M M x I Mr2 r I=small r r = dist from axis of rotation I=large (same M) easy to turn harder to turn Moment of inertia Angular vs “linear” quantities Linear quantity distance velocity acceleration mass Force symb. Angular quantity symb. d angle q v angular vel. w a angular accel. a m moment of inertia I F (=ma) torque t (=I a) Sameforce; force; Same bigger torque even bigger torque torque = force x moment-arm Teeter-Totter His weight produces a larger torque F Forces are the same.. but Boy’s moment-arm is larger.. F Torque = force x moment-arm t=Fxd F “Moment Arm” = d “Line of action” Opening a door d small d large F difficult F easy Angular vs “linear” quantities Linear quantity symb. distance velocity acceleration mass Force momentum d v a m F (=ma) p (=mv) p x L= p x moment-arm = Iw Angular momentum is conserved: L=const Angular quantity angle symb. q angular vel. w angular accel. a moment of inertia I torque t (=I a) angular mom. L (=I w) Iw = Iw Conservation of angular momentum w I Iw Iw High Diver Iw w I Iw Conservation of angular momentum Iw w I Conservation of angular momentum Angular momentum is a vector Right-hand rule Torque is also a vector example: pivot point another right-hand rule F t is out of the screen Thumb in t direction F wrist by pivot point Fingers in F direction Conservation of angular momentum L has no vertical component No torques possible Around vertical axis vertical component of L= const Girl spins: net vertical component of L still = 0 Turning bicycle L These compensate Spinning wheel t F wheel precesses away from viewer Angular vs “linear” quantities Linear quantity symb. distance velocity acceleration mass d v a m Force momentum F (=ma) p (=mv) kinetic energy ½ mv2 I w V Angular quantity angle symb. q angular vel. w angular accel. a moment of inertia I torque t (=I a) angular mom. L (=I w) rotational k.e. ½ I w2 KEtot = ½ mV2 + ½ Iw2 Hoop disk sphere race Hoop disk sphere race hoop I I disk I sphere Hoop disk sphere race hoop KE = ½ mv2 + ½ v KE = ½ m disk I 2 + ½ v KE = ½ m sphere w2 Iw 2 2+½ Iw 2 Hoop disk sphere race Every sphere beats every disk & every disk beats every hoop Kepler’s 3 laws of planetary motion Johannes Kepler 1571-1630 • Orbits are elipses with Sun at a focus • Equal areas in equal time • Period2 r3 2 2 ( 2 r ) ( 2 r ) 1 2 v 2 3 P r r 1 r Basis of Kepler’s laws Laws 1 & 3 are consequences of the nature of the gravitational force F G M sunM planet r2 The 2nd law is a consequence of conservation of angular momentum v2 r2 A1=r1v1T L1=Mr1v1 r1 v1 A2=r2v2T L2=Mr2v2 L1=L2 v1r1 =v2r2 Symmetry and Conservation laws Lect 4a Hiroshige 1797-1858 36 views of Fuji View 4 View 14 Hokusai 1760-1849 24 views of Fuji View 18 View 20 Temple of heaven (Beijing) Snowflakes 600 Kaleidoscope Start with a random pattern Include a reflection Use mirrors to repeat it over & over The attraction is all in the symmetry Rotational symmetry qq2 1 No matter which way I turn a perfect sphere It looks identical Space translation symmetry Mid-west corn field Timetranslation symmetry in music Prior to Kepler, Galileo, etc God 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 Newton’s Laws Law of Inertia (1st Law): only works in inertial reference frames. Circular Logic!! What is an inertial reference frame?: a frame where the law of inertia works. Newton’s 2nd Law F = ma ????? But what is F? whatever gives you the correct value for ma Is this a law of nature? or a definition of force? But Newton’s laws led us to discover Conservation Laws! • Conservation of Momentum • Conservation of Energy • Conservation of Angular Momentum Newton’s laws implicitly assume that they are valid for all times in the past, present & future Processes that we see occurring in these distant Galaxies actually happened billions of years ago Newton’s laws have time-translation symmetry The Bible agrees that nature is time-translation symmetric Ecclesiates 1.9 The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun 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. Newton’s laws have space-translation symmetry rotational symmetry F=ma F Same rule for all directions a (no “preferred” directions in space.) a F Newton’s laws have rotation symmetry Symmetry recovered Symmetry resides in the laws of nature, not necessarily in the solutions to these laws. Emmy Noether Conserved Symmetry: Conservation quantities: something laws are stay the same that stays consequences throughout the same a of symmetries process a throughout process 1882 - 1935 Symmetries Conservation laws Conservation law Symmetry Angular momentum Space translation Momentum Time translation Energy Rotation Noether’s discovery: Conservation laws are a consequence of the simple and elegant properties of space and time! Content of Newton’s laws is in their symmetry properties