Conservation of angular momentum Relation between conservation laws

-Conservation of angular momentum -Relation between conservation laws & symmetries Lect 4

Rotation

Rotation d 1 d 2 The ants moved different distances: d 1 is less than d 2

Rotation q q 1 q 2 Both ants moved the Same angle: q 1 = q 2 (=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 d v a mass 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 (= mr 2) 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 Mr 2 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; Same bigger torque even bigger 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 symb. angle q angular vel. w angular accel. a moment of inertia I torque t (=I a) angular mom. L (=I w) I w = Iw

Conservation of angular momentum I Iw w Iw

High Diver Iw I w Iw

Conservation of angular momentum Iw I w

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 Girl spins: net vertical component of L still = 0 L has no vertical component No torques possible Around vertical axis vertical component of L= const

Turning bicycle These compensate L L

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 mv 2 I w V Angular quantity symb. angle 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 w 2 KEtot = ½ m. V 2 + ½ Iw 2

Hoop disk sphere race

Hoop disk sphere race hoop I I disk I sphere

Hoop disk sphere race I disk I hoop KE = ½ mv 2 + ½ KE = ½ m v KE = ½ m sphere I I 2 w 2 + ½ v 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 • Period 2 r 3

Basis of Kepler’s laws Laws 1 & 3 are consequences of the nature of the gravitational force The 2 nd law is a consequence of conservation of angular momentum A 1=r 1 v 1 T L 1=Mr 1 v 1 r 2 r 1 v 2 A 2=r 2 v 2 T L 2=Mr 2 v 2 L 1=L 2 v 1 r 1 =v 2 r 2

Symmetry and Conservation laws Lect 4 a

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 te 0 a t ro 45 by The attraction is all in the symmetry

Rotational symmetry qq 2 1 No matter which way I turn a perfect sphere It looks identical

Space translation symmetry Mid-west corn field

Timetranslation symmetry in music r in a g a t a pe re t a e p e & in a g a

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 Law of Inertia (1 st 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 2 nd 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! e r a tal e n s e e h m T da n u f • Conservation of Momentum • Conservation of Energy t s a • Conservation of e l Angular Momentumt (A e w k n i h t s ). o

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 stayconsequences the same that stays the throughout a ofsame 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