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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 ).

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## -Conservation of angular momentum -Relation between conservation laws & symmetries

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**-Conservation of angular momentum-Relation between**conservation laws & symmetries Lect 4**Rotation**d2 d1 The ants moved different distances: d1 is less thand2**Rotation**q q2 q1 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 symb. Angular quantity symb. distance d angle q velocity v angular vel. w change in d elapsed time = change in q elapsed time =**Angular vs “linear” quantities**Linear quantity symb. Angular quantity symb. distance d angle q velocity v angular vel. w acceleration a angular accel. a change in w elapsed time change in v elapsed time = =**Angular vs “linear” quantities**Linear quantity symb. Angular quantity symb. distance d angle q velocity v angular vel. w acceleration a angular accel. a mass m Moment of Inertia I (= mr2) resistance to change in the state of (linear) motion resistance to change in the state of angular motion moment arm M Moment of inertia = mass x (moment-arm)2 x**Moment of inertial**M M x I Mr2 r r r = dist from axis of rotation I=small I=large (same M) easy to turn harder to turn**Angular vs “linear” quantities**Linear quantity symb. Angular quantity symb. distance d angle q velocity v angular vel. w acceleration a angular accel. a mass m moment of inertia I Force F (=ma) torque t (=I a) Same force; bigger torque Same force; 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 = F x d F “Moment Arm” = d “Line of action”**Opening a door**d small d large F F difficult easy**Angular vs “linear” quantities**Linear quantity symb. Angular quantity symb. distance d angle q velocity v angular vel. w acceleration a angular accel. a mass m moment of inertia I torque t (=I a) Force F (=ma) angular mom. L(=I w) momentum p (=mv) p Iw = Iw x L= p x moment-arm = Iw Angular momentum is conserved: L=const**Conservation of angular momentum**Iw Iw Iw**High Diver**Iw Iw Iw**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 wrist by pivot point F 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 wheel precesses away from viewer F**Angular vs “linear” quantities**Linear quantity symb. Angular quantity symb. distance d angle q velocity v angular vel. w acceleration a angular accel. a mass m moment of inertia I torque t (=I a) Force F (=ma) momentum p (=mv) angular mom. L(=I w) kinetic energy ½mv2 rotational k.e. ½I w2 w I V KEtot = ½mV2 + ½Iw2**Hoop disk sphere race**I hoop I disk I sphere**Hoop disk sphere race**KE = ½mv2 + ½Iw2 I hoop KE = ½mv2 + ½Iw2 I disk KE = ½mv2+½Iw2 I sphere**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**Basis of Kepler’s laws**Laws 1 & 3 are consequences of the nature of the gravitational force The 2nd law is a consequence of conservation of angular momentum v2 r2 A2=r2v2T A1=r1v1T r1 L2=Mr2v2 L1=Mr1v1 L1=L2 v1r1 =v2r2 v1**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**Snowflakes**600**Kaleidoscope**Start with a random pattern Include a reflection rotate by 450 The attraction is all in the symmetry Use mirrors to repeat it over & over**Rotational symmetry**q1 q2 No matter which way I turn a perfect sphere It looks identical**Space translation symmetry**Mid-west corn field**Time-translation symmetry in music**repeat repeat again & again & again**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**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 These are fundamental (At least we think so.)**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 a F Same rule for all directions (no “preferred” directions in space.) a Newton’s laws have rotation symmetry F**Symmetry recovered**Symmetry resides in the laws of nature, not necessarily in the solutions to these laws.

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