Spring 2010PHYS 172: Modern Mechanics Lecture 21 ? Angular Momentum Read 11.8 ? 11.11 Predicting Position with Rotation F A light string is wrapped around disk of radius R and moment of intertia I that ca freely spin around its axis. The string is pulled with force F during time Dt. Assume that the disk was initially at rest (wi=0) 1) What will be the angular speed wf ? q R I tot net dL dt t= tot netL ttD = D Solution: f i fI I I R F tw w w- = = · D fI RF tw = D f RF t Iw D= Predicting Position with Rotation F A light string is wrapped around disk of radius R and moment of inertia I that ca freely spin around its axis. The string is pulled with force F during time Dt. Assume that the disk was initially at rest (wi=0) 1) What will be the angular speed wf ? 2) How far (Dx) will the end of string move? q R I tot net dL dt t= Solution: f RF t Iw D= aver t qw D? Daver tq wD = D w changes linearly with time: 2 2 i f f aver w w ww += = ( )2 2 2 f RF tt I wq DD = D = ( )2 2 F R tx R Iq DD = D = See also examples in chapter 11.8 Angular momentum quantization Elementary particles may behave as if they have rotational angular momentum Electron may have translational (around nucleus) and rotational angular momenta Angular momentum is quantized Angular momentum quantum = 341.05 10 J s2hp -= = · Whenever you measure a vector component of angular momentum you get either half-integer or integer multiple of [ ] 2 -1J s kg m s = The Bohr model of the hydrogen atom Niels Bohr 1885-1962 1913: electron can only take orbits where its translational angular momentum is integer multiple of ,trans CL r p N= · = r p The Bohr model: allowed radii and energies Allowed Bohr radii for electron orbits: ( ) 0 2 2 2 10 21 4 0.53 10 mNr N Ne m pe -= » · Bohr model energy levels: 22 4 0 2 2 2 1 4 13.6 eV , N=1,2,3,... 2 NN e e m rE N pe = - = - See derivation on page 444-446 The Bohr model: and photon emission 2 13.6 eV NNE = - Refined quantum mechanical model Probability density of the electron for the ground and two excited states of Hydrogen. http://astro.unl.edu/naap/hydrogen/levels.html Translational angular momentum of 1st level (N=1) is 0 of 2nd level (N=2) is 0 or a0 (z component) Particle spin Rotational angular momentum Electron, muon, neutrino have spin 1/2 : mesurements of a component of their angular momentum yields ?˝a0 Quarks have spin ˝ Protons and neutrons (three quarks) have spin ˝ Mesons: (quark+antiquark) have spin 0 or 1 Macroscopic objects: quantization of L is too small to notice! Two lowest energy electrons in any atom have total angular momentum 0 Fermions: spin ˝, Pauli exclusion principle Bosons: integer spin Cooper pairs: superconductivity Rotational energies of molecules are quantized Quantum mechanics: Lx, Ly, Lz can only be integer or half-integer multiple of a0 Quantized values of ( )2 21L l l= + where l is integer or half-integer Gyroscopes Precession and nutation Precession W w cm R Ft = · Gyroscopes rot RMg RMg L IwW = = rot cm dL dt t= A B CLICKER: What is the direction of , A or B?rotL rotL rotL R Mg NF CLICKER: What is the direction of ? A) Left B) Right R CLICKER: What is the direction of ? A) Down C) into the page B) Up D) out of the page cmt cm NR Ft = · cm NRF RMgt = = M rot rot dL L dt = W For rotating vector: RMg= i>clicker In which of the two gyroscopes the disk spins faster? A B RMg IwW = Precession phenomena (see book) Magnetic Resonance Imaging (MRI) Precession of spin axes in astronomy Tidal torques Felix Bloch 1905-1983 NMR - nuclear magnetic resonance Edward Mills Purcell 1912-1997 Independently discovered (1946) Nobel Price (1952) B.S.E.E. from Purdue electrical engineering NMRI = MRI sergei Microsoft PowerPoint - Lect22-Phys172s10-(11.8-11.11-Angular_momentum).ppt