PHYS 172: Modern Mechanics Lecture 15 – Multiparticle Systems Read 9.1 – 9.2 Summer 2012

Quantizing two interacting atoms Any value of A is allowed And any E is possible. Classical harmonic oscillator:Quantum harmonic oscillator: U = (1/2)k s s EE  EE  Energy levels: N EN  Js 2 h   =  ｴ 0 s k m  E  equidistant spacing ground state

Time to Throw Things BALL We need to understand Center of Mass BATON

The Center of Mass where This is a weighted average of the positions -- each position appears in proportion to its mass

cm mrmrmr r mmm  =  … … The Center of Mass

Motion of the Center of Mass 1) Take one time derivative: Same as: (Good!)

Motion of the Center of Mass 1) Take a second time derivative: This says that the motion of the center of mass looks just like what would happen if all forces were applied to the total mass, as a point particle located at the center of mass position!

 F net, ext =M total a cm = dP total dt This says that the motion of the center of mass looks just like what would happen if all forces were applied to the total mass, as a point particle located at the center of mass position. Motion of the Center of Mass Center of Mass

Same Tension. Which puck will move faster? The centers of mass experience the same acceleration! HOWEVER: Hand #2 has to pull the string farther: W 2 > W 1. Where does this energy go? Rotational energy. The bottom spool is spinning. Center of Mass Motion

Question for Discussion

Through what distance did the force act on the Point Particle system? A) 0.03 m B) 0.04 m C) 0.07 m D) 0.08 m E) 0.10 m Clicker Question Equal masses

Through what distance did the force act on the Real system? A) 0.03 m B) 0.04 m C) 0.07 m D) 0.08 m E) 0.10 m Equal masses Clicker Question

Which is the energy equation for the Point Particle system? Equal masses Clicker Question A) Δ K trans = F*(0.07 m) B) Δ K trans = F*(0.08 m) C) Δ K trans + Δ K vib + Δ U spring = F*(0.07 m) D) Δ K trans + Δ K vib + Δ U spring = F*(0.08 m)

Kinetic energy of a multiparticle system Translational, motion of center of mass Vibration Rotation about center of mass Motion of parts relative to center of mass

Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws the bottle away as shown. The velocity of the center of mass v cm will be: A)Zero B)Directed to right C)Directed to left

Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws a bottle away as shown. The translational kinetic energy of the system will be: A)Zero B)> 0 C)< 0

Vibrational kinetic energy - Net momentum = 0 - Energy is constant (sum of elastic energy and kinetic energy)

Rotational kinetic energy - Net momentum = 0 - Energy is constant Motion around of center of mass

Rotation and vibration CM Rotation and vibration and translation

Gravitational potential energy of a multiparticle system Gravitational energy near the Earth’s surface M y cm

Example: Rotation and translation Assume all mass is in the rim Energy principle: =0 EXAMPLE: