PHYS 172: Modern Mechanics Summer 2012 Lecture 11 – The Energy Principle Read 6.8 – 6.14

TODAY Multiparticle Systems and Potential Energy Relationship of Force and Potential Energy Energy Graphs START TIME HERE:

Last Time: Single Particle System Energy principle (single particle system): where energy is and work is = 0 for now How do we generalize these results to multiparticle systems?

Example: Energy in 2-Particle System = 0 1 2 system Let’s find the energy change of each particle: For a single particle system, only the second term (external force dot displacement) exists. For a multiparticle system, work is done when the position of one particle changes with respect to the other due to the interactions between the particles. We’re counting the work done by internal forces.

Example: Energy in 2-Particle System 1 2 Put system on left side, surroundings on right side: E1, etc refer to single particle energies (rest and kinetic). Now define the change in potential energy as U  – Wint :

Potential Energy (in 2-Particle System) 1 2 The potential energy U represents a sum of interaction energies between all pairs of particles inside the system. NOTE: U is defined to take into account both terms above. U is related to a system changing shape.

Energy of a Multiparticle System sum of single particle energies sum of interaction energies of all pairs Energy of system = + We now write Now W is about external forces only (internal forces show up in U).

Connection: Force and Potential Energy 1 2 Equal and opposite The combination is independent of coordinate system. Attractive for U<0.

Connection: Force and Potential Energy 1 2 For Gravity: Attractive for U<0. To see this:

Gravitational Potential Energy Ug=0 It is negative! r=0

Example: Planet and Star Each of these is constant So these together must be constant

Example: Planet and Star Energy K K+U r U

Application: Escape Speed What does it take to launch a rocket so it leaves the Earth's gravitational well? Minimal condition for escape: Assume: planet is stationary (choose frame of planet) 13

Application: Escape Speed What does it take to launch a rocket so it leaves the Earth's gravitational well? Bound state: Unbound state: 14

Gravitational U Near Earth's Surface Are these the same? They are the same near the Earth's Surface. Taylor Expansion 15

WHAT WE DID TODAY Multiparticle Systems and Potential Energy Relationship of Force and Potential Energy Energy Graphs