PHYS 172 Modern Mechanics Lecture 11 The Energy

PHYS 172: Modern Mechanics Lecture 11 – The Energy Principle Summer 2012 Read 6. 8 – 6. 14

TODAY • Multiparticle Systems and Potential Energy • Relationship of Force and Potential Energy • Energy Graphs

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

Example: Energy in 2 -Particle System =0 system 2 Let’s find the energy change of each particle: 1 We’re counting the work done by internal forces.

Example: Energy in 2 -Particle System 2 1 Put system on left side, surroundings on right side: Now define the change in potential energy as U – Wint :

Potential Energy (in 2 -Particle System) 2 1 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 sum of interaction Energy of system = + particle energies of all pairs We now write Now W is about external forces only (internal forces show up in U).

Connection: Force and Potential Energy Equal and opposite 2 1 The combination is independent of coordinate system.

Connection: Force and Potential Energy 2 1 To see this: For Gravity:

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

Energy Example: Planet and Star 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)

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

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

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