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

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

3. 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?

4. 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.

5. Example: Energy in 2-Particle System1 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 :

6. 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.

7. 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).

8. Connection: Force and Potential Energy1 2
Equal and opposite
The combination is independent
of coordinate system.
Attractive for U<0.

9. Connection: Force and Potential Energy1 2
For Gravity:
Attractive for U<0.
To see this:

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

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

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

13. 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

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

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

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