1. PHYS 172: Modern Mechanics Summer 2012
Lecture 12 – The Energy Principle Read 6.15 – 6.17

2. TODAY Mass and Energy Mass of a Multiparticle System Binding Energy
Energy of a Mass-Spring System 2

3. Mass and Energy Example: neutron decay electron neutron antineutrino
proton
antineutrino
neutron
Mass of products < mass of neutron!
Missing mass is converted into kinetic energy
Example: electron and positron (antielectron) annihilation:
Remind them of neutron decay – mass changes into energy. Here is another example where mass disappears entirely
It may appear that this is an example which has no application – but this reaction is used in positron tomography – radioactive compound is administered to a patients blood and it produces positrons….
Radioactive compound is ejected into patient. Radioactivity creates positrons that immediately annihilate emitting two gamma-photons in opposite directions (momentum conservation).
All mass is converted into electromagnetic radiation – gamma rays
Positron tomography 3

4. Mass and energy are the same thing!
Remind them of neutron decay – mass changes into energy.
1905: special theory of relativity 4

5. Mass of a Multiparticle System
The System
Mass of a multiparticle system:
If the system is at rest, K=0 and: 5

6. Two ways of thinking
The energy of a multiparticle system consists of the individual particle energies plus their pair-wise interactions
A system itself has energy, like a single particle, and if the system is at rest (not individual objects within the system!) its energy E = Mc2, where M is mass of the system. 6

7. Binding energy: nuclear reactions
Adding these, get u
Difference is u
Binding energy: mc2
(1 eV = 110-19 J)
Proton: m = u
Neutron: m = u
Deuteron: m = u
The unified atomic mass unit (u), or dalton (Da),
The chemist John Dalton was the first to suggest the mass of one atom of hydrogen as the atomic mass unit. Francis Aston, inventor of the mass spectrometer, later used 1⁄16 of the mass of one atom of oxygen-16 as his unit.
Before 1961, the physical atomic mass unit (amu) was defined as 1⁄16 of the mass of one atom of oxygen-16, while the chemical atomic mass unit (amu) was defined as 1⁄16 of the average mass of an oxygen atom (taking the natural abundance of the different oxygen isotopes into account). Both units are slightly smaller than the unified atomic mass unit, which was adopted by the International Union of Pure and Applied Physics in 1960 and by the International Union of Pure and Applied Chemistry in Hence, before 1961 physicists as well as chemists used the symbol amu for their respective (and slightly different) atomic mass units. One still sometimes finds this usage in the scientific literature today. However, the accepted standard is now the unified atomic mass unit (symbol u), with: 1 u = amu (physical scale) = amu (chemical scale). 7

8. Nuclear fission
slow neutron
This process generates more than 1 neutron in product, they can initiate another reaction: chain reaction
0.454 kg * c^2 = 4.1 e16 J
Today’s world consumption: 17 TW = 1.7e13 W
4.1e16J/1.7e13 = 2411 s = 40 minutes
13140 pound per year, about 6 tons.
36 pound/day
----- one household ----
About 300 GJ/year = 3e11 J/year = 3 mg/year
One pound will power a house for more than 100,000 years!
Convert 1 pound into energy – enough to power one household for 100,000 years!
Nuclear Power Plants and Nuclear Weapons 8

9. Energy versus forces Energy: predict some aspects of motion
Limits of possible
Less details 9

10. Potential energy of a springx
Potential energy of a spring:
Assume Us = 0 for relaxed spring 10

11. Real Life Physics: Potential energy of a spring11

12. Energy of an oscillating spring-mass system
Neglect friction:
Maximum speed:
-xmax
xmax x
Speed at any point:
Note – mass is not changing, so change in energy is associated with K and U only, this is why we dropped mc2.
Alternative way – using the momentum principle. More complex – since force is changing we will have to integrate to find speed…
Where is K max, U max? 12

13. Energy of an oscillating spring-mass systemx
xmax
-xmax
Same as
Note – mass is not changing, so change in energy is associated with K and U only, this is why we dropped mc2.
Alternative way – using the momentum principle. More complex – since force is changing we will have to integrate to find speed… 13