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

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

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

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

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

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

Binding energy: nuclear reactions Adding these, get 2.015 941 4 u Difference is 0.002 388 6 u Binding energy: mc2 (1 eV = 110-19 J) Proton: m = 1.007 276 5 u Neutron: m = 1.008 664 9 u Deuteron: m = 2.013 552 8 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 1961. 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 = 1.000 317 9 amu (physical scale) = 1.000 043 amu (chemical scale). 7

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

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

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

Real Life Physics: Potential energy of a spring 11

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

Energy of an oscillating spring-mass system x 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