Fall 2010 PHYS 172: Modern Mechanics Lecture 11 – The Energy Principle Chapter 6.15 – 6.17, 7.1 – 7.2 ANNOUNCEMENT The Final Exam in Phys 172H is scheduled for Wednesday, December 15th, from 8:00 am to 10:00 am in Room 114. Note that this is 1.5 hours earlier than our lectures

Mass and energy Example: electron and positron (antielectron) annihilation: neutron electron proton antineutrino Example: neutron decay Mass of products < mass of neutron! Missing mass is converted into kinetic energy Positron tomography All mass is converted into electromagnetic radiation – gamma rays

Mass and energy Mass and energy are the same thing! There is no “mass conservation law”! Energy, not mass is conserved! But only if you do COMPLETE bookkeeping

Mass of a multiparticle system system Mass of a multiparticle system: For example, the mass of a proton is MOSTLY kinetic energy – the three “valence” quarks in the proton have rest masses totaling less than 100 MeV/c 2, while the proton’s mass is 938 MeV/c 2.

i>Clicker question 1 O 2 molecule is oscillating: Mass of the molecule is: A)Larger than total mass of two noninteracting oxygen atoms B)Smaller than total mass of two noninteracting oxygen atoms C)The same as mass of two noninteracting oxygen atoms

i>Clicker question 2 O 2 molecule is oscillating: Mass of the molecule is: A)Larger when atoms are further apart B)Larger when atoms are closer to each other C)Larger when atoms are moving faster D)It does not change during oscillation

Two ways of thinking 1.The energy of a multiparticle system consists of the individual particle energies plus their pair-wise interactions 2.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 = Mc 2, where M is mass of the system.

Clicker 1. What happens with the mass of an object when you heat it up? A)It increases B)It decreases C)It does not change Why don’t we notice it? What is the change in mass when we heat up 1 kg of water from 0C to 100C? Later we will learn that energy Q = 4.2  10 5 J is needed to heat it 0 C: kg 100 C: kg

Binding energy: nuclear reactions Proton: m = u Neutron: m = u Deuteron: m = u Adding these, get u Difference is u Binding energy,  mc 2 =2.2MeV (1 eV = 1  J)

The Curve of Nuclear Binding Energy The strong nuclear force attracts nucleons to each other. Nucleons are the generic name for neutrons and protons. The Pauli Exclusion Principle says building block particles (like nucleons or electrons) can’t occupy exactly the same states, so the way nucleons like to combine is with two neutrons and two protons in the ground state (where spin up and spin down distinguish one neutron from the other, ditto for the protons.) So Helium (ppnn) is especially tightly bound, by about 7 MeV per nucleon. This also works well for C-12 and O-16, for example, which are very stable isotopes made from three or four helium nuclei stuck together, speaking loosely. And so on up to Iron, with 56 nucleons. Iron has the greatest binding energy, ~8.7 MeV

The Curve of Nuclear Binding Energy BUT the electrical Coulomb repulsion of the protons from each other tends to work AGAINST binding. Also, this electrical UN- binding energy grows as the SQUARE of the number of protons [each PAIR of protons has its own POSITIVE electrical potential energy.] This requires an excess of neutrons whose strong attraction counteracts the Coulomb repulsion and helps bind the nucleus. Even so, the binding energy per nucleon for elements above Iron in the periodic table is LESS than it is for Iron, which is the most tightly bound nucleus known. The Uranium-235 nucleus has 92 protons and the rest are neutrons (143 in number). It is very weakly bound, and acts like a large liquid drop. Adding one more neutron makes it unstable against “fission” – the drop breaks apart into two droplets, or “fission fragments”, Since the binding energy per nucleon is LARGER for the fragments than for the parent Uranium, the excess is released as Kinetic Energy. This is the source of nuclear power.

Fission and fusion The strong force causes nucleons to attract each other. When nucleons combine into nuclei, energy is released. This is called the binding energy. Iron is the most tightly bound so when elements up to Iron are made by the fusion of light elements energy is released. Similarly if we can break a very heavy atom into lighter elements there is kinetic energy released. E = mc 2 Mass is another form of energy a nucleus is lighter than the sum of the constituent nucleon masses Fusion processes are how the sun produces energy and fission processes are how nuclear reactors produce energy

Nuclear fission slow neutron This process generates more than 1 neutron. These neutrons can initiate further fissions in the Uranium: chain reaction Convert 1 pound into energy – enough to power one household for 100,000 years! 0.454kg x(3E8m/s) 2 = 4.1E16J Power = 4.1E16J/(piE7xE5s) = 1.3E4 Watts = 13kW

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

Potential energy of a spring x Potential energy of a spring: Take U s = 0 for relaxed spring y

Interatomic potential energy between two atoms Spring-mass model: Problem: energy becomes infinite at large distances and there are no bound states! The Morse potential energy function is a realistic approximation Spring approximation:

Energy of an oscillating spring-mass system x Neglect friction: y x max -x max Maximum speed: Speed at any point:

HOME STUDY: Example, “a rebounding block” k s = 2000 N/m System: block, spring, Earth Assume: no interaction with surroundings  = 1 1. What is the lowest point reached by the block? L0=L0= One equation, one unknown 2. What is the highest point reached by the block? One equation, one unknown See details in the book (7.3), p What facts do we apply to get Eq. 2 ? --- since v i is downward in (1) ??