http://www.nearingzero.net (work059.jpg)

1. http://www.nearingzero.net (work059.jpg)

2. Where we stopped last time... Guess what? Non-science majors get to write your definition of momentum on a piece of paper and turn it in. Science majors, please define the term that the non-science majors have chosen and turn it in.

3. Implicature is... …an English term that no one actually uses. …I don't know. …Shirley Temple's favorite word after she had a few Shirley Temples. …a definition that implies something else. …a caricature of a monkey eating raspberry jello. …the association of a person or object with another person or object, or action based on theory. …the implication of an ature. Momentum is... …a force that is waiting to be used. …a term defining the transfer of potential to kinetic energy. …the amount of force in a direction an object has. …a built-up resistance of inertia. …the force of movement. …(blank paper).

4. Momentum is related to the "mass in motion part" of Newton's first law. Remember, that law says an object in motion remains in motion unless acted on by external forces. Momentum is a measure of a body's "desire" to remain in motion.

5. In physics, momentum is calculated using the equation p=mv. (I guess since m was already used for mass, "p" is as good a symbol as any for momentum.) The more mass you have, or the faster you are going, the greater your momentum. There is an important law of physics, known as the Law of Conservation of Momentum. This law says that momentum is conserved in any process, as long as no external forces are acting.

6. If your car and an 18-wheeler collide, what does the Law of Conservation of Momentum have to say? graphics from http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/

7. It has been observed experimentally and verified over and over that in the absence of external forces, the total momentum of a system remains constant. The above is a verbal expression of the Law of Conservation of Momentum. It sounds like an experimental observation, which it is… …which implies maybe we just haven’t done careful enough experiments, and that maybe some day we will find the “law” is not true after all. But the Law of Conservation of Momentum is much more fundamental than just an experimental observation.

8. In 1905, mathematician Emmy Noether proved the following theorem: For every continuous symmetry of the laws of physics, there must exist a conservation law.For every conservation law, there must exist a continuous symmetry. The laws of physics are invariant under coordinate transformations. That’s a fancy way of saying you and I can pick different coordinate systems for measurements and experiments, but we will still arrive at equivalent results.

9. If the laws of physics were not invariant under coordinate transformations, then everybody would have his own version of the laws of physics. Not much point in doing physics, if that’s true! The Law of Conservation of Momentum is a mathematical consequence of the invariance of the laws of physics under coordinate transformations. Any violation of the Law of Conservation of Momentum would be as revolutionary (if not more so) as Einstein’s relativity. But even if a violation were found, any “new” laws of physics would contain all our “old” ones, which would still work under “normal” circumstances.

10. Conservation laws are fundamental, powerful, and beautiful. We will see another one soon. In contrast, Newton’s laws work only in the macroscopic world, and are only an approximate description of nature.

11. Before I leave momentum, have you ever thought of racing a train across the crossing? What is your car’s mass? How far does it take you to stop your car when you are going 60 miles per hour? Some of the distance traveled is due to your reaction time, and the rest is due to the time needed from the frictional forces applied by your brakes to "use up" your car's momentum. What is a train’s mass? How far does a train traveling at 60 miles per hour go before the friction forces from its brakes cause it to stop?

12. Density First, a little experiment… Please make for me a cubic meter of air. No peeking at the next slide! “The body is most fully developed from thirty to thirty-five years of age, the mind at about forty-nine.”—Aristotle

13. Air has a density of about 1.3 kilograms per cubic meter. That much air weighs about 0.6 pounds (near the surface of the earth). Air is not very dense, but I still am surprised at how much a cubic meter of air weighs. The density of water is about 1 gram per cubic centimeter. It takes 100 centimeters to make a meter, so it takes 100x100x100=106 cubic centimeters to make a cubic meter.

14. A cubic meter of water has a mass of 106 grams, or 103 kilograms, or 1000 kilograms. An object having a mass of 2.2 kilograms weighs a pound at the surface of the earth, so the cubic meter of water weighs about 454 pounds. I’ve been using the word “density” without defining the term, so… What happens when you put something in water. Well, yes, it either sinks or floats; but the water also pushes on it. Jump in the pool and feel the water trying to push you out.

15. Who remembers the story of Archimedes* and the king’s crown? http://www.engineering.usu.edu/jrestate/workshop/buoyancy.htm: “As the story goes, the king of Syracuse had given a craftsman a certain amount of gold to be made into an exquisite crown.” *If Archimedes was born and raised in Syracuse, Sicily, why do we consider him a Greek?

16. “When the project was completed, a rumor surfaced that the craftsman had substituted a quantity of silver for an equivalent amount of gold, thereby devaluing the crown and defrauding the king.” “Archimedes was tasked with determining if the crown was pure gold or not. The Roman architect Vitruvious relates the story…”

17. ‘While Archimedes was considering the matter, he happened to go to the baths. When he went down into the bathing pool he observed that the amount of water which flowed outside the pool was equal to the amount of his body that was immersed.’

18. ‘Since this fact indicated the method of explaining the case, he did not linger, but moved with delight, he leapt out of the pool, and going home naked, cried aloud that he had found exactly what he was seeking.’ ‘For as he ran he shouted in Greek: Eureka! Eureka! (Eureka translated is "I have found it").’ “Although there is speculation as to the authenticity of this story, it remains famous.” “Probably no other tale in all of science combines the elements of brilliance and bareness quite so effectively.” “Whether the story is true or not, there is no doubt to the truth of Archimedes understanding of buoyancy.”

19. Archimedes’ death also makes an interesting story: “In 212 BC Syracuse surrendered to Rome. Before sending his men to sack the city Marcellus told them ‘Spare that mathematician.’ Plutarch records what happened next:” ‘As fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he [Archimedes] never noticed the incursion of the Romans, nor that the city was taken.’

20. “‘In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.’ ”

21. Archimedes is remembered by most of us as a mathematician, but he also invented fabulous war machines.

22. News Flash! Archimedes invents Death Ray that sets enemy ships on fire! My Physics 24 text and Mythbusters say it’s impossible!

23. News Flash! MIT students set wooden ships on fire with death rays! Details here!

26. Archimedes also overcame our 34-foot straw problem (remember the last lecture) by inventing the Screw of Archimedes. www.nearingzero.net What is not so well known about Archimedes is that he had a career record of 40 wins and 28 losses while pitching for the Cosmic Ionians. For proof, see here. (on the web, so it must be true)

27. Finally, you can go here to read why the traditional story of Archimedes and the king’s crown is probably not true: http://www.mcs.drexel.edu/~crorres/Archimedes/Crown/CrownIntro.html What is important for us, and what Archimedes understood, is that an object immersed in a fluid experiences a buoyant force equal in magnitude to the weight of the fluid displaced. Archimedes could have done this experiment: • The craftsman lives. • The craftsman dies.

28. Now let's think about sinking and floating. If I put a very dense object in water, it pushes water aside and the water pushes up. But if the object is more dense than water, the upward push of the water is less than the weight of the object. Which way does the object go?

29. If I put a low-density object in water, it floats because the upward push of the water is greater than the downward pull of gravity. That doesn't mean the water pushes up so hard it launches the object into the air. What happens is that the object sinks partially, until the upward push of the water displaced equals the downward pull on the object by gravity.

30. So, if an object sinks in water, it is more dense than water. If it floats, it is less dense. Humans are made up of more water than anything else, so it is not surprising that humans have about the same density as water. Some of us barely float, some of us barely sink. Also, some of us are more full of hot air than others, which also helps flotation. Steel is about 9 times more dense than water? How come steel boats can float. Let's experiment to find out.

31. Is glass more or less dense than water? I'm an experimentalist, so let's do the experiment. Is rubber more or less dense than water? Again, experiment. Is rubber stuck on to glass more or less dense than water? Experiment...

32. http://www.exploratorium.edu/snacks/descartes_diver.html I won’t repeat their misquote of Descartes here. The toy we just made is called a Cartesian Diver (after Descartes). The density of glass and rubber are greater than water, but when the glass/bulb have some air in them, the density of the combined system can be greater or less than water, depending on how much air is in the system.

33. We put the "diver" in a sealed flexible bottle with some air also trapped in the bottle. When we squeeze the plastic bottle, the water doesn't compress very much (water is virtually incompressible; its molecules "want" to be just so far apart, and no closer). The air does compress, a lot. Pressure inside the bottle is transmitted uniformly through all fluids inside (both air and water) so the air inside the diver feels the pressure and compresses. When the air compresses, water comes in, the diver gets more dense, because water is far more dense than the compressed air, and it sinks. When the pressure is released, the air inside the diver goes back to normal, and it floats. You can make yourself into a diver like this and float all day, expending virtually no energy.

34. Steel boats float, then, because they are designed to "hold" lots of air. Here’s a neat applet. Click the picture to go to the web page. Push on the red plunger to operate the diving chamber. Don’t ask me why!

35. That's the end of forces and motion. I spent a lot of time on density. Most adults don’t understand density. It also turns out that you need to understand density to understand lots of important phenomena, like the weather, some alternative energy sources work, and El Niño.