- Exam1
- Exam2
- Exam3
-
Exam4
(Final)
Sat, 18 Dec 1999
Sat, 18 Dec 1999Wed, 8 Dec 99: I reviewed topics that were not been reviewed before Exam 3.
Mon, 6 Dec 99: Lecture 39. I lecture on
damped free and forced vibrations. My lecture notes can be found here:
1
, 2
, 3
, 4
, 5
, 6
, 7
, 8
.
Thu, 9 Dec 1999
Fri, 3 Dec 99: Exam 3. Recall some useful tips for exam takers.
Wed, 1 Dec 99: Review (lecture 38).
I reviewed for Exam 3, and in particular discussed #19.58 of HW#13.
You can find the scratch of my discussion on #19.58 with Andrew Tatsch
here: 1
, 2
, 3
, 4
, 5
.
Wed, 1 Dec 1999
Mon, 29 Nov 99: Lecture 37. I lectured
on undamped forced vibration, showed how to solve the equation of motion,
and finished with an application (#19.40). My
lecture notes can be found here: 1
, 2
, 3
, 4
.
Wed, 1 Dec 1999
Wed, 24 Nov 99: Lecture 36. I finished #19.96. See my lecture notes posted on Mon, 22 Nov 99, below. The solution for #19.96 from the manual can be found here. Happy Thanksgiving.
Mon, 22 Nov 99: Lecture 35. I lectured
on a quick method to obtain the equation for free vibration, the use of
the conservation of energy, and as application discussed the solution of
#19.96. My lecture notes can be found here: 1
, 2
, 3
, 4
, 5
, 6
, 7
, 8
, 9
, 10
, 11
, 12
, 13
, 14
.
Wed, 1 Dec 1999
Have you heard? If you have some notion of the German language, you may enjoy the following writing of Mark twain: "Whenever the literary German dives into a sentence, that is the last time you are going to see of him until he emerges on the other side of his Atlantic with his verb in his mouth." A Connecticut Yankee in King Arthur's Court, Mark Twain. (I got this hilarious quotation after I booted up my Linux laptop.)
Fri, 19 Nov 99: Lecture 34. I completed #19.40, and started #19.96 using conservation of energy. My lecture notes can be found here: 1 , 2 , 3 , 4 , 5 . The solution for #19.40 from the manual can be found here.
Wed, 17 Nov 99: Lecture 33. I solved #19.31 and discussed the related problem #19.40. My lecture notes can be found here: 1 , 2 , 3 , 4 , 5 , 6 , 7 .The solution for #19.31 from the manual can be found here.
Have you heard? This morning, while driving to school, I heard on the radio the following inspiring lines: "Work as if you don't need money, love as if you've never been hurt, and dance as if no one is watching." And ... I thought it would be good to extend this inspiring thought into: study as if you don't care about grades (just for the sake of knowledge, for the joy of learning).
Mon, 15 Nov 99: Lecture 32. I completed #18.6, and explained how to derive the expression for the mass moment of inertia of a disk. My lecture notes can be found here: 1 , 2 , 3 . I then lectured on vibration. We'll solve a challenging problem in my next lecture.
Did you know? Tibetan monks are performing sand paintings in the Florida Museum of Natural History (near the Harn museum).
Fri, 12 Nov 99: Lecture 31. I completed my detailed lectures on impulse, momentum, and impact analysis of rigid bodies. I lectured on 3-D kinetics, in particular the story of angular momentum vectors in 3-D. My lecture notes can be found here: 1 , 2 .
Did you know? The moving and inspiring story of Ray Charles, who will perform in Gainesville soon.
Wed, 10 Nov 99: Lecture 30. I continue to explain in detail the topic of solve impulse, momentum, and impact analysis of rigid bodies, using #17.117 and #17.118 as starting point. The lectures went beyond what is required in these problems. My lecture notes can be found here ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9). The solution for #17.117 and #17.118 from the manual can be found here.
Mon, 8 Nov 99: Lecture 29. I lectured on Section 17.11 and 17.12, impulse, momentum, and impact analysis of rigid bodies, and discussed #17.117 and #17.118. I'll post my lecture notes and the solution for the manual soon; those who were not present today are not covered by the covenant (except for those who already talked to me before class).
Did you know? Yesterday, 7 Nov, was the birthday of Marie Curie, the co-discoverer of radioactive materials, and the first woman whose body was laid to rest under the "mighty dome of the Pantheon" in Paris, an honor based solely on her own merit (and not of her husband). The Pantheon is where "Great Men" (and now also great women) of France--such as Victor Hugo, J.J. Rousseau, etc., and Marie Curie--lay rested. Another beautiful view of the Pantheon and the Latin Quarter. I once lived in the Latin Quarter in Paris, for two years, just a 5-minute walk to the Pantheon.
Fri, 5 Nov 99: HOMECOMING. NO CLASS.
Did you know? A typhoon in the Pacific ocean is what a hurricane is in the Atlantic ocean. Both are tropical storms with a cyclone.
Wed, 3 Nov 99: Lecture 28: I discussed conservation of angular momentum, and solved #17.89. A copy of my lecture notes on this problem can be found here: 1, 2, 3, 4, 5, 6. The solution from the manual can be found here. I returned Exam 2 papers, discussed the performance and the covenant between you, the students, and myself to improve your learning.
Did you know? The picture on the left below (larger size) shows Hurricane Floyd (lower left) at the coast of Florida on 15 Sep 99 and Hurricane Gert (lower right). (When you click at the picture of Hurricane Floyd, you'll see that Gainesville in North Florida was at the edge of the huge cyclone of Hurricane Floyd. The university was closed down at 3pm on 14 Sep 99, and reopened on 16 Sep 99, since fortunately Floyd spared North Florida, but not other parts of the country.) The picture on the right below (larger size) shows Hurricane Osea in South Pacific on 24 Nov 97; you can see the island of Tahiti in the lower right of the larger size photo. One picture was taken in the Northern Hemisphere, whereas one picture was taken in the Southern Hemisphere. I have explained the reason for the direction of rotation of hurricanes using concepts of dynamics taugth in the course. This topic is part of the course contents.
You can find additional explanation here or here or here or here , but nowhere can you find a clearer explanation than what I gave in class.
Mon, 1 Nov 99: Lecture 27: I explained in detail #17.32. A copy of my lecture notes on this problem can be found here: 1 , 2 , 3 , 4, 5 , 6 , 7. The solution from the manual can be found here.
Fri, 29 Oct 99: Lecture 26: I explained the direction of hurricanes (this topic is part of the course) using the concepts of dynamics taught in the course, lectured on the theory of Chap 17, and discussed #17.32; I'll continue with this problem next time.
Tired of studying after Exam 2? Take
a break on some math problems (relaxation guaranteed!).
(This e-mail was forwarded to me by a classmate,
now teaching at the University of Maryland.)
Wed, 27 Oct 99: Exam 2. Recall some useful tips for exam takers.
Mon, 25 Oct 99: I reviewed the lectures
after Exam 1 until last week. The teams filled out the Intermediate
Course Evaluation 2 to give me feedback on the course conduct since
Exam 1. If you did not agree with your team, you can always fill out the
evaluation form for dissenting opinion, and put it in my mailbox anonymously.
I will also discuss my
evaluation of the students in class (anonymously).
Fri, 22 Oct 99: I lectured on Section 16.8, contrained motion, and discussed #16.125.
Did you know? "Mathematics is a deep way of expressing nature... It is too bad that it has to be mathematics, and that mathematics is hard for some people. It is reputed---I do not know if it is true---when one of the kings was trying to learn Geometry from Euclid, he complained that it was difficult. And Euclid said, `There is no royal road to Geometry.'" (R. Feynman, The Character of Physical Law.) And...
Wed, 20 Oct 99: Lecture 23. I lectured on planar motion of rigid modies, and discussed completely #16.18, p.1007. I also continued to discuss #15.193, p.962. I completed the computation of the velocity of A using two methods, and showed that the two methods lead to the same result; the computation of the acceleration of A is similar, and is left as a home exercise.
Did you know? Once the universal law of gravitation was discovered, many other phenomena and laws in nature were discovered as a consequence of accepting the law of gravitation as correct. You already saw that the discovery that light has a finite speed, and the measurement of the speed of light, was a consequence of Newton's law of gravitation.
Newton was the first to explain correctly tidal movement. This sketch of the tidal movement and the influence by the moon's gravity was given by the late Richard Feynman (Nobel laureate in Physics, 1965) in his lecture on the Character of Physical Law at Cornell University in 1964. In this sketch, three explanations were given. The top two subfigures give a single tide, which is not correct, since the period of tidal movement is 12 hours. The bottom subfigure gives a double tide, and is the actual situation, where both water and Earth are pulled by the moon, but with different magnitude of gravitational force that depends in the inverse square law (force is proportional to the inverse of the square of the distance): "... the water at y is closer to the moon, and the water at x is farther from the moon than the rigid Earth. The water is pulled more towards the moon at y, and at x is less towards the moon than the Earth, so there is a combination of those two pictures that makes a double tide."
Another discovery is the planet Neptune, which was discovered on paper, before being observed! The story goes as follows: Actually, Kepler's laws are only valid for two celestial bodies alone, withtout the presence of other celestial bodies. In the solar system, the planets are not only attracted by the Sun, but each planet also exerts gravitational pull on other planets. So the planets should not move on exact ellipses, but with some deviations. Jupiter, Saturn, and Uranus were known to be big planets (before the discovery of Uranus). Calculations were made to see how their orbits deviate from perfect ellipses predicted by Kepler. Calculation and observation agreed with each other for Jupiter and Saturn, but not for Uranus. Two astronomers, J.C. Adams and U. Leverrier, made the calculations independently at about the same time, proposed that the motion of Uranus was due to an unseen planet. They requested their respective observatories to turn their telescopes to certain part of the sky to find a planet. "`How absurd', said one observatory, `some guy sitting with pieces of paper and pencils can tell us where to look to find some new planet'" (R. Feynman, The Character of Physical Law). The other obsevatory, more cooperative, actually found Neptune (close-up view).
But the law of gravitation is also valid beyond our solar system. Three pictures of a double star were taken a few years apart, showing that the two stars attracted each other, and that they went around in an ellipse. Note that the larger star was not at the focal point of the ellipse, but was off; the reason was because the orbit was tilted, and that we were looking at the projection of the orbit of the star on the plane perpendicular to the line of sight (R. Feynman, The Character of Physical Law). One can then see how deep the insights of Kepler were, when he deduced the elliptic character of the orbits of planets based on observations of the projected orbits.
Recently, a student just mentioned to me that there was a program on the Biography channel about the 100 most influential persons in the Millennium 1000-2000. The most influential person was Gutenberg, the inventor of the printing press. The sesond most influential person was Newton. (See also The Rise of Calculus.)
Mon, 18 Oct 99: Lecture 22. I lectured on 3-D motion of a particle, acceleration in 3-D, and discussed #15.233, p.976, and #15.193, p.962. The solution of #15.233 is completed.
Fri, 15 Oct 99: Lecture 21. I completed the solution of #15.154, p.948. I lectured on the computation of velocity and acceleration in general 3-D motion, and discussed #15.192, p.962.
Wed, 13 Oct 99: Lecture 20. I completed the solution of #15.129, presented the topic of rotating frames and the computation of velocity and acceleration wrt rotating frames. As application, I discussed the solution of #15.154. I will come back to #15.154 in the next lecture.
Mon, 11 Oct 99: Lecture 19. I explained the theory of absolute and relative acceleration, and discussed #15.129. I'll continue to discuss the solution of #15.129 on Wed.
Fri, 8 Oct 99: Lecture 18. I explained the theory of instantaneous center of rotation, and showed you how to apply this technique to solve #15.66.
Wed, 6 Oct 99: I discussed the theory
of general plane motion, and solve #15.66. My lecture notes for #15.66
can be found here ( 1
, 2
, 3
, 4
, 5
, 6
). Another way to solve the problem using geometry can be found
here.
Did
you know?
The the speed of light was discovered and measured very early
after the discovery of the universal law of gravitation by Newton.
In 1676, astronomer Olaus Roemer observed the moon Io of Jupiter (see the
black dot on Jupiter in the Figure on the left; that's the shadow of Io
on Jupiter; a larger picture can be seen here
; for a nice artistic, i.e., not real, family picture of Jupiter and its
four moons, click here
where moon Io is at the top; a close-up
view of Io), and noticed that the time between the eclipses of the
Jupiter moon Io was smaller when Jupiter is closer to the Earth, and that
time was larger when Jupiter was farther away from the Earth. Based
on a still inaccurate estimation, made at his time, of the distance between
planets, Roemer correctly suggested that light had a finite velocity, and
computed its velocity to be 214,000 km/s. The adopted value of the
velocity of light in 1983 is 299,792.458 km/s. For more information, see
Fri, 1 Oct 99: I lectured on Energy and Impulse for systems of particles, and discussed in detail #14.43. The solution can be found here. I returned Exam 1, and discussed the class performance. I also gave you some time to give me your feedback on the conduct of the course; see Intermediate Course Evaluation 1 .
Wed, 29 Sep 99: I lectured on momentum methods for systems of particles. We did #14.19 in detail. Check the solution here.
Mon, 27 Sep 99: I lectured on impact problem. We solved #13.171 in detail. Check the solution here.
Fri, 24 Sep 99: Exam 1
Wed, 22 Sep 99: I reviewed everything discussed in class from the beginning. I'll post interesting things about the universal law of gravitation soon.
Mon, 20 Sep 99: No more comments.
Fri, 17 Sep 99: I discussed #13.61 in detail, and the method to solve #13.69. Finish the detailed solution for #13.69; the solution can be found here: page1 , page2 .
Wed, 15 Sep 99: Hurricane Floyd, class cancelled. I will shift the contents of the lectures, leaving the exam dates the same.
Mon, 13 sep 99: I discussed various aspects related to work and energy, including the conservation of energy. Problem #13.61 was chosen to illustrate the lesson. I showed to you the method of solution; try to complete the solution. We'll continue to discuss #13.61 and other problems in the next lecture.
Fri, 10 Sep 99: I discussed the method of solving the differential equation in Question (b) of #12.73; the solution for #12.73 can be found here: page1 , page2 . I also discussed #13.6 using energy and work. You can also solve #13.6 using Newton's second law. Check the solution here. In the meantime, think about #12.101; the solution can be found here.
Wed 8 Sep 99: I discussed #12.96. You know the answer, and should fill in the algebraic manipulation. The solution can be found here.
Did you know? Today's lecture was about space mechanics. Before Newton, in early 17th century, Kepler deduced his three laws of planatery motion from the observations obtained fromTycho Brahe , after Brahe's death. Since the Earth does not lie in the plane of the orbit of any other planet in the Solar System, one only observes the projection of the orbit of a planet, i.e., the apparent orbit. It was a great intellectual triumph of Kepler to have arrived at his three laws based on the apparent orbits of the planets as viewed from the Earth. Einstein wrote in 1927 (Ideas and Opinions, 1954): "Today everybody knows what prodigious industry was needed to discover these laws from the empirically ascertained orbits. But few pauses to reflect on the brilliant method by which Kepler deduced the real orbits from the apparent ones---i.e., from the movement as they were observed from the Earth".
Did Tycho Brahe use a telescope for his observations? Surprisingly, the answer is no. He used a host of some 30 instruments that he designed himself. Brahe's instruments (pictures and explanations) were major achievements in astronomical science.
This figure is an example of Brahe's instrument. There were no optical
elements. It is called the Zodiacal
Armillary Instrument, and was used to observe the altitudes and
azimuths of stars.
Kepler's laws are, however, concerned with the movement as a whole, and not with the question "how the state of motion of a system gives rise to that which immediately follows it in time". Kepler's laws are integral laws, and not differential laws, which are needed to predict the motion of objects (causality). Newton's greatest achiement is the establishment of the clear concept of the differential law of motion (F=ma).
Fri, 3 Sep 99: We discussed #12.48 and #12.73. The solution for #12.48 can be found here: page1 , page2 . I also post here the solution to #12.72, which is related to #12.73, but easier. Regarding #12.73: Question (a) is easy, while Question (b) requires some ingenuity in solving the differential equation [try to express the second time derivative of the acceleration (r^{..}, i.e., "r double dot") in terms of the radial velocity v_r (i.e., "v sub r") and the radial position r]. I will discuss briefly the method in a future lecture, and will post the solution soon.
Wed, 1 Sep 99: Complete the numerical answer to #12.43 that we discussed in class on your own. The solution of #12.43 can be found here: page1 , page2 , page3 , page4 , page5 , page6 .
Mon, 30 Aug 99: Complete #11.140 by yourself before looking at the solution, which is posted here .
Did you know? In the course, we use both the overhead dot and the notation d/dt to denote time differentiation; the "overhead dot" notation was introduced by Newton, while the d/dt notation was introduced by Leibniz. "We need not consider the question here whether Leibniz hit upon the same mathematical methods [calculus] independently of Newton, or not. In any case, it was absolutely necessary for Newton to perfect them, since they alone could provide him with the means of expressing his ideas," A. Einstein [1927], Ideas and Opinions, p.278.
Today, you learned about tangent vectors and normal vectors to the trajectory of a particle. In 1666, at the age of 24, Newton established a rational method, which he called the "fluxion method", to compute these quantities. We now call this fluxion method calculus, or differential calculus; this terminology was introduced by Leibniz in 1675, before Newton published his work in 1704, according to a different account of the contribution of Newton and Leibniz, click here. Warning: The web page on Leibniz stated, on the other hand, that Leibniz "published his results slightly after Newton" !?? (thus contradicting the previous statement). Which account is correct? Could some of you try to find out the answer in this web site , or elsewhere (books, other web sites, etc.), and bring the result of your search up in class. It is also stated that Leibniz's "notation was by far superior" to Newton's, "and is still in use today" (which is true, and we did use it in class today).
Fri, 27 Aug 99:
Mon, 23 Aug 99: We discussed the method of solution for problem #11.29. Try to do this problem by yourself first. After that, you can check your method of solution and answers.
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