Projectile HW

Chapter 4:

38
26
41
13

Don't forget the earlier vector problems.

Let's have a take-home test soon, covering one-dimensional motion, relativity and the intro to vectors (but not projectile motion).

hw for Friday

Chapter 3

27

31

37

42

71

On Friday, we will discuss motion in 2 dimensions - projectiles! Feel free to start reading chapter 4.

Thanks!

Vector HW

1. Consider a set of vectors:

A = 2i - 4j + 12k

B = 6i + 2j - 8k

Find the following:

A + B
B - A
the magnitude of each vector
the angle between the vectors

2. We didn't get into cross products seriously. Here's the brief definition: the cross product of 2 vectors is a third vector (perpendicular to the plane of the original 2 vectors; described by the right hand rule). The vector has magnitude given by:

A X B = A B sin(phi)

When you take the cross product of 2 vectors, you foil them (as we did for the dot product). However, doing the cross product of unit vectors (i, j, k) is trickier to determine. There is a simple way to remember this, but it's hard to describe in print - I'll show you Wednesday. Until then, here's the general answers:

i x j = k
j x k = i
k x i = j
i x k = -j
k x j = -i
j x i = -k

With this in mind, take the cross product of the two vectors above - the answer will be a vector (with i, j, and k components).

3. For fun:

If you understand the right hand rule, try these:

UP X WEST

BACKWARDS X DOWN

4. As always, be caught up on reading, etc.

vector notes 2

vector notes

Oh, and another thing....

To let you know where we're headed next.....

I'm thinking that another take-home quiz is in order - maybe next week?

I want to cover projectile motion - we didn't do it in great detail in 9th grade.

There will be a formal lab on projectiles.

After this, it's forces - this will have some of my "rocket science" ideas. I have parts for you to build your own model rockets (either alone or in pairs, as you wish). I'd like us to investigate the air resistance associated with high-speed flight.

If you have time, investigate ways to build a small wind tunnel. We may want to use this as a class.

Thanks!

Vector HW

Hi there!

I'd like you to prep for Monday's class by reading the chapter on vectors. Focus on the following:

vector magnitude
i, j, k components
vector multiplication
dot and cross products

I'd assign a couple of vector addition problems at this point, but I suspect that yesterday's stuff isn't too challenging yet. If you like, create a triangle (knowing 3 side, or 2 sides and the angle between) and use the laws of signs and cosines to determine the other information. Alternately, create your own river or plane problem to solve. Again - do this if it is helpful. It is more important to prep for Monday's class by learning about i, j, and k components

Grazie!

Homework

Nothing new

Just get caught up with the applets, etc.

Relativity fun!

Applets worth your time:

http://chair.pa.msu.edu/applets/travel/a.htm

http://www.its.caltech.edu/~phys1/java/phys1/Einstein/Einstein.html

https://parkscience.pbworks.com/w/page/43576403/Modern%20Physics%202011
See the video.

http://physics.ucsc.edu/~snof/Tutorial/index.html
Work through the tutorial, if you like.

http://www.kcvs.ca/site/projects/specialRelativity.html

http://physics.ucsc.edu/~snof/Rally/index.html
http://physics.ucsc.edu/~snof/Game/index.html
Games?

Relavitity problems

1. What are the Lorenz factors for velocity values of:
a. 0.1 c
b. 0.9 c
c. 0.99 c

2. Consider two twins, Spacey and Stacey. Spacey travels at 0.9 c to a planet (Planet Lally!) that is (measured before travel) 6 LY (light-years, or c-years) away. Stacey stays home. Spacey travels to the planet and quickly turns around once arriving. Find the following:
a. how much Stacey ages during the trip
b. how much Spacey ages during the trip
c. the odometer reading on Spacey's spaceship upon arriving at Planet Lally

3. In the problem above, the spaceship is measured to be 100-m long before flight. How long does it appear to:
a. those aboard it
b. those viewing it from Earth

4. Consider a group of pi-mesons (pions) traveling at 0.8 c. In the lab, the lifetime is measured to be 3 x 10^-8 seconds with no travel. How "old" can the pions become traveling at 0.8 c? How long a distance could they traverse?

Special relativity notes

more Einstein reading

http://en.wikipedia.org/wiki/Special_relativity

Read up until this part:

Reference frames, coordinates and the Lorentz transformation

Einstein redux

Apparently the recent link did not work - thanks, Jeremy. Try reading this:

http://www.pitt.edu/~jdnorton/Goodies/Einstein_and_S/index.html

How Einstein Changed the Way We Think About Science

John D. Norton
Department of History and Philosophy of Science, University of Pittsburgh
Pittsburgh PA 15260. Homepage: www.pitt.edu/~jdnorton
This page is available at www.pitt.edu/~jdnorton/goodies

This is an expanded version of my "What Is Einstein's Legacy to Me? Philosophy of Science" appearing in Imagine (Johns Hopkins Center for Talented Youth) Vol. 13, Issue 1, September/October, 2005.



We all know of Einstein's contribution to modern physics. Through his theories of relativity he showed us that there is a fastest possible speed and that light moves at it. He showed us that gravity is a curvature of spacetime. And he laid the foundations of modern quantum mechanics when he proposed that light really comes in little bundles of energy he called quanta. Any philosopher of science interested in the nature of space, time and matter has to take notice, for our accounts of all three were changed fundamentally by Einstein's hand.

What is less well recognized is how Einstein's work altered our understanding of the nature of science itself. To begin, he changed our ideas of how to do theoretical science. In 1905, he showed us how to make sense of the odd fact that light always propagates at exactly the same speed c, no matter how fast we go. The trick was to see that when we change our state of motion, we change our judgments of which events are simultaneous. His lucid analysis, laid out in the opening pages of his famous 1905 special relativity paper, was especially vivid. He used a thought experiment, in which asked us to imagine two clocks exchanging light signals and observers in different states of motion. The signals bounce, the clocks tick and altogether too quickly the final, astonishing result emerges.

The apparent ease with which ordinary thought experiments could yield extraordinary results was inspiring. It led many to try to copy his method They sought new theories, not in novel experiments, but in astonishing revisions of familiar notions like space and time, through carefully crafted thought experiments. These efforts rarely succeeded for those who are not Einsteins.

Another notion was read from Einstein's analysis of 1905. It was a view of which concepts may be used in science. They must be defined by the operations needed to measure the concept. So we can only use the concept of the simultaneity of two events if we can specify just how we may determine their simultaneity, by, for example, operations with light signals. This stringent demand is very effective if our goal is to force critical re-evaluation of some dubious concept. However it is as likely to cause problems where there should be none, for few of our concepts really conform to its standards.

Einstein invented neither the notion of the thought experiment nor the operational definition of a concept. He merely used them more perfectly than any who had gone before him.

The most enduring change brought by Einstein's work was to shake our sense of certainty. When Einstein entered science at the start of the 20th century, there was a strong sense of its stability. In antiquity, Euclid had described perfectly how space really is. In the 17th century, Newton had discovered the dynamics that govern time and matter. It is only from the perspective of this certainty that we can now understand the project of the influential eighteenth century philosopher Immanuel Kant. He felt compelled to devise an explanation of why all our experience must conform to the geometry of Euclid and the mechanics of Newton. The project now seems misplaced. Einstein showed us that both theories can fail when we enter the realms of the cosmically large, the very heavy, the atomically small and the very fast.

Einstein was not the only one to show us the fragility of our knowledge. But he was the first, the most effective, and the best remembered. He showed us that the old certainties had failed. So, we concluded, surely anything that replaces them could fail again.

The old confidence in our knowledge was based on the notion that experiment and experience could bear quite directly on our science. Some, like Ernst Mach, thought that all of science was or should be nothing more than compact summaries of experience. While this idea was always somewhat dubious, it could survive because even the most complicated theory of the era, Maxwell's electrodynamics, appeared to remain close to experience. For each of Maxwell's equations, one could point to experiments that seemed to be expressed just by that equation.

This sense of the closeness of theory to experience was shattered by Einstein's general theory of relativity. It required a new and complicated mathematics then unfamiliar to most physicists. Yet most of its predictions were no different than those of Newton's much simpler theory. If theories were merely summaries of experience and did not add to them, how could two theories, so much in agreement on experience, differ so much in structure?

Einstein's physics and the new physics developed by others in the twentieth century led to a sense of the fragility of theories and the powerlessness of evidence to pick out the unique truths of nature. Philosophers of science struggled to accommodate this new sense within their systems, all the while seeking to fit their ideas with Einstein's theories.

Einstein's own diagnosis of this gap between experience and theory was extreme. He proclaimed that concepts and theories are "free inventions of the human spirit" and that no method could assuredly take us from experience to the true theory. Here he contradicted the optimism of the nineteenth century during which many felt that scientific discovery could be reduced to simple recipes. John Stuart Mill continued a tradition extending back to the seventeenth century Francis Bacon. To identify the cause of some effect, he believed, one merely needed to collect cases in which the effect was present and those in which it was not. The cause could then be read directly from the systematic differences between the cases.

Later in life, Einstein came to a radical solution of the problem of responsibly practicing science while still believing that its core concepts are free inventions. Drawing on his discovery of general relativity, he concluded that the right concepts and theories could be found merely by seeking the mathematically simplest theories.

In my view, Einstein's response was too optimistic in his confidence that mathematical simplicity could be the guide to the truths of nature. Einstein was able to make no major discovery using this principle during the decades of his legendary and ultimately barren search for a unified field theory.

And Einstein's notion that concepts and theories are free inventions not fixed by experience seems too pessimistic, for science seems time and again to be able to determine the right theory on the basis of evidence. In the face of this commonplace of science, Einstein too seems to have had some difficulty maintaining his notion of free invention. He likened nature to a "well-designed word puzzle." While we may try to solve it with many words, only one "really solves the puzzle in all its parts."

This last view seems to me to capture much better the real power of evidence to point to a definite theory. If the equivalence of energy and matter expressed by E=mc2 is based on free inventions, why is there no alternative that enjoys equally powerful support from our experiences and experiments?



Some Reading

Einstein, Albert. Relativity: The Special and the General Theory. Methuen & Co., 1920.

Einstein, Albert. Ideas and Opinions. New York: Bonanza Books, 1954.

Howard, Don A., "Einstein's Philosophy of Science", The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.) http://plato.stanford.edu/archives/spr2004/entries/einstein-philscience/.


John D. Norton is a licensed and certified dilettante whose hobby is the study of Einstein's work and thought. This hobby has so overtaken his life that he has little time for anything else and is Professor of History and Philosophy of Science and (Sept. 2005-) Director of the Center for Philosophy of Science at the University of Pittsburgh.

Copyright John D. Norton, May 14, 2005.

>

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/index.html

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http://plato.stanford.edu/archives/spr2004/entries/einstein-philscience

Einstein!

Einstein notes

Worth a read (from an Einstein scholar and former prof of mine):

http://www.pitt.edu/~jdnorton/Goodies/Einstein_and_S/index.html

Notes from class:

And then there was Einstein…

Albert Einstein 1879-1955

http://www.aip.org/history/einstein/index.html

What’s happening around the turn of the 20th century? Physics was set to explode with 30 brilliant years of excitement and unprecedented activity

X-rays – Roentgen

Radioactivity – Becquerel, Marie & Pierre Curie

Blackbody radiation (and the quantum discontinuity) – Planck

1905/6 – Einstein publishes 6 major papers:

a) “On the electrodynamics of moving bodies”

b) “Does the inertia of a body depend upon its energy content?”

c) “On a heuristic point of view about the creation and conversion of light”

d) “On the theory of the Brownian movement”

e) “On the movement of small particles suspended in stationary liquid demanded by the molecular-kinetic theory of heat”

f) “A new determination of molecular dimensions”

What are these about anyway?

a. Special relativity (SR)

b. E = m c2 (actually, L = m c2)

c. Photoelectric effect, light quanta, fluorescence

d. Same as title

e. Brownian motion agan

f. Avagadro’s number, etc.

Now, these are interesting (and very different fields of study), but is this why we revere Uncle Al? Not necessarily. Others (Poincare, Lorentz) were working on what would become SR. Planck had introduced the quantum discontinuity (E = h f) and quantum mechanics (QM) would have many contributors. The photoelectric effect had also several investigators (Lenard, et.al.).

Mostly, Einstein’s legend grows because of General Relativity (GR), which appears 1912-1915 and later and on which he worked largely alone with pad and pen. He forced us to re-examine how we see ourselves in the universe; indeed, how we think of gravitation. All of this around the time his marriage was falling apart (he married young after fathering 1 illegitimate child) and he began an affair with his cousin (whom he would later marry). Also, between 1902 and 1909, Einstein held a modest post in Bern, Switzerland as a Patent Clerk. By 1914, he would be director of the Kaiser Wilhelm Institute (later Max Planck Institute).

Special Relativity

Spaceship – Inside an inertial reference frame (constant velocity), you can’t tell whether or not you’re moving (“Principle of Relativity”)

Biographical notes

1879 – born in Ulm, Germany

1884 – receives first compass

1895 – attempts to gain entrance to Swiss Polytechnic (and finish high school early), but is rejected

1896 – begins Federal Polytechnic (ETH) in Zurich, Switzerland

1898 – meets Mileva Maric

1900 – graduates from ETH

1901 – Einstein becomes Swiss citizen and moves to Bern; Mileva becomes pregnant

1902 – Lieserl born (put up for adoption); Hermann dies

1903 – Albert and Mileva marry

1904 – Hans Albert born

1905 – Einstein’s “Annus Mirabilus”, his miracle year; Ph.D. (Zurich)

1919 – divorces Mileva (having lived apart for 5 years); marries Elsa; GR verified

1921 – awarded the Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect".

1933 – settles in Princeton, NJ

1936 – Elsa dies

1939 – E. writes FDR

1940 – E. becomes American citizen

1949 – Mileva dies

1955 – E. dies

http://www.aip.org/history/einstein/index.html

http://www.albert-einstein.org/

http://einstein.stanford.edu/

http://en.wikipedia.org/wiki/Einstein

General Relativity

By 1907, E. wanted to advance the SR theory to include non-inertial (accelerated) frames of reference. Around this time, E. has the “happiest thought of my life”. In a uniformly accelerated spaceship, a stationary thing (ball, etc.) would appear to be falling (accelerating) down – it would be indistinguishable from normally accelerated motion. Light, too, follows this idea. Two clocks at different ends of an accelerated spaceship would be out of sync. Gravity is the result of the curvature of space and time?

Why does the Earth follow the Sun? Gravity – the very presence of the Sun causes Earth to veer from its otherwise straight (Newtonian inertial) path. With the Sun, it takes an elliptical path as its natural motion. Getting to this point, and showing that mass alters space and time is the real genius.

There is a breakdown of the observed geometry (Euclidean). E. must use non-Euclidean geometry (Gauss, surfaces, infinitesimal geometry) to consider the behavior of things (rods, etc.) on surfaces. He also considers the shortest distance between 2 points on a sphere (geodesic, great circle). He obtains the mathematical advice of his friend Marcel Grossman and studies (at great length) tensor calculus, differential geometry, Riemann and Minkowski math … it’s all puzzle solving. Soon, the principle of equivalence emerges. Eventually, the gravitational field equations appear (to show how matter “produces” gravity. GR was mostly worked out by 1913