Selamat datang di http://prigell-zone.blogspot.com . Blog ini bersifat umum; isi artikel multi-katagori sesuai mood dan apa yang sedang dipikirkan penulis. Anda dipersilakan untuk meng-copy, mem- paraphrase dan quote sesuai kaidah HAKI yang berlaku. Segala saran positif akan sangat penulis hargai. [EN: Welcome to http://prigell-zone.blogspot.com. This blog is general; multi-category content of the article according to mood and what the author was thinking. You are welcome to copy, to paraphrase and quote according to the rules of intellectual property laws. Any positive advice would be greatly appreciated writer.] -Prigell Priya Ragil-
Tampilkan postingan dengan label Fisika. Tampilkan semua postingan
Tampilkan postingan dengan label Fisika. Tampilkan semua postingan

Kamis, 14 Juli 2011

Testimony About MAssachusetts Institute of Technology

harjoko.jpg
Triatno Yudo Harjoko uses MIT OpenCourseWare content as a guide in redesigning the teaching model at University of Indonesia.

"I was amazed that a university such as MIT would freely give access to its educational information."

Triatno Yudo Harjoko has a long and close association with the University of Indonesia in Depok, Indonesia. A graduate of the institution, Harjoko has also been a professor of architecture at the school since 1979, and is currently head of the architecture department. In addition, Harjoko has designed several of the university's buildings, including the expansive Faculty of Engineering complex, and the stunning University Mosque.

Redesigning the teaching model

Harjoko characterizes the learning atmosphere at the University of Indonesia as primarily a one-way street, in which professors are assumed to be knowledge-bearers, and students are expected to master a predetermined knowledge base.

This approach, while typical of many Indonesian institutions, is something that Harjoko has been attempting to change in his department. Together with his colleagues, Harjoko is redesigning the teaching model, moving toward an active, student-centered learning process.

Harjoko describes the main goal in this transition as "encouraging students to learn by themselves, and to be both critical and creative."

Adapting MIT content

In the redesign process, MIT's OpenCourseWare – to which Harjoko was introduced by a colleague several years ago – has served as an immense comparative database for Harjoko and his fellow professors. Rather than directly transposing OCW syllabi to University of Indonesia courses, Harjoko and his colleagues have been scrutinizing MIT's courses to better understand how they were designed and developed.

"We try to understand how the MIT courses are formulated," Harjoko explains, "and what the expected outcomes are. This gives us an important perspective on the learning process."

Two courses for which this approach has been particularly helpful, he adds, are architectural theory and structure.

"I was amazed that a university such as MIT would freely give access to almost all of its educational information to the world," says Harjoko. "This is important, because critical thinking and creativity demand the liberalization of learning and information. But I also believe that it's not simply the information that's valuable, but also the glimpse it offers into how MIT has structured its teaching and research to become such a prestigious institution."

Source: http://ocw.mit.edu/about/ocw-stories/triatno-yudo-harjoko/

Selasa, 12 Juli 2011

The Grand Design Synopsis of Atheis Stephen Hawking

[lıƃäɹ àʎıɹd lléƃıɹd]


The book examines the history of scientific knowledge about the universe. It starts with the Ionian Greeks, who claimed that nature works by laws, and not by the will of the gods. It later presents the work of Nicolaus Copernicus, who advocated the concept that the Earth is not located in the center of the universe.

The authors then describe the theory of quantum mechanics using, as an example, the probable movement of an electron around a room. The presentation has been described as easy to understand by some reviewers, but also as sometimes "impenetrable," by others.

The central claim of the book is that the theory of quantum mechanics and the theory of relativity together help us understand how universes could have formed out of nothing.

The authors write:
"Because there is a law such as gravity, the universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist. It is not necessary to invoke God to light the blue touch paper and set the universe going."
The authors explain, in a manner consistent with M-theory, that as the Earth is only one of several planets in our solar system, and as our Milky Way galaxy is only one of many galaxies, the same may apply to our universe itself: that is, our universe may be one of a huge number of universes.

The book concludes with the statement that only some universes of the multiple universes (or multiverse) support life forms. We, of course, are located in one of those universes. The laws of nature that are required for life forms to exist appear in some universes by pure chance, Hawking and Mlodinow explain (see Anthropic Principle).


Positive reactions

Evolutionary biologist and advocate for atheism, Richard Dawkins, welcomed Hawking's position and said that "Darwinism kicked God out of biology but physics remained more uncertain. Hawking is now administering the coup de grace."


Canada Press journalist, Carl Hartman, said: "Cosmologists, the people who study the entire cosmos, will want to read British physicist and mathematician Stephen Hawking's new book. The Grand Design may sharpen appetites for answers to questions like 'Why is there something rather than nothing?' and 'Why do we exist?' — questions that have troubled thinking people at least as far back as the ancient Greeks."

James Trefil, a professor of physics at George Mason University, reviews this book for The Washington Post: "I've waited a long time for this book. It gets into the deepest questions of modern cosmology without a single equation. The reader will be able to get through it without bogging down in a lot of technical detail and will, I hope, have his or her appetite whetted for books with a deeper technical content. And who knows? Maybe in the end the whole multiverse idea will actually turn out to be right!"


Writing in the Los Angeles Times, Michael Moorcock praised the authors: "their arguments do indeed bring us closer to seeing our world, universe and multiverse in terms that a previous generation might easily have dismissed as supernatural. This succinct, easily digested book could perhaps do with fewer dry, academic groaners, but Hawking and Mlodinow pack in a wealth of ideas and leave us with a clearer understanding of modern physics in all its invigorating complexity."

Cosmologist Dr. Lawrence Krauss, in his article "Our Spontaneous Universe", wrote that "there are remarkable, testable arguments that provide firmer empirical evidence of the possibility that our universe arose from nothing. ... If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero. And when we measure the total energy of the universe, which could have been anything, the answer turns out to be the only one consistent with this possibility. Coincidence? Maybe. But data like this coming in from our revolutionary new tools promise to turn much of what is now metaphysics into physics. Whether God survives is anyone's guess."


German daily Süddeutsche Zeitung spent the whole opening page of its culture section on The Grand Design. CERN physicist and novelist Ralf Bönt reviews the history of the Theory of everything from the 18th century to M-theory, and takes Hawking's conclusion on God's existence as a very good joke which he obviously welcomes very much.

Critical reactions
Roger Penrose in the FT doubts that adequate understandings can come from this approach, and points out that "unlike quantum mechanics, M-Theory enjoys no observational support whatsoever". Joe Silk in Science suggests that "Some humbleness would be welcome here...A century or two hence...I expect that M theory will seem as naïve to cosmologists of the future as we now find Pythagoras's cosmology of the harmony of the spheres"

The Bishop of Swindon, Dr. Lee Rayfield said, "Science can never prove the non-existence of God, just as it can never prove the existence of God." Anglican priest, Cambridge theologian and psychologist Rev. Dr. Fraser N. Watts, said "a creator God provides a reasonable and credible explanation of why there is a universe, and ... it is somewhat more likely that there is a God than that there is not. That view is not undermined by what Hawking has said."


Dwight Garner in the New York Times was critical of the book, saying: "The real news about The Grand Design is how disappointingly tinny and inelegant it is. The spare and earnest voice that Mr. Hawking employed with such appeal in A Brief History of Time has been replaced here by one that is alternately condescending, as if he were Mr. Rogers explaining rain clouds to toddlers, and impenetrable."


Craig Callender, in the New Scientist, was not convinced by the theory promoted in the book: "M-theory ... is far from complete. But that doesn't stop the authors from asserting that it explains the mysteries of existence ... In the absence of theory, though, this is nothing more than a hunch doomed – until we start watching universes come into being – to remain untested. The lesson isn't that we face a dilemma between God and the multiverse, but that we shouldn't go off the rails at the first sign of coincidences.


Paul Davies, in The Guardian, wrote: "The multiverse comes with a lot of baggage, such as an overarching space and time to host all those bangs, a universe-generating mechanism to trigger them, physical fields to populate the universes with material stuff, and a selection of forces to make things happen. Cosmologists embrace these features by envisaging sweeping “meta-laws” that pervade the multiverse and spawn specific bylaws on a universe-by-universe basis. The meta-laws themselves remain unexplained – eternal, immutable transcendent entities that just happen to exist and must simply be accepted as given. In that respect the meta-laws have a similar status to an unexplained transcendent god." Davies concludes "there is no compelling need for a supernatural being or prime mover to start the universe off. But when it comes to the laws that explain the big bang, we are in murkier waters."


Dr. Marcelo Gleiser, in his article 'Hawking And God: An Intimate Relationship', stated that "contemplating a final theory is inconsistent with the very essence of physics, an empirical science based on the gradual collection of data. Because we don’t have instruments capable of measuring all of Nature, we cannot ever be certain that we have a final theory. There’ll always be room for surprises, as the history of physics has shown again and again. In fact, I find it quite pretentious to imagine that we humans can achieve such a thing. ... Maybe Hawking should leave God alone."

Physicist Peter Woit, of Columbia University, has criticized the book: "One thing that is sure to generate sales for a book of this kind is to somehow drag in religion. The book's rather conventional claim that "God is unnecessary" for explaining physics and early universe cosmology has provided a lot of publicity for the book. I'm in favor of naturalism and leaving God out of physics as much as the next person, but if you're the sort who wants to go to battle in the science/religion wars, why you would choose to take up such a dubious weapon as M-theory mystifies me."

In Scientific American, John Horgan is not sympathetic to the book: "M-theory, theorists now realize, comes in an almost infinite number of versions, which "predict" an almost infinite number of possible universes. Critics call this the "Alice's Restaurant problem," a reference to the refrain of the old Arlo Guthrie folk song: "You can get anything you want at Alice's Restaurant." Of course, a theory that predicts everything really doesn't predict anything... The anthropic principle has always struck me as so dumb that I can't understand why anyone takes it seriously. It's cosmology's version of creationism.

The physicist Tony Rothman, with whom I worked at Scientific American in the 1990s, liked to say that the anthropic principle in any form is completely ridiculous and hence should be called CRAP. ... Hawking is telling us that unconfirmable M-theory plus the anthropic tautology represents the end of that quest. If we believe him, the joke’s on us."


The Economist is also critical of the book: Hawking and Mlodinow "...say that these surprising ideas have passed every experimental test to which they have been put, but that is misleading in a way that is unfortunately typical of the authors. It is the bare bones of quantum mechanics that have proved to be consistent with what is presently known of the subatomic world. The authors’ interpretations and extrapolations of it have not been subjected to any decisive tests, and it is not clear that they ever could be. Once upon a time it was the province of philosophy to propose ambitious and outlandish theories in advance of any concrete evidence for them. Perhaps science, as Professor Hawking and Mr Mlodinow practice it in their airier moments, has indeed changed places with philosophy, though probably not quite in the way that they think."


Leading British scientist Baroness Greenfield also criticized the book in an interview with BBC Radio: "Of course they can make whatever comments they like, but when they assume, rather in a Taliban-like way, that they have all the answers, then I do feel uncomfortable." She later claimed her Taliban remarks were "not intended to be personal", saying she "admired Stephen Hawking greatly" and "had no wish to compare him in particular to the Taliban"

References:

Rabu, 06 Juli 2011

Benar kah Fenomena Mencairnya Es Kutub menyebabkan Peningkatan Tinggi permukaan Air Laut?

Bismillah . Be balance between Mind, Body & Soul

Baru-baru ini masyarakat dunia dihebohkan dengan fenomena mencairnya es di kutub, tetapi benar kah apabila es di kutub tersebut mencair dapat menenggelamkan beberapa kawasan di dunia seperti Jakarta??? Saya berpendapat, jawabannya TIDAK. Jawaban saya bukan sekedar jawaban asal-asalan tetapi jawaban ini datang dari ilmu Fisika. Adapun alasan penarikan jawaban tersebut akan saya uraikan pada uraian di bawah ini:

1. Siklus Hidrologi
Dalam ilmu Hidrologi kita mengetahui bahwa kandungan air di bumi tetap atau kekal, wujud air dapat berupa tiga fase yaitu fase padat, fase cair, dan fase uap. Sehingga dapat disimpulkan bahwa air yang terdapat di bumi dalam tiga wujud tersebut baik yang terdapat di laut, di darat, di dalam mahluk hidup, dan di atmosfer tetap.

2. Hukum Fisika
Sebelum membahas kaitan antara mencairnya es dikutub terlebih dahulu akan saya ceritakan beberapa eksperimen sederhana. Jika kita disuruh untuk menentukan volume sebuah kotak apa yang akan kita lakukan. Untuk mencari volume sebuah kotak kita harus mengukur panjang, lebar, dan tinggi dari kotak tersebut, dan secara matematika volume kotak tersebut dapat kita cari dengan mengalikan panjang, lebar dan tingginya. Namun, bagaimana jika kita disuruh untuk menentukan volume sebuah kalung yang berbentuk tidak beraturan??

Ok. Jika Anda pandai matematika berarti kita akan menggunakan teknik integrasi atau dengan menimbang massanya kemudian dibagi dengan rapat massa benda tersebut yang sudah ada pada tabel tetapan zat padat, tapi ada cara yang lebih mudah dan cara ini telah dilakukan oleh Archimedes.

Suatu ketika Archimedes disuruh oleh raja untuk menentukan volume dari mahkota yang dipakai raja. apa yang Archimedes lakukan? Archimedes pada waktu itu sangat stress, mungkin Anda juga, hehehe. Suatu ketika dia mandi pada sebuah bak mandi yang penuh dengan air. Ketika tubuhnya masuk kedalam bak tersebut, sebagian air dari bak tersebut tumpah lalu beliau menjerit kegirangan karena hal ini menginspirasi perhitungan volume mahkota tersebut(air pada bak tersebut tidak akan tumpah jika bak tersebut masih dapat menanpung peningkatan volume akibat Archimedes masuk kedalam bak tersebut).

Kenapa saya menceritakan kisah Archimedes tersebut?? Baik, sekarang kita masuk ke bahasan tentang fenomena es mencair tersebut. Sebelumnya kita harus mengetahui kondisi es di kutub. Es di kutub mengapung diatas air laut hal ini dikarenakan massa jenis es lebih kecil dari massa jenis air laut, tapi es dikutub sudah ada sejak dulu (bukan es yang kita datangkan ke air laut, jika es tersebut kita datangkan ke air laut maka ketinggian air laut akan meningkat hal ini analog dengan badan Archimedes yang dimasukan ke dalam bak), sehingga jika demikian es yang mencair tersebut tidak akan berpengaruh terhadap jumlah air laut karena pada peristiwa ini hanya terjadi perubahan bentuk dari fase padat menjadi cair.

Sebagai percobaan, masukan lah es batu ke dalam gelas yang berisi air dan amati volume air di dalam gelas ketika  es batu tersebut baru dimasukan dan ketika es batu tersebut telah mencair).

Kalau demikian penjelasannya, kenapa kita harus khawatir dengan fenomena es mencair tersebut??? Ya, kita memang harus hawatir karena fenomena tersebut mengindikasikan bahwa suhu bumi kita mengalami peningkatan (es akan mencair jika diberi kalor atau panas). Peningkatan suhu bumi tersebut biasa kita sebut pemanasan global, sedangkan gejala pemanasan global ini mirip green house effect ( efek rumah kaca) yang disebabkan oleh gas-gas seperti uap air, metana, nitrogen oksida dll.

Harus diketahui juga bahwa pemanasan tersebut bukan karena banyaknya kaca atau rumah yang dibuat dari kaca. Green house merupakan tempat percobaan dimana suhu didalam ruangan tersebut dijaga stabil hal ini bertujuan untuk mengamati spesies dan biasanya tanaman yang ada di dalam tempat tersebut. Kaitannya dengan gas-gas yang dapat menimbulkan pemanasan global, karena gas-gas tersebut berprilaku sebagai perangkap sebagian sinar matahari yang dipancarkan kebumi tidak dapat dipantulkan ke luar angkasa dan malah terus dipantulkan ke bumi lagi. Hal inilah yang menyebabkan terjadinya pemanasan global.

Sebenarnya, bumi kita telah dilapisi  oleh pelindung sinar matahari dan akan menahan sinar matahari supaya mencapai bumi dalam tingkat aman untuk kehidupan di bumi, lapisan pelindung ini disebut lapisan Ozon(O3). Ozon merupakan senyawa berbahaya jika berada di permukaan bumi karena jika terurai akan menghasilkan oksigen radikal,  namun akan bermanfaat dan menjadi pelindung jika berada di tempatnya yaitu di lapisan atas atmosfer bumi.

Menurut pengamatan satelit bahwa lapisan ozon di bumi kita telah menipis dan bahkan di daerah kutub telah bolong (inilah penyebab es di kutub mencair disamping gas-gas rumah kaca) dikarenakan penggunaan senyawa CFC yang biasa kita gunakan dalam alat pendingin ruangan dan lemari es. Kita harus mengetahui persamaan dan perbedaan es dikutub mencair akibat penipisan lapisan ozon oleh senyawa CFC dan atau oleh gas-gas rumah kaca. Es dikutub mencair sebagian besar dikarenakan bolongnya lapisan ozon di kawasan tersebut dan sebagian kecil diakibatkan oleh peningkatan suhu daerah sekitar (suhu bumi secara global akibat sifat gas-gas rumah kaca yang telah dijelaskan diatas), sedangkan untuk mengetahui akibat gas-gas rumah kaca kita dapat melihatnya dari es yang terdapat pada puncak gunung seperti rangkaian pegunungan Evrest yang makin berkurang dimana gletser (aliran air dari es yang mencair) di kaki gunung semakin sedikit. kesimpulannya selain bumi kita terus dipanasi akibat adanya gas-gas rumah kaca kita juga akan kehilangan lapisan pelindung(lapisan ozon) sehingga akibatnya akan terjadi pemanasan global.

Akibat dari pemanasan global adalah terjadinya perubahan iklim yang sangat ekstrim sehingga salah satu akibatnya akan mengganggu rantai makanan dalam suatu ekosistem yang nantinya akan berdampak kepada kehidupan manusia dan kehidupan di bumi.

Apa yang harus kita lakukan untuk menyelamatkan bumi kita dari pemanasan global?
  • Pertama kita harus mengurangi atau menghentikan penggunaan senyawa CFC dan mengurangi terbuangnya gas-gas rumah kaca akibat aktivitas rumah kaca. 
  • Kedua dengan cara menanam pepohonan untuk mengurangi senyawa CO2 di atmosfer karena tumbuhan atau pepohonan akan menggunakan gas tersebut bersama senyawa air dengan bantuan klorofil dan sinar matahari untuk menghasilkan energi dan oksigen. 
  • Namun bagaimana dengan lapisan ozon yang telah bolong?
    Lapisan ozon yang telah bolong secara alami dapat pulih kembali namun memerlukan waktu yang lama tergantung katalisnya (zat yang dapat mempercepat pembentukan ozon tersebut). Bagi yang tertarik ilmu kimia coba cari senyawa katalisnya supaya dapat menutup lapisan ozon yang bolong dan ganjarannya pasti dapat hadiah Nobel.. :)
*) Terima kasih untuk Fatih Khusno Satrio , 10205026, Fisika ITB untuk sharing dan diskusinya tahun kemarin ^_^

Selasa, 05 Juli 2011

ITB-Tekno 2011



Waktu
09 Juli · 8:00 - 15:00

Tempat
Aula Timur - Institut Teknologi Bandung
Jalan Ganesa No. 10
Bandung, Indonesia

Dibuat oleh

Info Selengkapnya
Pameran ini diselenggarakan sebagai sarana sosialisasi karya-karya ITB dengan target pengunjung pameran adalah Industri dan masyarakat Indonesia. Dengan adanya pameran ini diharapkan tercipta komunikasi dan kerjasama antara civitas akademika ITB dan Industri menghasilkan karya-karya tepat guna bagi kebutuhan masyarakat. Pameran dilakukan dengan pemanfaatan multimedia untuk mempresentasikan karya.

Zona Pameran dibagi menjadi 8 (delapan) zona:
1. Infrastruktur, Mitigasi Bencana, dan Kewilayahan
2. Energi
3. Teknologi Informasi dan Komunikasi
4. Pangan, Kesehatan dan Obat-obatan
5. Produk Budaya dan Lingkungan
6. Teknologi Nano dan Kuantum
7. Bioteknologi
8. Karya Unggulan Mandiri

Sekretariat Pameran:
Pusat Informasi Kampus (Rotunda)
Institut Teknologi Bandung
Jl. Ganesa No.10
 
Source : ITB-Tekno 2011
 



Sabtu, 02 Juli 2011

How do you measure the distance between Earth and the Sun?

How do we calculate the distance of the Sun from the Earth, or the actual size of the Sun, or the speed of travel of Earth in its orbit around the Sun? Clearly, from an answer to one of these questions one can find out the answers to the others. But how do we find the first answer?

It’s pretty amazing how well we know the distance between Earth and Moon so accurately

The first step in measuring the distance between the Earth and the Sun is to measure the distance between Earth and another planet in terms of the distance between Earth and the Sun. So, let us assume that the distance between Earth and the Sun is "a". Now, let us consider the orbit of Venus. To first approximation, the orbits of Earth and Venus are perfect circles around the Sun.

Take a look at the diagram below (not to scale). From the representation of the orbit of Venus, it is clear that there are two places where the Sun-Venus-Earth angle is 90 degrees. At these points, the line joining Earth and Venus will be a tangent to the orbit of Venus. These two points indicate the greatest elongation of Venus and is the farthest that Venus will get away from the Sun in the sky.

Another way to understand this is to look at the motion of Venus in the sky relative to the Sun: as Venus orbits the Sun, it gets further away from the Sun in the sky, reaches a maximum separation from the Sun (corresponding to the greatest elongation) and then starts going towards the Sun again. This by the way is the reason why Venus is never visible in the evening sky for more than about three hours after sunset and in the morning sky more than 3 hours before sunrise. 


How to measure distance to Sun

Now, by making observations of Venus in the sky, one can determine the point of greatest elongation. One can also measure the angle between the Sun and Venus in the sky at the point of greatest elongation. In the diagram, this angle will be the Sun-Earth-Venus angle marked as "e" in the right angled triangle. Now, using the trigonometry, one can determine the distance between Earth and Venus in terms of the Earth-Sun distance:

distance between Earth and Venus = a * cosine(e).

Now, the distance to Venus can be measured by radar measurements, where a radio wave is transmitted from Earth and is received when it bounces off Venus and comes back to Earth. By measuring the time taken for the pulse to come back, the distance can be calculated as radio waves travel at the speed of light. Once this is known, the distance between Earth and Sun can be calculated.

Historically, the first person to do this measurement was Aristarchus (310-230 BC). He measured the angular separation of the Sun and the Moon when its phase was first or third quarter to derive the distance between Earth and Sun in terms of the distance between the Earth and the Moon.

Eratosthenes (276-194 BC) also measured the distance between Earth and Sun as 804,000,000 stadia. The first scientific measurement of the Earth-Sun distance was made by Cassini in 1672 by parallax measurements of Mars (he observed Mars from two places simultaneously).

As you have indicated, once the distance between Earth and Sun is known, one can calculate all the other parameters. We know that the Sun subtends an angle of 0.5 degrees. Again, using trigonometry, the radius/diameter of the Sun can be calculated from the distance between Earth and Sun, d, as Rsun = tan(0.5 degrees) * d. Also, since we know the time taken by the Earth to go once around the Sun (P = 1 year), and the distance traveled by the Earth in this process (2*pi*a), we can calculate the average orbital speed of Earth as v = P/(2*pi*a).

Anyway, the relevant numbers are:

Earth-Sun distance, d = roughly 150 million km (defined as 1 Astronomical Unit)
 
Radius of the Sun, Rsun = roughly 700,000 km  
Orbital speed of Earth, v = roughly 30 km/s

References:

Kamis, 30 Juni 2011

Enam Gerhana Akan Terjadi Pada Tahun 2011

Diperkirakan, enam gerhana akan terjadi pada tahun 2011: Empat gerhana matahari sebagian dan dua gerhana bulan total akan terjadi di 2011. Kombinasi empat dan dua gerhana dalam satu tahun ini adalah peristiwa yang jarang terjadi.

Gerhana matahari sebagian pada 2011 akan terjadi pada 4 Januari, 1 Juni, 1 Juli, dan 25 November. Sedangkan gerhana bulan total akan terjadi pada 15 Juni dan 10 Desember. Kombinasi empat gerhana matahari dan dua gerhana bulan dalam setahun hanya akan terjadi enam kali sepanjang abad ke-21, yakni pada tahun 2011, 2029, 2047, 2065, 2076, dan 2094.

Sayangnya, seluruh gerhana matahari tidak akan dapat dilihat dari Indonesia. Tapi, dua gerhana bulan total akan dapat diamati di Nusantara. Seluruh fase gerhana bulan total Juni akan dapat diamati di Indonesia bagian barat, sementara wilayah Indonesia lainnya mengalami gerhana bulan sebagian. Pada penghujung tahun 2011, seluruh fase gerhana bulan total akan dapat diamati dari Sabang hingga Merauke.

Gerhana terdekat, yakni gerhana matahari sebagian pada 4 Januari, akan dapat diamati di sebagian besar wilayah Eropa, Afrika utara, dan Asia Tengah. Di kota-kota Eropa seperti Madrid, Paris, London, dan Copenhagen, akan menjadi lokasi terbaik jepretan foto gerhana sebagian, saat matahari baru terbit.

Puncak gerhana matahari sebagian ini akan terjadi pada 08:50:35 waktu universal (UT) dan lokasi terbaik adalah di wilayah utara Swedia. Warga di Kairo, Jerusalem, Istanbul, dan Teheran juga akan mendapati gerhana matahari sebagian dengan magnitud besar.

Pemandangan indah gerhana matahari sebagian menjelang surya tenggelam akan dapat diamati di kawasan tengah Rusia, Kazakhstan, Mongolia, dan kawasan barat laut China. Gerhana matahari sebagian resmi berakhir saat penumbra meninggalkan bumi pada 11:00:54 UT.

Sumber data: NASA via National Geographic Indonesia

Rabu, 29 Juni 2011

Deconvolusi

Deconvolusi adalah proses pengolahan data seismik yang bertujuan untuk meningkatkan resolusi temporal (baca: vertikal) dengan cara mengkompres wavelet seismik.

Deconvolusi umumnya dilakukan sebelum stacking akan tetapi dapat juga diterapkan setelah stacking.

Selain meningkatkan resolusi vertikal, deconvolusi dapat mengurangi efek 'ringing' atau multiple yang mengganggu interpretasi data seismik.

Deconvolusi dilakukan dengan melakukan konvolusi antara data seismik dengan sebuah filter yang dikenal dengan Wiener Filter .

Filter Wiener diperoleh melalui permasaan matriks berikut:

a x b = c

a adalah hasil autokorelasi wavelet input (wavelet input diperoleh dengan mengekstrak dari data seismik), b Filter Wiener dan c adalah kros korelasi antara wavelet input dengan output yang dikehendaki.

Output yang dikehendaki
terbagi menjadi beberapa jenis [Yilmaz, 1987]:

  1. Zero lag spike (spiking deconvolution)
  2. Spike pada lag tertentu.
  3. Time advanced form of input series (predictive deconvolution) 
  4. Zero phase wavelet.
  5. Wavelet dengan bentuk tertentu (Wiener Shaping Filters)

Zero lag spike memiliki bentuk [1 , 0, 0, 0, ..., 0] yakni amplitudo bukan nol terletak para urutan pertama. Jika Output yang dikehendaki memiliki bentuk [0 , 0, 1, 0, ..., 0] maka disebut spike pada lag 2 (amplitudo bukan nol terletak para urutan ketiga) dan seterusnya.

Dalam bentuk matrix, Persamaan Filter Wiener dituliskan sbb:
dimana n adalah jumlah elemen.

Matriks a diatas merupakan matriks dengan bentuk spesial yakni matriks Toeplitz, dimana solusi persamaan diatas secara efisien dapat dipecahkan dengan solusi Levinson. Dengan demikian operasi Deconvolusi jenis ini seringkali dikenal dengan Metoda Wiener-Levinson.

Untuk memberikan kestabilan dalan komputasi numerik diperkenalkan sebuah Prewhitening (e) yakni dengan memberikan pembobotan dengan rentang 0 s.d 1 pada zero lag matriks a (sehingga elemen a0 matrix diatas menjadi a0(1+e).

Gambar dibawah ini menunjukkan diagram alir proses Deconvolusi:

8th International Symposium on Modern Optics and Its Applications


When
 
4th - 7th July 2011
 
Where
 
Auditorium Campus Center Timur ITB
 
Organizer
 
Physics of Magnetism and Photonics Group Institut Teknologi Bandung Physics Building Jl. Ganesa 10, Bandung 40132, INDONESIA
 
Category
 
ITB
 
Participant
 
Umum, Dosen, Peneliti, Mahasiswa
 
Link agenda
 
http://fismots.fi.itb.ac.id/ismoa
 
Description
 
Since its first commencement in 2001, the International Symposium on Modern Optics and Its Applications has been regularly organized to bring together experts in the field of photonics science and technology to share and discuss new results achieved in the field and the potential impacts on their applications as well as their further advances. This meeting will also provide the opportunity to foster international scientific networkings and cooperations. This meeting is also aimed at introducing the important topics in the field of optics to the students and young scientists, as well as bringing together important advances in areas of information, sensing and biomedical technologies. The subject matters to be presented and discussed in this Symposium cover a variety of topics listed below (but not restricted to it).
  • New Photonic Materials, Metamaterials and Liquid Crystals
  • Nano Optical Science and Technology
  • Surface Plasmon, Near Field Optics and Super Lens
  • Nonlinear Optical Phenomena, Spectroscopy and Optical Solitons
  • Modelling and Fabrication of Photonic Devices and Integrated Optics
  • Optical Periodic Systems and Photonic Crystals
  • Biophotonics and Biomedical Optics
File pdf dapat diunduh di http://www.itb.ac.id/agenda/itb_agenda_1168.pdf

Selasa, 28 Juni 2011

Conceptual Physics 11th Edition by Paul G. Hewitt (ebook)

Di dalam edisi 11 Conceptual Physics ini terdapat konten yang berfokus pada aplikasi fisika dan fitur pedagogical terkini. Diantara topik yang ada dalam buku ini addalah: seputar Science, Newton’s First Law of Motion: Inertia, Linear Motion, Newton’s Second Law of Motion: Force and Acceleration Newton’s Third Law of Motion: Action and Reaction, Momentum, Energy, Rotational Motion, Gravity, Projectile and Satellite Motion Atomic Nature of Matter, Solids, Liquids, Gases and Plasmas, Temperature, Heat and Expansion, Heat Transfer, Change of Phase Thermodynamics, Vibrations and Waves, Sound, Musical Sounds, Electrostatics, Electric Current, Magnetism, Electromagnetic Induction, Properties of Light, Color, Reflection and Refraction, Light Waves, Light Emission, Light Quanta, The Atom and the Quantum, Atomic Nucleus and Radioactivity, Nuclear Fission and Fusion, Special Theory of Relativity, General Theory of Relativity Appendices Market: Intended for those interested in learning the basics of conceptual physics.
 
Conceptual Physics 11th edition by Paul G. Hewitt
Sampul Buku Conceptual Physics

E-book ini menggunakan software djvu (bukan pdf). Bila komputer Anda belum terinstal djvu, software djvu file bisa Anda download di sini.

Sedangkan untuk e-booknya bisa Anda download di link Conceptual Physics.

Teori Fisika tentang Lubang Putih

Rumus-rumus relativitas umum memiliki sifat matematika yang menarik: simetri waktu. Itu artinya kita dapat memasukkan nilai apapun ke persamaannya dan membayangkan waktu mengalir mundur bukannya maju, dan kita bisa mendapatkan jawaban valid lainnya dari rumus tersebut.

Bila kita menerapkan aturan ini pada solusi yang menjelaskan lubang hitam, kita akan mendapatkan objek yang disebut lubang putih. Karena lubang hitam adalah daerah ruang dimana segalanya tidak dapat lepas, versi balikan waktu dari lubang hitam adalah daerah ruang dimana segalanya tidak dapat jatuh. Karena lubang hitam hanya menyerap segalanya, maka lubang putih hanya memuntahkan segalanya.

Apakah kita dapat menemukan lubang putih? Well, lubang hitam memang sudah ditemukan. Masalahnya, menurut Stephen Hawking, lubang hitam sendiri bersifat acak dan simetri waktu. Karenanya, ia tidak dapat dibedakan dari lubang putih.

Cara untuk mengetahui keberadaan lubang putih menurut Roger Penrose, adalah dengan menyelam ke dalam lubang hitam. Sayangnya, secanggih apapun, hal ini sama artinya dengan melompat dari puncak gedung bertingkat 100 tanpa bantuan apapun untuk melihat apa yang ada di balik sebuah lubang di jalan raya. Kita pasti akan berderai di jalan raya, sebelum bisa mengintip di lubang jalan, belum lagi harus menembus lubang itu untuk pergi ke bawah tanah.

Ilustrasi lubang hitam, cacing dan putih
Seandainya kita sebuah partikel hipotesis yang masuk ke dalam lubang hitam, apa yang kita lihat?
  1. Dari alam semesta, kita masuk ke dalam lubang hitam lewat cakrawala peristiwa (event horizon) dan tersedot menuju cakrawala lubang cacing.
  2. Kita kemudian terbawa ke cakrawala lubang cacing (wormhole) atau cakrawala dalam (inner horizon) lubang hitam. Disini kita melihat masa lalu tak terhingga dari alam semesta kita yang tercermin dari singularitas repulsif gravitasi. 
  3. Kita masuk ke dalam lubang cacing dan tiba pada cakrawala dalam lubang putih. Disini kita melihat masa depan tak terhingga dari alam semesta kita. 
  4. Kita masuk ke lubang putih dan terlontar menuju cakrawala peristiwa lubang putih. Di cakrawala ini kita akan melihat masa lalu tak terhingga dari alam semesta yang baru.
Skenario diatas hanya berlaku untuk lubang hitam Reissner-Nordstrom dan lubang hitam Schwartzchild. Lubang hitam jenis Reissner-Nordstrom adalah lubang hitam yang memiliki massa dan muatan listrik, tapi tidak memiliki spin (putaran). Sementara itu lubang hitam Schwartzchild memiliki massa saja, tanpa muatan listrik maupun spin. Sayangnya, lubang hitam yang ada di alam semesta kita, tampak memiliki spin atau lubang hitam Kerr.  Lebih parah lagi lubang hitam nyata tidak memiliki muatan listrik. Para ilmuan mengatakan alam semesta kita tampaknya netral secara listrik, dan lubang hitam yang bermuatan akan segera menjadi netral.

Karenanya, jangankan lubang putih, lubang cacing saja hanyalah mitos ilmiah. Mitos ilmiah artinya ia dibangun dari seperangkat hitungan matematis yang konsisten namun ternyata tidak terbukti ada.

Tapi Michio Kaku berpendapat lain. Hal di atas hanya memperhitungkan teori relativitas umumnya Einstein. Kita baru tahu kalau teori relativitas umum Einstein itu sangat mendekati kenyataan, tapi kita belum tahu apakah teori string lebih mendekati kenyataan daripada relativitas umum. Kita belum dapat menguji teori string hingga sekarang, tapi bila benar, kemungkinan skenario di atas berlaku bagi lubang hitam biasa. Teori string memprediksikan juga kalau alam semesta ini seperti gelembung sabun yang mengembang dan mati. Miliaran tahun ke depan bintang akan mati; langit malam akan gelap dan samudera akan mendingin. Tapi kita bisa lari. Gelembung sabun kita eksis bersama gelembung sabun lainnya. Setiap kali lubang hitam lahir, ia menciptakan alam semesta bayi. Materi yang tersedot ke dalamnya akan di muntahkan di sisi lain, menciptakan sebuah lubang putih di alam semesta kembaran kita. Lubang putih ini akan mengembang sangat cepat, seperti Big Bang.

Marcelo Samuel Berman bahkan lebih ekstrim lagi. Ia mengatakan kalau alam semesta kita sendiri adalah lubang putih. Kita hidup di dalam lubang putih dan pengembangan alam semesta adalah buktinya.

Stephen Hawking menambah keruh suasana. Ia sebelumnya bilang kalau lubang hitam dan lubang putih tak terbedakan, dan sekarang mengajukan kalau lubang hitam dan lubang putih sebenarnya sama. Karenanya pernyataan alam semesta berada di dalam lubang putih, sama saja dengan mengatakan kalau alam semesta kita berada di dalam lubang hitam.

Terbentuknya Jembatan Einstein-Rosen
Well, di alam semesta kita sendiri ada lubang hitam toh? Dan di dalam lubang hitam ada lubang cacing, istilah ilmiahnya Jembatan Einstein-Rosen, dan di luar lubang cacing ini ada lubang putih. Saat sebuah lubang hitam terbentuk, maka di alam semesta lain terbentuklah lubang putih. Nah karena kita berada di dalam lubang hitam yang lebih besar, otomatis;

  1. Ada alam semesta yang lebih besar lagi dimana alam semesta kita merupakan sebuah lubang hitam di alam semesta tersebut. 
  2. Ada alam semesta kembaran kita yang terbentuk bersamaan dengan alam semesta kita
Gagasan ini disimpulkan oleh Nikodem Poplawski dan cukup membuat semakin anehnya dunia ini. Beda lagi dengan Valeri Pavlovich Frolov dan  Igor Dmitrievich Novikov. Bagi mereka lubang putih tidak pernah dapat stabil. Dalam waktu cepat setelah ia terbentuk, lubang putih akan menjadi lubang hitam. Ia runtuh.

Diagram Ruang Waktu Penrose

Kalau kita ingin kompromi, bisa saja kan alam semesta kita sedang berada dalam fase menjelang runtuh? Masalahnya, alam semesta kita mengembang dipercepat, bukan diperlambat. Ini memang belum menutup kemungkinan sih. Soalnya bisa jadi kita mengembang semakin cepat dan tiba-tiba terhenti dan mengerut menjadi lubang hitam.

Gagasan lainnya lebih aneh lagi. Kita memang berada di alam semesta yang mengerut. Hanya saja fisika kita yang terbalik. Lawrence Krauss mengatakan kalau alam semesta yang mengerut dan mengembang pada dasarnya tidak terbedakan. Ia hanya dapat dibedakan bila kita membandingkan alam semesta kita dengan pasangan alam semesta kita. Dalam isolasi ini, kita mungkin hidup di dunia yang waktunya berjalan mundur, tapi kita merasa waktu berjalan maju.

Semuanya terdengar mengagumkan dan membuat kita terpana. Bagaimana mungkin para manusia kecil ini bisa mempelajari alam semesta seolah ia adalah gundu mainan anak-anak. Begitu hebatkah mereka? Mereka memang hebat, tapi mereka hebat karena sains. Sains membuat kita lebih besar dari alam semesta dan membuat kita terpesona dengan keindahannya. Belum pernah dalam sejarah manusia kita berhadapan dengan kenyataan betapa rumitnya alam ini. Alam semesta lebih aneh daripada fiksi.

Referensi
1.  Ted Bunn. 1995.     Black Holes FAQ
2. S. W. Hawking. 1976. Black holes and thermodynamics. Phys. Rev. D 13, 191–197 (1976)
3. Andrew Hamilton. 2009. Penrose diagrams
4. Andrew Hamilton. 2006. Collapse to a Black Hole
5. John Crace. 2005. Michio Kaku: Mr Parallel Universe
6. Marcelo Samuel Berman. 2007. IS THE UNIVERSE A WHITE-HOLE?
Astrophysics and Space Science. Volume 311, Number 4, 359-361
7. Igor R. Klebanov. 2006. TASI Lectures: Introduction to the AdS/CFT Correspondence. Lectures at TASI ’99, Boulder, June 1999
8. Nikodem J. Poplawski. Radial motion into an Einstein-Rosen bridge. Physics Letters B, 2010; 687 (2-3): 110
9. Valeri Pavlovich Frolov, Igor Dmitrievich Novikov. 1998. Black hole physics: basic concepts and new developments. Springer Science & Business
10. Wikipedia. 2010. White hole

Artikel ini saya salin dari Fisika Ceria dengan editing seperlunya.

The Physics of Vibration and Waves, Author H.J.Pain (ebook)

Teman-teman yang ingin belajar tentang Gelombang tapi belum ada bukunya, disini saya berikan link download e-book gelombang karangan H.J.Pain. Berikut ini sekilas daftar isinya.



ISBN: 0470012951
Title: The Physics of Vibrations and Waves
Author: H. J. Pain
Publisher: John Wiley & Sons
Publication Date: 2005-06-13
Number Of Pages: 576

Editorial Description:

The main theme of this highly successful book is that the transmission of energy by wave propogation is fundamental to almost every branch of physics. Therefore, besides giving students a thorough grounding in the theory of waves and vibrations, the book also demonstrates the pattern and unity of a large part of physics. This new edition has been thoroughly revised and has been redeisgned to meet the best contemporary standards. It includes new material on electron waves in solids.

1. Simple Harmonic Motion.
2. Damped Simple Harmonic Motion.
3. The Forced Oscillator.
4. Coupled Oscillations.
5. Transverse Wave Motion.
6. Longitudinal Waves.
7. Waves on Transmission Lines.
8. Electromagnetic Waves.
9. Waves in More than One Dimension.
10. Fourier Methods.
11. Waves in Optical Systems.
12. Interference and Diffraction.
13. Wave Mechanics.
14 Non-linear Oscillations and Chaos.
15 Non-linear Waves, Shocks and Solitons.
Appendix 1: Normal Modes, Phase Space and Statistical Physics.
Appendix 2: Kirchhoffs Integral Theorem.
Appendix 3: Non-Linear Schringer Equation.

Untuk yang ingin mendownloadnya silahkan disini

Happy Physics

Prigell Priya Ragil

 


Sabtu, 25 Juni 2011

Parametric Equation

Now we introduce yet another way of representing curves, known as parametric equations, as well as how to graph them. We will graph some familiar curves, such as circles and ellipses using their parametrized forms, as well as some as yet unfamiliar ones, such as cycloids and Lissajous Figures.

A parametrized curve is a curve written represented by the following system of equations:
  • (1a) x = f(t)
  • (1b) y = g(t)
where f and g are specified functions of the variable t, and t has a specified range, i.e. an interval such as [0,1], [0,2π), or (-∞, ∞). The variable t is known as the parameter of the system. Equations (1a) and (b) are known as parametric equations for the coordinates x and y respectively.

Unit Circle
As a first example of a parameterized curve, consider the unit circle. As we have seen, the unit circle is parametrized by the polar angle θ. Specifically, it has the following parametrization:
  • (2a) x = cos θ
  • (2b) y = sin θ
where θ lies in the interval [0,2π).

Example 1: Graph the unit circle using the parametrization given by Equations (2).
Solution: To make a graph of a parametrized curve, we first need to tabulate x and y for as many values of the parameter as we can. We may use Table 1, which we reproduce below with x replaced by θ and the last two columns interchanged, for this purpose. Note that for the purpose of graphing, we no longer need to know the value of the parameter θ accurately.)

θ x = cos θ y = sin θ
0
1.000
0.000
π/6
3/2 ≈ 0.866
1/2 = 0.500
π/4
2/2 ≈ 0.707
2/2 ≈ 0.707
π/3
1/2 = 0.500
3/2 ≈ 0.866
π/2
0.000
1.000
2π/3
-1/2 = -0.500
3/2 ≈ 0.866
3π/4
-√2/2 ≈ -0.707
2/2 ≈ 0.707
5π/6
-√3/2 ≈ -0.866
1/2 = 0.500
π
-1.000
0.000
7π/6
-√3/2 ≈ -0.866
-1/2 = -0.500
5π/4
-√2/2 ≈ -0.707
-√2/2 ≈ -0.707
4π/3
-1/2 = -0.500
-√3/2 ≈ -0.866
3π/2
0.000
-1.000
5π/3
1/2 = 0.500
-√3/2 ≈ -0.866
7π/4
2/2 ≈ 0.707
-√2/2 ≈ -0.707
11π/6
3/2 ≈ 0.866
-1/2 = -0.500
1.000
0.000

Now we just plot these points and draw a smooth curve through them, obtaining the following graph:

graph of parametrized circle

Figure 1: Graph of a Parametrized Unit Circle

Ellipse
It is also useful to know how to parametrize an ellipse. As it turns out, an ellipse centered at the origin with semi-major axis a and semi-minor axis b has the following parametrization:
  • (4a) x = a cos t
  • (4b) y = b sin t
where t ranges over the interval [0,2π).

Example 2: Use the parametric equations of an ellipse to graph an ellipse with semi-major axis 5, semi-minor axis 4, and centered at the origin.
Solution: It is easy to tabulate x and y in this case; we simply take our previous table and multiply all values of x by 5 and all values of y by 4. We obtain the following table:

t x = 5 cos t y = 4 sin t
0
5.00
0.00
π/6
4.33
2.00
π/4
3.54
2.83
π/3
2.50
3.46
π/2
0.00
4.00
2π/3
-2.50
3.46
3π/4
-3.54
2.83
5π/6
-4.33
2.00
π
-5.00
0.00
7π/6
-4.33
-2.00
5π/4
-3.54
-2.83
4π/3
-2.50
-3.46
3π/2
0.00
-4.00
5π/3
2.50
-3.46
7π/4
3.54
-2.83
11π/6
4.33
-2.00
5.00
0.00

Once again, we plot a smooth curve through these points, obtaining the following graph:
parametrized ellipse

Figure 2: Graph of a Parametrized Ellipse

Now we introduce some more exotic curves, which are nevertheless easy to graph, due to their simple parametrizations.

Cycloid
A cycloid is a curve obtained by marking the position of a point on the rim of a wheel of a moving vehicle, such as a bicycle. A cycloid generated by a wheel of radius R has the following parametrization:
  • (6a) x = R (θ - sin θ)
  • (6b) y = R (1 - cos θ)
Here, the parameter θ represents the angle by which the wheel has turned since the marked point was touching the ground. Here the parameter θ ranges over all real numbers, i.e. from -∞ to -∞.

Example 3: Graph two full cycles of a cycloid with R = 1, using the above parametrization.
Solution: First we need to compute and plot the values of x and y for θ ranging over an interval of length 4π. The simplest interval to use for this purpose is [-2π, 2π]. The following table gives values of x and y as a function of the parameter θ ranging over this interval:

θ x = θ - sin θ y = 1 - cos θ θ x = θ - sin θ y = 1 - cos θ
-2π ≈ -6.283 -6.283 0.000
0 0.000 0.000
-11π/6 ≈-5.760 -6.260 0.134
π/6 ≈ 0.524 0.024 0.134
-7π/4 ≈-5.498 -6.205 0.293
π/4 ≈ 0.785 0.078 0.293
-5π/3 ≈ -5.236 -6.102 0.500
π/3 ≈ 1.047 0.181 0.500
-3π/2 ≈ -4.712 -5.712 1.000
π/2 ≈1.571 0.571 1.000
-4π/3 ≈ -4.189 -5.055 1.500
2π/3 ≈2.094 1.228 1.500
-5π/4 ≈ -3.927 -4.634 1.707
3π/4 ≈ 2.356 1.649 1.707
-7π/6 ≈-3.665 -4.165 1.866
5π/6 ≈2.618 2.118 1.866
-π ≈ -3.142 -3.142 2.000
π ≈ 3.142 3.142 2.000
-5π/6 ≈-2.618 -2.118 1.866
7π/6 ≈ 3.665 4.165 1.866
-3π/4 ≈ -2.356 -1.649 1.707
5π/4 ≈3.927 4.634 1.707
-2π/3 ≈ -2.094 -1.228 1.500
4π/3 ≈ 4.189 5.055 1.500
-π/2 ≈ -1.571 -0.571 1.000
3π/2 ≈ 4.712 5.712 1.000
-π/3 ≈ -1.047 -0.181 0.500
5π/3 ≈ 5.236 6.102 0.500
-π/4 ≈ -0.785 -0.078 0.293
7π/4 ≈5.498 6.205 0.293
-π/6 ≈ -0.524 -0.024 0.134
11π/6 ≈5.760 6.260 0.134
0 0.000 0.000
2π ≈ 6.283 6.283 0.000

Below is a graph of two cycles of the cycloid, based on this data.
cycloid

Figure 3: Graph of a Cycloid

Lissajous Figures
Lissajous figures are some of the most interesting parametrized curves. A Lissajous figure has the following parametrization:
  • (8a) x = cos at
  • (8b) y = sin bt
where a and b are positive integers and t ranges over the interval [0, 2π). Lissajous figures can be generated on an oscilloscope from two sinusoidal inputs of different frequencies. When a=b, the figure is a circle.
Below are some examples of Lissajous figures with given vales of a and b.

lissajous figures

Figure 4: Lissajous Figures
(Image taken from Wikipedia)

Example 4: Use the parametrization given by Equations (8) to plot the Lissajous figure with a=3 and b=2.
Solution: First we tabulate x = cos 3t and y = sin 2t against t, for t ranging from 0 to 2π.

t x = cos 3t y = sin 2t t x = cos 3t y = sin 2t
0
1.000
0.000

π
-1.000
0.000
π/12
0.707
0.500

13π/12
-0.707
0.500
π/6
0.000
0.866

7π/6
0.000
0.866
π/4
-0.707
1.000

5π/4
0.707
1.000
π/3
-1.000
0.866

4π/3
1.000
0.866
5π/12
-0.707
0.500

17π/12
0.707
0.500
π/2
0.000
0.000

3π/2
0.000
0.000
7π/12
0.707
-0.500

19π/12
-0.707
-0.500
2π/3
1.000
-0.866

5π/3
-1.000
-0.866
3π/4
0.707
-1.000

7π/4
-0.707
-1.000
5π/6
0.000
-0.866

11π/6
0.000
-0.866
11π/12
-0.707
-0.500

23π/12
0.707
-0.500
π
-1.000
0.000

1.000
0.000

Below is a graph of this Lissajous figure, based on the data.

lissajous figure

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