text:
User talk:Tecmp18

 Hello Tecmp18, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:19, 23 September 2010 (UTC)

title: User talk:Tecmp18


text:
User talk:Zarathustra1214

 Hello Zarathustra1214, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:19, 23 September 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Zarathustra1214


text:
User talk:Sushantbs

 Hello Sushantbs, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:20, 23 September 2010 (UTC)

title: User talk:Sushantbs


text:
User talk:Oteng~enwikiversity

 Hello Oteng, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:20, 23 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Oteng. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Oteng~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:46, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:Oteng~enwikiversity


text:
User talk:Jennysindel

 Hello Jennysindel, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:20, 23 September 2010 (UTC)

title: User talk:Jennysindel


text:
User talk:Bloom10

 Hello Bloom10, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 23:21, 23 September 2010 (UTC)

title: User talk:Bloom10


text:
User talk:Tanat S.

 Hello Tanat S., and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 04:00, 24 September 2010 (UTC)

title: User talk:Tanat S.


text:
User:Waris p

My name is Waris Panyadhanya and I go by Pop. I were born in 30 November 1990. I used to go to Assumption College for 12 years as a high school graduation. Now I am studing in the second years of Mahidol University International College(MUIC). My major is TIM.

title: User:Waris p


text:
User:Bussabong Preeyaporn

Hello, I am Bussabong Preeyaporng. I am now 17 years old and currently a Freshman of Mahidol Univerity International College. I am majored in Tourism Industry Management. I usually spend most of my free time reading sport news and watching soccer. I love soccer. It is one of the most important parts of my life. Liverpool is my favourite soccer club of all time. Even though I love soccer to bits, I don't really do any sports or any types of exercises. I just enjoy watching all the matches. I also like listening to music. I am in love with Alternative Rock and Metal genres songs. I don't have much of the knowledges about the computer. I only know things like how to type things up and edting photos, that is all. Thank you for reading!--Bussabong Preeyaporn 05:50, 24 September 2010 (UTC)

This is my 'Welcome to MUIC' video, http://www.youtube.com/watch?v=jbFTDZq-laQ

title: User:Bussabong Preeyaporn


text:
User talk:Praewploy

My name is Praewploy Taweechaiwat (Ploy).

 * /Youtube Link/
 * /Specs on Mobile Equipment/

title: User talk:Praewploy


text:
User:May Haas

I'm May Haas My ID:5280127 I'm majoring in TIM.

Second Homework: youtube video This is the url of my "Welcome to MUIC" video on youtube : http://www.youtube.com/watch?v=jC7nfpaCjCU

iPad specs: CPU& Speed: Apple A4 custom-designed, high-performance, low-power system-on-a-chip 1GHz Bus Speed: N/A Cache size: L1 cache 32 KB Instruction + 32KB Data, L2 cache 640 KB Storage type& amount: Flash memory 16GB, 32GB or 64GB

My partner's work Yoon, 3GiPhone Specs: CPU & Speed : 600 MHz CPU, a significant increase over the 412 MHz with its 128MB CPU found earlier models, faster 7.2 Mbps Bus speed : 100mhz, 103 mhz, 256 MB up from 128 MB, 600 Mhz from 412 Mhz in iphone 3Gs. I phone cache size: L1 cache size of 32 KB I-Cache and 32 KB D-Cache, and an L2 cache size of 256 KB in iphone 3GS Storage type and amount: 8GB flash drive, The iPhone 3GS released in 2009 includes 16 GB or 32 GB of storage and the version released in 2010 includes 8 GB of storage.

In conclusion: The ipad is technologically more advanced than the iphone, beacause it is newer and for instance the battery last longer than that of the iphone, giving the user up to 10hrs. to watch videos. It offers many new applications and gadgets and is more suitable for daily use, as it has a bigger screen and you can even read like in real books.Ipad CPU and Speed is 1GHz but Iphone CPU and Speed is 600MHz. Also when we compare with Cache size, both has same L1 cache size, but Ipad L2 cache size 384 KB bigger than Iphone3G. Lastly, iphone’s biggest storage amount is 32GB but Ipad’s biggest storage amount is 64GB which is double bigger than Iphone.

title: User:May Haas


text:
User:Elixer vocal

My name is Thitipak Khuptisuwan 5180173. You can call me Up.

http://www.youtube.com/watch?v=knJb2cEPUfk

Computer Essentials Homework #3 (2 people in group)

CPU & Speed: iPhone 3G: SMID, 412 MHz. iPad: Apple A4 custom-designed, 1GHz

Bus speed: iPhone 3G: 103 MHz. iPad: not found

Cache size: iPhone 3G: 16Kb Instruction + 16 Kb Data. iPad: 32 Kb Instruction + 32 Kb Data

Storage: iPhone 3G: Flash storage, 8 GB.            iPad: Flash storage, 32 GB

According to information given above Ipad is apparently the most technological advance. This is because it has a 1 GHz CPU, which can process approximately 2 times faster than the other two devices. Ipad also have the highest cache size that will allow it to run application faster or reopen the recent used application faster. Another advantage that iPad has is the storage facility which is about 4 times bigger storing size comparing with iPhone 3G. To conclude, iPad is the most technological advance as it has better CPU, cache size and bigger storing capacity. Reference http://sio2interactive.forumotion.net/sio2-engine-f3/iphone-3d-cpu-bus-speed-t594.htm http://www.anandtech.com/show/3633/apples-a4-soc-faster-than-snapdragon http://www.apple.com/ipad/specs/

title: User:Elixer vocal


title: User:Fizz16


text:
User talk:Mantanee

My name is Mantanee Burananusorapong ID 4980583,studying in Food Science major My account here is "Mantanee"

My video
my welcoming video http://www.youtube.com/watch?v=JhWuBskUpGo

CPU comparison
Blackberry 8900, CPU:BlackBerry OS, CPU speed:512 MHZ, Cache size: 256 kB, Storage: ROM 256 MB iphone, CPU: ARM 11, CPU speed:400 MHz, BUS speed:100MHz, Cache size: 256 kB, Storage (flash drive) 32 GB Netbook,CPU:Intel® Atom™ Processor N450, CPU speed:1.66 GHz, BUS speed:667 MHz, Cache size:512 kB, Storage:PROM type 160 GB

Which technology is more advanced and why ? According to specifications of each device, Netbook is the most advanced one among others because it ranks the highest amount of storage, CPU speed and BUS speed due to the figure in the table above.However, if the advancement is technically compared by type of CPU architecture, it cannot absolutely be compared and ranked. This is because each type of CPU is designed to properly fit the device and its application. For example, ARM is required less power supply and able to run several applications then it fits small portable device like PDA or iphone. Other CPU like Intel Atom is larger than ARM and more power consumption.Then it is suitable for netbook, mostly applying for running more applications and programs.

title: User talk:Mantanee


text:
User:Navavit

http://www.youtube.com/watch?v=PPLyBS3kSzg

Iphone4 (5180511 Choochon) 16GB or 32GB flash drive 620 MHz Arm Chips 16k/16k cache 256MB RAM

iPad (Apple A4 chip) (5180070 Navavit) 16, 32, 64 GB flash drive 1Ghz Apple A4 custom-designed, high performance, low power system Cores: 1 L1 Cache 32KB instruction + 32 KB data L2 Cache 640KB

Blackberry bold 9700 (5080710 Thitipon) CPU 624 Mhz processor (PXA930) 65nm tech 256MB storage, 256 MB RAM Micro SD, up to 16GB, 2GB inclu. Primary (Ram) Data Bus: 32 bit

iPad A4 proves to be the best in terms of speed and processing, while the iPad also offers the most memory and storage.

Team members: 5080710, 5180511, 5180070

title: User:Navavit


text:
User:Peerasin46

hello everyone my name is Peerasin Rattanakoson ID: 5280105 Mahidol University International College ICNS141 Computer Essential email address: peerasin46@hotmail.com Facebook: Peerasin Nott Rattanakoson Thank you

These is some problem occured with my wikiversity undated writing My first Project for Youtube video: Welcome to MUIC The link is :http://www.youtube.com/watch?v=CnX-AuNmBfw This video was updated on October 3rd, 2010 sorry for corrupted in my upload

hello everyone my name is Peerasin Rattanakoson ID: 5280105 Mahidol University International College ICNS141 Computer Essential email address: peerasin46@hotmail.com Facebook: Peerasin Nott Rattanakoson Thank you

These is some problem occured with my wikiversity undated writing My first Project for Youtube video: Welcome to MUIC The link is :http://www.youtube.com/watch?v=CnX-AuNmBfw This video was updated on October 3rd, 2010 sorry for corrupted in my upload

Best Operating System for a Netbook The operating system that I might use for my net-book is Windows XP. According to my research, i found out that Window XP is the most suitable one for my net book because net book has low capacity and could be attach a little ram, to use more advance operating system such as Window 7 might be to tough for the net book. However, linux is not widely use and might cause some problem with linking the information between computer that use the different system. Window XP is also the most suitable one to run on net book. Window XP give a good speed and well constant operating system. Otherwise, Window XP might consume lots of energy use to run, but Window XP is quite good in other facility such as Window office, Internet, and connecting between system world wide. Please visit http://smidgenpc.com/2010/03/23/choosing-the-best-operating-system-for-a-netbook/ (by MATTHEW SMITH on MARCH 23, 2010) for more information to make a decision about the operating system that you would like to install on your net book.

''' iPad Specification iPad CPU: 1GHz Apple A4 custom-designed, high-performance, low-power system-on-a-chip, 256 MB DRAM BUS Speed: Cache size: 5 MB storage type and amount: 16GB, 32GB, or 64GB flash drive operating system: iOS For more information please visit :http://gizmodo.com/5508864/ipad-test-notes-speed

iPhone 4 Specification CPU: 1Ghz Apple A4 processor Bus speed: ARM's 64-bit-wide AMBA 3 AXI bus Cache size: 4 MB Storage type and amount: 16/32 GB storage, 512 MB RAM Operating system: iOS4 For more information please visit: http://www.gsmarena.com/apple_iphone_4-3275.php

Blackberry Storm2 9550 CPU & Speed : Dual Core CPU ARM11 400 MHz and ARM9 274 MHz Bus Speed : 32-bit Cache Size : Storage Type & Amount : On-board 2GB and MicroSD Card up to 32GB Information From : http://en.wikipedia.org/wiki/Blackberry_9550

For me, I think that iPad is the most suitable and the best out of three. iPad has a very good specification and very high capacity. Out of those three, iPad can run the fastest CPU out of three and with high capacity, ipad can contain very much data, much larger than BB. However, BB can insert the external capacity such as microSD, however, the capacity of MicroSD id much lover than the capacity of the iPad. Although the iPad and the iPhone run with the same operation system (iOS), iPad has a better RAM that will push the operating system to reach it maximum performance.

title: User:Peerasin46


text:
User:Samat698

Yo! my name is Nine. http://www.youtube.com/watch?v=Owpe_pDWTws CPU : The BlackBerry Curve 8520 features a 512 MHz processor, while the Bold 9700 features a newer version of the Bold 9000's processor, but is clocked at the same speed. Display: Half VGA resolution 480 × 320 pixel color display backlit transmissive TFT LCD Cache Size : the maximum number of converted documents that can be located in the document cache (as DOM) for a single conversion process. The default value is 32. Range 1 through 128 Storage :Different BlackBerry® devices support different types of memory. The following types of memory are possible on BlackBerry devices: • Application storage is internal to the BlackBerry device. It contains the operating system, Java® Virtual Memory (JVM), and an internal file system. Application storage is also called flash memory and onboard memory. Application storage is the only place on a BlackBerry device from which applications can be run. All BlackBerry devices have application storage. • External media card storage is a microSD card that BlackBerry device users can insert to extend the storage on their device. It is optional and removable. A File Allocation Table (FAT) file system is mounted on the media card. MicroSD cards are supported on all devices running BlackBerry® Device Software 4.2 or later, with the exception of the BlackBerry® 8700 Series. • Built-in media storage is an embedded multimedia card (eMMC). It is not removable. A FAT file system is mounted on the built-in media card. Built-in media storage is also called internal media memory and on-board device memory. Built-in media storage is included on many newer devices.

Iphone4 CPU Apple A4 (ARM Cortex-A8) Storage capacity 16 GB or 32 GB flash memory Cache Size - L1 cache size of 32 KB I-Cache and 32 KB D-Cache, and an L2 cache size of 256 KB. Price 16GB $199 32GB $299

Iphone is better ,even the price is higher but the quality and options are more useful. Memory storage is also bigger.

title: User:Samat698


text:
User talk:Numnimjeaztii

 Hello Numnimjeaztii, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:17, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 24 September 2010 (UTC)

title: User talk:Numnimjeaztii


text:
User talk:Egm6321.f10.team2.sahin

 Hello Egm6321.f10.team2.sahin, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:17, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 24 September 2010 (UTC)

title: User talk:Egm6321.f10.team2.sahin


text:
User talk:Pabloruiz

 Hello Pabloruiz, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 24 September 2010 (UTC)

cleaning up your image uploads
You have requested deletion of some of your uploads, but they are in use. I did delete one and removed the usage, because it seems not at all important. But then, looking at the next, -->

the image contains important information. That information might be replaced with a link to a page elsewhere with the information, avoiding questions of copyright (assuming that we don't know the linked page to be a copyvio!). Or the information could be assembled, etc. What would you like to do? I'm sure you can fix this much faster than I!

Please look over your images in Category:Candidates for speedy deletion, you can follow the "What links here" tool, and remove unneeded images from resource pages. That way I can quickly, no problem, delete the images, as you have requested. Otherwise I'll have to use a brute force approach. Thanks. --Abd 14:15, 18 January 2011 (UTC)

About all these deletion notices
I'd like to let you know that when a file is deleted, it can easily be recovered by any custodian. I've noticed that quite a number of files that you uploaded have been deleted, and that may be necessary as a routine matter where appropriate license information hasn't been provided on request, but all this can be fixed if needed. Ask me, and I assume I'll be able to help. Thanks. --Abd 23:37, 26 October 2011 (UTC)

What is the copyright of File:Compressibility.png?
Hi! Thanks for uploading the file File:Compressibility.png. Unfortunately, when you did so you didn't include information about the license, and the file will need a license or other copyright information in order to stay at Wikiversity.

You can easily fix this by adding on the page, and filling out who the owner is and the copying terms. In particular, we need to know who owns the rights to the file and what the terms of use are. In regard to the latter, Wikiversity accepts open content, public domain, and fair use works (see Wikiversity's Copyright policy). You can place a on the file page to signify the copyright terms. If you need any help, feel free to leave me a message, and I'll do what I can to assist.Bilby 04:30, 2 November 2011 (UTC)

title: User talk:Pabloruiz


text:
User talk:Mise~enwikiversity

 Hello Mise, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 24 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Mise. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Mise~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:38, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:16, 19 April 2015 (UTC)

title: User talk:Mise~enwikiversity


text:
User talk:Sucheera

 Hello Sucheera, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 24 September 2010 (UTC)

title: User talk:Sucheera


text:
User talk:Navavit

 Hello Navavit, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 24 September 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 24 September 2010 (UTC)

title: User talk:Navavit


title: User:Durex24


text:
Talk:Educational Media Awareness Campaign/Sports/POTD 4

Football has 11 people on the team. 22 on the field. Normally 5 offense, 2 forward, 3 defense, 1 goal. The offense people face each other at the center line at the start. When deemed one offense can be pulled back as defense.--Martin Lenoar 11:14, 24 September 2010 (UTC)


text:
User talk:Polypeppu

 Hello Polypeppu, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:54, 24 September 2010 (UTC)

title: User talk:Polypeppu


text:
User talk:Jinwah50

 Hello Jinwah50, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:54, 24 September 2010 (UTC)

title: User talk:Jinwah50


text:
User talk:Harshini06

 Hello Harshini06, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:55, 24 September 2010 (UTC)

title: User talk:Harshini06


text:
User talk:Urpriyansundar

 Hello Urpriyansundar, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Urpriyansundar


text:
User talk:Bwanga.mbula

 Hello Bwanga.mbula, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Bwanga.mbula


text:
User talk:Hrhodes3~enwikiversity

 Hello Hrhodes3, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:56, 24 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Hrhodes3. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Hrhodes3~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:38, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:11, 19 April 2015 (UTC)

title: User talk:Hrhodes3~enwikiversity


text:
User talk:Natthida

 Hello Natthida, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:56, 24 September 2010 (UTC)

title: User talk:Natthida


text:
User talk:Osamasidd

 Hello Osamasidd, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:56, 24 September 2010 (UTC)

title: User talk:Osamasidd


text:
User talk:Annerberg

 Hello Annerberg, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:57, 24 September 2010 (UTC)

title: User talk:Annerberg


text:
User talk:Sriharsha.astroguy

 Hello Sriharsha.astroguy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:57, 24 September 2010 (UTC)

title: User talk:Sriharsha.astroguy


text:
User talk:Justiceguiter

 Hello Justiceguiter, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Justiceguiter


text:
User talk:Mukitshah

 Hello Mukitshah, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:58, 24 September 2010 (UTC)

title: User talk:Mukitshah


text:
User talk:Deepak k~enwikiversity

 Hello Deepak k, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:58, 24 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Deepak k. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Deepak k~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:30, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:05, 19 April 2015 (UTC)

title: User talk:Deepak k~enwikiversity


text:
User talk:Mrulopa

 Hello Mrulopa, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:59, 24 September 2010 (UTC)

title: User talk:Mrulopa


text:
User talk:Tabi~enwikiversity

 Hello Tabi, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:59, 24 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Tabi. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Tabi~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:44, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:23, 19 April 2015 (UTC)

title: User talk:Tabi~enwikiversity


text:
User talk:Janet19

 Hello Janet19, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 11:59, 24 September 2010 (UTC)

title: User talk:Janet19


text:
User talk:Trezzor

 Hello Trezzor, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 12:00, 24 September 2010 (UTC)

title: User talk:Trezzor


text: File:2010 09 23 13 57 17.djvu

title: File:2010 09 23 13 57 17.djvu


text: File:2010 09 23 14 52 54.djvu

title: File:2010 09 23 14 52 54.djvu


text:
User:Aklenhayal

'''Buraya gelmekten gerçekten çok mutluyum. Herkese iyi vikilemeler.Bu site sayesinde bir çok güncel haber artık çok kolay alınır. Herkese çok teşekkürler.

Aklenhayal kullanıcısı Vikiversity'de amaçladığı birçok hedefi vardır.Bunlardan birkaçı da aşağıda yazmaktadır. 1: Vikiversityde bilgilerin sorgulanarak doğruya ulaşılabileceğini savunur.

2: Vikiversity kullanıcılarının gerekli zamanlarda vikitatile çıkmalarını önerir.

3: Bir sayfayı değiştirmeden önce konu ile ilgili yeterli bilgiye sahip olunması ya da yeterli araştırmayı yapmış olması gerektiğini düşünür.

4: Vikiversity de Vandalizmi kesinlikle reddeder.

5: Rastgele sayfa silindiğinde önce silen kişiyi uyarmalı ve sayfayı eski durumuna geri getirme gereğini uygulamalı.

6: Sayfa yazmak isteyen yeni kullanıcıların deneme tahtasında kendisini göstermesi önerir.

7: Vikiversityde tarafsız bir bakış açısının izlenmesinin gerek bilir.

8: Katkıda bulunmak isteyen yeni kullanıcılar topluluk portalından seçtikleri sayfaları biçimlendirebilme imkanı bulabileceğini söyler.

9: Vikiversity'i asla istediğiniz gibi bağırabileceğiniz bir yer olarak algılamamalısınız.

title: User:Aklenhayal


text:
User talk:Aklenhayal

 Hello Aklenhayal, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 16:53, 24 September 2010 (UTC)

title: User talk:Aklenhayal


text:
User talk:Egm6321.f10.team5.oh/hw3

= Problem 1 - Derive EOM (SC-N1-ODE) = From Meeting 13, p. 13-1 ~ p. 13-2

Given
Figure shows the Trajectory of a projectile (ex:Rocket):

Find

 * $$\left( 1 \right)$$ Drive Equation of Motion (EOM)
 * $$\left( 2 \right)$$ Particular case $$\displaystyle k = 0 $$ : Verify $$y\left( x \right)$$ is parabolla
 * $$\left( 3 \right)$$ Consider $$k\ne 0$$, $$\displaystyle {{v}_{x}}=0$$,

 * $$ (3.1) $$ Find $${{v}_{y}}\left( t \right)$$, $$\displaystyle y(t)$$ for $$ \displaystyle m$$ = constant
 * $$ (3.2) $$ Find $${{v}_{y}}\left( t \right)$$, $$\displaystyle y(t)$$ if $$m=m\left( t \right)$$

Solve
Part 1 Consider the trajectory of a projectile (ex. Rocket)

 * Various forces acting on the projectile at time 't' are:

 * 1) Weight of the projectile

 * {| style="width:100%" border="0"

 * $$\displaystyle W=mg$$
 * }
 * }
 * }

 * 2) Inertia force

 * {| style="width:100%" border="0"

 * $${{F}_{a}}=ma=m\cdot \frac{dv}{dt}$$ for particle with constant mass
 * $${{F}_{a}}=ma=m\cdot \frac{dv}{dt}$$ for particle with constant mass

 * }
 * }

 * 3) Air resistance which is proportional to the velocity of particle

 * {| style="width:100%" border="0"

 * $$\displaystyle {{F}_{D}}=k{{v}^{n}}$$
 * }
 * }
 * }

Now consider the force equilibrium in both horizontal and vertical direction

 * a) Force Equilibrium in horizontal direction:

 * {| style="width:100%" border="0"

 * $$\sum{{{F}_{H}}=0}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$m\cdot \frac{d{{v}_{x}}}{dt}+k{{v}^{n}}\cos \alpha =0$$,
 * $$m\cdot \frac{d{{v}_{x}}}{dt}+k{{v}^{n}}\cos \alpha =0$$,

 * where $$\displaystyle {{v}_{x}}\to $$ horizontal component of velocity

 * }
 * }

 * {| style="width:100%" border="0"

$$m\cdot \frac{d{{v}_{x}}}{dt}=-k{{v}^{n}}\cos \alpha $$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 1.1)
 * }
 * }

 * b) Force Equilibrium in the vertical direction:

 * {| style="width:100%" border="0"

 * $$\sum{{{F}_{V}}=0}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

$$m\cdot \frac{d{{v}_{y}}}{dt}+k{{v}^{n}}\sin \alpha +mg=0$$,

 * where $$\displaystyle {{v}_{y}}\to $$ vertical component of velocity

 * }
 * }

 * {| style="width:100%" border="0"

$$m\cdot \frac{d{{v}_{y}}}{dt}=-k{{v}^{n}}\sin \alpha -mg$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 1.2)
 * }
 * }

Part 2

 * Particular case: When $$\displaystyle k=0$$

Eq.1.1 reduces to

 * {| style="width:100%" border="0"

 * $$m\frac{d{{v}_{x}}}{dt}=0\Rightarrow \frac{d{{v}_{x}}}{dt}=0$$
 * }
 * }
 * }

Integrating the above equation gives:

 * {| style="width:100%" border="0"

$$
 * $${{v}_{x}}\left( t \right)={{c}_{1}}$$
 * $$\displaystyle (Eq. 1.3)
 * $$\displaystyle (Eq. 1.3)
 * }
 * }

Apply 'initial condition' to determine integration constant,$$\displaystyle{{c}_{1}}$$

 * {| style="width:100%" border="0"

 * $${{v}_{x}}\left( t=0 \right)={{v}_}={{c}_{1}}$$
 * }
 * }
 * }

Now Eq.1.3 becomes:
 * {| style="width:100%" border="0"

 * $${{v}_{x}}\left( t \right)={{v}_}$$
 * }
 * }
 * }

Integrate the above equation to obtain ,$$\displaystyle x$$

 * {| style="width:100%" border="0"

 * $$x\left( t \right)={{v}_}t+{{c}_{2}}$$
 * }
 * }
 * }

Then 'Initial condition' is applied to determine ,$$\displaystyle{{c}_{2}}$$

 * {| style="width:100%" border="0"

 * $$x\left( t=0 \right)={{v}_}\times 0+{{c}_{2}}\Rightarrow {{c}_{2}}={{x}_{0}}$$
 * }
 * }
 * }

Therefore,

 * {| style="width:100%" border="0"

 * $$x\left( t \right)={{v}_}t+{{x}_{0}}$$
 * }
 * }
 * }

Now $$\displaystyle t$$, can be expressed in terms of $$\displaystyle x$$,

 * {| style="width:100%" border="0"

$$
 * $$t=\frac{x-{{x}_{0}}}$$
 * $$\displaystyle (Eq. 1.4)
 * $$\displaystyle (Eq. 1.4)
 * }
 * }

Similarly When $$\displaystyle k=0$$, Eq.1.2 reduces to

 * {| style="width:100%" border="0"

 * $$m\frac{d{{v}_{y}}}{dt}=-mg\Rightarrow \frac{d{{v}_{y}}}{dt}=-g$$
 * }
 * }
 * }

Integrate the above equation to evaluate, $$\displaystyle {{v}_{y}}$$

 * {| style="width:100%" border="0"

$$
 * $${{v}_{y}}\left( t \right)=-gt+{{c}_{3}}$$
 * $$\displaystyle (Eq. 1.5)
 * $$\displaystyle (Eq. 1.5)
 * }
 * }

Apply 'initial condition' to obtain $$\displaystyle {{c}_{3}}$$

 * {| style="width:100%" border="0"

 * $${{v}_{y}}\left( t=0 \right)={{v}_}={{c}_{3}}$$
 * }
 * }
 * }

Now Eq.1.5 becomes,

 * {| style="width:100%" border="0"

 * $${{v}_{t}}\left( t \right)=-gt+{{v}_}$$
 * }
 * }
 * }

Then integrate the above equation to determine, $$\displaystyle y $$

 * {| style="width:100%" border="0"

 * $$y\left( t \right)=\frac{-g{{t}^{2}}}{2}+{{v}_}t+{{c}_{4}}$$
 * }
 * }
 * }

$$\displaystyle {{c}_{4}}$$ is determined using 'initial condition' as:

 * {| style="width:100%" border="0"

 * $$y\left( t=0 \right)=\frac{-g\times 0}{2}+{{v}_}\times 0+{{c}_{4}}\Rightarrow {{c}_{4}}={{y}_{0}}$$
 * }
 * }
 * }

Therefore,

 * {| style="width:100%" border="0"

 * $$y\left( t \right)=\frac{-g{{t}^{2}}}{2}+{{v}_}t+{{y}_{0}}$$
 * }
 * }
 * }

Now Substitute Eq.1.4 for $$\displaystyle t$$ in the above equation;

 * {| style="width:100%" border="0"

$$y\left( x \right)=-\frac{g}{2}{{\left( \frac{x-{{x}_{0}}} \right)}^{2}}+{{v}_}\left( \frac{x-{{x}_{0}}} \right)+{{y}_{0}}$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 1.6)
 * }
 * }

Eq.1.6 is in the form of a parabolic equation. Therefore $$\displaystyle y\left( x \right)$$ is parabola.

Part 3

With $$v_{xo}=0$$, and $$\cos\alpha=\frac{v_x}{v}$$ (from the geometry)

Which presents an interesting fact, once $$v_x$$ is equal to zero it remains zero as its derivative is also zero (except when $$v=0$$ and $$n-1<0$$). Thus, if $$v_{xo}=0$$ the $$x$$ velocity remains zero and the equations of motion reduce to equation 1.2.

Part 3.1
Solving

To show exactness, we first put into a form that shows the first condition of exactness is met

so that

The second condition of exactness

Which is met if $$m$$ is not a function of time

Thus, the equation is non-exact.

The equation can be made exact through the integrating factor method

Then expressing as a total derivative and testing for exactness

Letting $$\partial h / \partial v_y = 0$$

Then

Combining

Though the equation is not integrable, by making it exact though the integrating factor method an expression was found.

Part 3.2
If $$m(t)\ne0$$ the integrating factor method is complicated in equation 1.22 as the partial of $$m$$ with respect to time remains, complicating the expression for $$h$$

Author and Proof-reader
[Author]

[Proof-reader]

= Problem 2 - Derive EOM (SC-L1-ODE) = From Meeting 13, p. 13-3

Given

 * {| style="width:100%" border="0" align="left"

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

$$\begin{align} {m}_{1}, {m}_{2} \end{align}$$ : mass of the each pendulum

$$\begin{align} {\theta}_{1}, {\theta}_{2} \end{align}$$ : the angle from the vertical to the each pendulum

$$\begin{align} {u}_{1}, {u}_{2} \end{align}$$ : applied forces to the each pendulum

$$\begin{align} l \end{align}$$ : length of the pendulum

$$\begin{align} k \end{align}$$ : force constant(or spring constant)

$$\begin{align} g \end{align}$$ : acceleration of gravity

 * }
 * }
 * }

Find
1. Derive (2-1) and (2-2)

2. Write (2.1) and (2-2) in form of (2-3)

where,

$$\begin{align} \textbf{x} := {\left [ {\theta}_{1}, \dot{{\theta}_{1}}, {\theta}_{2},\dot{{\theta}_{2}}\right]}^{T} \end{align}$$

$$\begin{align} \textbf{u} := {\left[ {u}_{1}l, {u}_{2}l\right]}^{T} \end{align}$$

Dimension of matrix

$$\begin{align} \dot{\textbf{x}} \ : \ 4 \times 1 \end{align}$$

$$\begin{align} \textbf{x} \ : \ 4 \times 1 \end{align}$$

$$\begin{align} \textbf{A} \ : \ 4 \times 4 \end{align}$$

$$\begin{align} \textbf{B} \ : \ 4 \times 2 \end{align}$$

$$\begin{align} \textbf{U} \ : \ 2 \times 1 \end{align}$$

Solve
'''Step 1. Derivation '''

 Background Knowledge

1. Torque

2. Hooke's Law

3. Pendulum

4. Moment

5. Moment of inertia

6. Angular acceleration

 Derive Using above background,

where,

$$\begin{align} \tau \end{align}$$ : torque

$$\begin{align} I=m{l}^{2}\end{align}$$ : moment of inertia

$$\begin{align} \alpha=\ddot{\theta} \end{align}$$ : angular acceleration

Therefore, left hand side is $$\ - > \ \begin{align} m {l}^{2} \ddot{\theta} \end{align}$$

Torque of the spring force

From the backgroud (Hooke's Law(wikipedia)),

where, $$\begin{align} F \end{align}$$ : restoring force $$\begin{align} k\end{align}$$ : spring constant $$\begin{align} x \end{align}$$ : displacement from the equilibrium position (in this case, x = a)

Therefore, Torque of the spring force is,
 * {| style="width:100%" border="0"

$$\begin{align} \vec{\tau}=\vec{r}\times\vec{F} =-k {a}^{2}(\mathrm{sin}{\theta}_{1}-\mathrm{sin}{\theta}_{2})\approx -k{a}^{2}({\theta}_{1}-{\theta}_{2})\\ \end{align}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * $$\displaystyle (2-6)
 * }

Torque of the gravity force
 * {| style="width:100%" border="0"

$$\begin{align} \vec{\tau}=\vec{r}\times\vec{F} = -{m}_{1}g\cdot l\mathrm{sin}{\theta}_{1}\approx -{m}_{1}g\cdot l{\theta}_{1} \end{align}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\displaystyle (2-7)
 * }

Torque of the applied force
 * {| style="width:100%" border="0"

$$\begin{align} \vec{\tau}=\vec{r}\times\vec{F} = {u}_{1}\cdot l \mathrm{cos}{\theta}_{1} \approx {u}_{1}\cdot l \end{align}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
$$\displaystyle (2-8)
 * }

Using (2-4) ~ (2-8)

 * {| style="width:100%" border="0" align="left"

$$ {m}_{1}{l}^{2}\ddot{{\theta}_{1}}=-k{a}^{2}({\theta}_{1}-{\theta}_{2})-{m}_{1}gl{\theta}_{1}+{u}_{1}l $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

(2-2) can be verified with same procedure.

 * {| style="width:100%" border="0" align="left"

$$ {m}_{2}{l}^{2}\ddot{{\theta}_{2}}=-k{a}^{2}({\theta}_{2}-{\theta}_{1})-{m}_{2}gl{\theta}_{2}+{u}_{2}l $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

'''Step 2. Find A, B and U'''

Let's make a equation of a matrix with the information that we already have.

rearrange the derived equations (2-1) and (2-2),

Let's put them to (2-9)

 * {| style="width:100%" border="0" align="left"

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

$$ \begin{bmatrix} \ \dot{{\theta}_{1}} \\ \ \ddot{{\theta}_{1}} \\ \ \dot{{\theta}_{2}} \\ \ \ddot{{\theta}_{2}} \\ \end{bmatrix} = \underbrace{ \begin{bmatrix} \ 0 & 0 & 0 & 0 \\ \ -\frac{k{a}^{2}}{{m}_{1}{l}^{2}}-\frac{g}{l} & 0 & \frac{k{a}^{2}}{{m}_{1}{l}^{2}} & 0 \\ \ 0 & 0 & 0 & 0 \\ \ \frac {k{a}^{2}}{{m}_{2}{l}^{2}}& 0 & -\frac{k{a}^{2}}{{m}_{2}{l}^{2}}-\frac{g}{l} & 0 \\ \end{bmatrix} }_{\mathrm{Matrix} \mathbf{A} (4\times4)} \begin{bmatrix} \ {\theta}_{1} \\ \ \dot{{\theta}_{1}} \\ \ {\theta}_{2} \\ \ \dot{{\theta}_{2}} \\ \end{bmatrix} + \underbrace{ \begin{bmatrix} \ 0 & 0 \\ \ \frac{1}{{m}_{1}{l}^{2}} & 0 \\ \ 0 & 0 \\ \ 0 & \frac{1}{{m}_{2}{l}^{2}} \\ \end{bmatrix} }_{\mathrm{Matrix} \mathbf{B} (4\times2)} \begin{bmatrix} \ {u}_{1}l \\ \ {u}_{2}l \\ \end{bmatrix}

$$

 * }
 * }
 * }

Given
Shown in figure are the two Pendulums connected by a spring:

Find

 * $$\left( 1 \right)$$ Derive equation of motion:

 * {| style="width:100%" border="0"

$$
 * $${{m}_{1}}{{l}^{2}}{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}}=-k{{a}^{2}}\left( {{\theta }_{1}}-{{\theta }_{2}} \right)-{{m}_{1}}gl{{\theta }_{1}}+{{u}_{1}}l$$
 *
$$\displaystyle (Eq. 2.1)
 *
$$\displaystyle (Eq. 2.1)

 * }
 * }

 * {| style="width:100%" border="0"

$$
 * $${{m}_{2}}{{l}^{2}}{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}}=-k{{a}^{2}}\left( {{\theta }_{2}}-\theta 1 \right)-{{m}_{2}}gl{{\theta }_{2}}+{{u}_{2}}l$$
 *
$$\displaystyle (Eq. 2.2)
 *
$$\displaystyle (Eq. 2.2)
 * }
 * }

 * $$\left( 2 \right)$$ Write Eq.2.1 and Eq.2.2 in the form of [[media:2010_09_21_14_56_48.djvu|Mtg 13 (c),page2 ]], of:

 * {| style="width:100%" border="0"

$$
 * $$\underset{-}{\overset{\centerdot }{\mathop{x}}}\,=\underset{-}{\mathop{A}}\,\left( t \right)\underset{-}{\mathop{x}}\,\left( t \right)+\underset{-}{\mathop{B}}\,\left( t \right)\underset{-}{\mathop{u}}\,\left( t \right)$$
 *
$$\displaystyle (Eq. 2.3)
 *
$$\displaystyle (Eq. 2.3)
 * }
 * }

 * {| style="width:100%" border="0"

$$\underset{-}{\mathop{x}}\,={{\left[ \begin{matrix} {{\theta }_{1}} & \overset{\centerdot }{\mathop{{{\theta }_{1}}}}\, & {{\theta }_{2}} & \overset{\centerdot }{\mathop{{{\theta }_{2}}}}\, \\ \end{matrix} \right]}^{T}}$$ and $$\underset{-}{\mathop{u}}\,={{\left[ \begin{matrix} {{u}_{1}}l & {{u}_{2}}l \\ \end{matrix} \right]}^{T}}$$
 * Given
 * Given
 * }
 * }

Solution

 * $$\left( 1 \right)$$ Derive equation of motion:
 * (a) Consider Free Body Diagram of left pendulum:

 * For small angle:
 * {| style="width:100%" border="0"

 * $$\displaystyle l\sin {{\theta }_{1}}=l{{\theta }_{1}}$$ and $$\displaystyle l\cos {{\theta }_{1}}=l$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$acceleration, {{a}_{1}}=\frac{{{d}^{2}}\left( l{{\theta }_{1}} \right)}{d{{t}^{2}}}=l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$ Inertia force ={{m}_{1}}l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$\displaystyle Spring force =ka{{\theta }_{1}}-ka{{\theta }_{2}}=ka({{\theta }_{1}}-{{\theta }_{1}})$$
 * }
 * }
 * }

Now using D'Alembert's_principle, sum of the moments about pivot(A)is equal to zero

 * {| style="width:100%" border="0"

 * $$\sum{{{M}_{A}}=0\Rightarrow }\left( {{m}_{1}}l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}} \right)\cdot l+\left( ka({{\theta }_{1}}-{{\theta }_{2}}) \right)\cdot a+\left( {{m}_{1}}g \right)\cdot l{{\theta }_{1}}-\left( {{u}_{1}} \right)\cdot l=0$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

$$\Rightarrow {{m}_{1}}{{l}^{2}}{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}}=-k{{a}^{2}}({{\theta }_{1}}-{{\theta }_{2}})-{{m}_{1}}gl{{\theta }_{1}}+{{u}_{1}}l$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 2.1)
 * }
 * }

 * (b) Consider Free Body Diagram of right pendulum:

 * For small angle:
 * {| style="width:100%" border="0"

 * $$\displaystyle l\sin {{\theta }_{2}}=l{{\theta }_{2}}$$ and $$\displaystyle l\cos {{\theta }_{2}}=l$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$acceleration, {{a}_{2}}=\frac{{{d}^{2}}\left( l{{\theta }_{2}} \right)}{d{{t}^{2}}}=l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$ Inertia force ={{m}_{2}}l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}}$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

 * $$\displaystyle Spring force =ka{{\theta }_{2}}-ka{{\theta }_{1}}=ka({{\theta }_{2}}-{{\theta }_{1}})$$
 * }
 * }
 * }

Using D'Alembert's_principle, sum of the moments about pivot(B)is equal to zero

 * {| style="width:100%" border="0"

 * $$\sum{{{M}_{B}}=0\Rightarrow }\left( {{m}_{2}}l{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}} \right)\cdot l+\left( ka({{\theta }_{2}}-{{\theta }_{1}}) \right)\cdot a+\left( {{m}_{2}}g \right)\cdot l{{\theta }_{2}}-\left( {{u}_{2}} \right)\cdot l=0$$
 * }
 * }
 * }

 * {| style="width:100%" border="0"

$$\Rightarrow {{m}_{2}}{{l}^{2}}{{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}}=-k{{a}^{2}}({{\theta }_{2}}-{{\theta }_{1}})-{{m}_{2}}gl{{\theta }_{2}}+{{u}_{2}}l$$ $$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 *
$$\displaystyle (Eq. 2.2)
 * }
 * }

 * $$\left( 2 \right)$$ Write Eq.2.1 and Eq.2.2 in the form of Eq.2.3(system of coupled equation):

 * Eq.2.1 can be rearranged as,

 * {| style="width:100%" border="0"

$$
 * $${{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}}=\frac{-\left( k{{a}^{2}}+{{m}_{1}}gl \right)}{m{}_{1}{{l}^{2}}}{{\theta }_{1}}+\frac{k{{a}^{2}}}{m{}_{1}{{l}^{2}}}{{\theta }_{2}}+\frac{{{u}_{1}}l}{m{}_{1}{{l}^{2}}}$$
 *
$$\displaystyle (Eq. 2.4)
 *
$$\displaystyle (Eq. 2.4)
 * }
 * }

 * {| style="width:100%" border="0"

$$
 * $${{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}}=\frac{k{{a}^{2}}}{m{}_{2}{{l}^{2}}}{{\theta }_{1}}+\frac{-\left( k{{a}^{2}}+{{m}_{2}}gl \right)}{m{}_{2}{{l}^{2}}}{{\theta }_{2}}+\frac{{{u}_{2}}l}{m{}_{2}{{l}^{2}}}$$
 *
$$\displaystyle (Eq. 2.5)
 *
$$\displaystyle (Eq. 2.5)
 * }
 * Now Eq.2.4 and Eq.2.5 can be put in the form of Eq.2.3 as:
 * Now Eq.2.4 and Eq.2.5 can be put in the form of Eq.2.3 as:

 * {| style="width:100%" border="0"

 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |

$$\left[ \begin{matrix} {{\overset{\centerdot }{\mathop{\theta }}\,}_{1}} \\ {{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{1}} \\ {{\overset{\centerdot }{\mathop{\theta }}\,}_{2}} \\ {{\overset{\centerdot \centerdot }{\mathop{\theta }}\,}_{2}} \\ \end{matrix} \right]=\underbrace{\left[ \begin{matrix} 0 & 1 & 0 & 0 \\   \frac{-\left( k{{a}^{2}}+{{m}_{1}}gl \right)}{m{}_{1}{{l}^{2}}} & 0 & \frac{k{{a}^{2}}}{m{}_{1}{{l}^{2}}} & 0  \\ 0 & 0 & 0 & 1 \\   \frac{k{{a}^{2}}}{m{}_{2}{{l}^{2}}} & 0 & \frac{-\left( k{{a}^{2}}+{{m}_{2}}gl \right)}{m{}_{2}{{l}^{2}}} & 0  \\ \end{matrix} \right]}_{\underset{-}{\mathop{A}}\,}\left[ \begin{matrix} {{\theta }_{1}} \\ {{\overset{\centerdot }{\mathop{\theta }}\,}_{1}} \\ {{\theta }_{2}} \\ {{\overset{\centerdot }{\mathop{\theta }}\,}_{2}} \\ \end{matrix} \right]+\underbrace{\left[ \begin{matrix} 0 & 0 \\   \frac{1}{m{}_{1}{{l}^{2}}} & 0  \\ 0 & 0 \\   0 & \frac{1}{m{}_{2}{{l}^{2}}}  \\ \end{matrix} \right]}_{\underset{-}{\mathop{B}}\,}\left[ \begin{matrix} {{u}_{1}}l \\ {{u}_{2}}l \\ \end{matrix} \right]$$

$$
 *
$$\displaystyle (Eq. 2.6)
 * }
 * }

 * Where:

 * {| style="width:100%" border="0"

0 & 1 & 0 & 0 \\   \frac{-\left( k{{a}^{2}}+{{m}_{1}}gl \right)}{m{}_{1}{{l}^{2}}} & 0 & \frac{k{{a}^{2}}}{m{}_{1}{{l}^{2}}} & 0  \\ 0 & 0 & 0 & 1 \\   \frac{k{{a}^{2}}}{m{}_{2}{{l}^{2}}} & 0 & \frac{-\left( k{{a}^{2}}+{{m}_{2}}gl \right)}{m{}_{2}{{l}^{2}}} & 0  \\ \end{matrix} \right]$$
 * $$\underset{-}{\mathop{A}}\,=\left[ \begin{matrix}
 * $$\underset{-}{\mathop{A}}\,=\left[ \begin{matrix}

 * }
 * }

 * {| style="width:100%" border="0"

0 & 0 \\   \frac{1}{m{}_{1}{{l}^{2}}} & 0  \\ 0 & 0 \\   0 & \frac{1}{m{}_{2}{{l}^{2}}}  \\ \end{matrix} \right]$$
 * $$\underset{-}{\mathop{B}}\,=\left[ \begin{matrix}
 * $$\underset{-}{\mathop{B}}\,=\left[ \begin{matrix}
 * }
 * }

Contributing Members
Solved and posted by Egm6321.f10.team3.Sudheesh 15:39, 4 October 2010 (UTC)

Author and Proof-reader
[Author]

[Proof-reader]

= Problem 3 - Derive (L1-ODE-CC) = From Meeting 14, p. 14-1

Find
Derive (3-2)

Solve
We can rearrange the eqn(3.1). As it is not time variable problem, let $$\begin{align}a(t)=a, b(t)=b\end{align}$$

Let's find the integrating factor first.

As the coefficient for the $$\begin{align}\dot{x}\end{align}$$ is 1,

Multiply the integrating factor to eqn(3.2) on both side.

let's integrate for the interval $$\begin{align} \left[{t}_{0},t\right]\end{align}$$

rearrange eqn(3.7),

Author and Proof-reader
[Author]

[Proof-reader]

= Problem 4 - Expand Taylor series(exponential and exponential matrix) = From Meeting 14, p. 14-2

Given

 * {| style="width:100%" border="0" align="left"

$$\displaystyle {e}^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + ... $$ $$\displaystyle \ = \sum_{k=0}^{\infty} \frac{x^k}{k!} $$
$$\displaystyle (4-1) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \underbrace{{e}^{\underline{A}}}_{\underline{A} \ : \ n \times n \ Matrix} = \underbrace{\underline{\mathit I }}_{n \times n \ Unit \ mat.} + \frac{\underline{A}}{1!} + \frac{\underline{A}^2}{2!} + ... $$ $$\displaystyle \ = \sum_{k=0}^{\infty} \frac{\underline{A}^k}{k!} $$
$$\displaystyle (4-2) $$
 * }
 * }

Find
1) Derive (4-1)

2) Derive (4-2)

Solve
Solution of 1)

Using Taylor series ,

$$f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. $$

which can be written in the more compact sigma notation as

 * $$ \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $$

In the particular case where a = 0, the series is also called a Maclaurin series
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$f(x) = f(0)+f'(0)x + \frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots .$$ $$ \ = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(0)}{n!} \, (x)^{n}$$
$$\displaystyle (4-3) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle f(x) := {e}^{x} $$
$$\displaystyle (4-4) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle f(0) = 1, \ f'(x) = {e}^{x}, \ f'(0) = 1, \ ... $$
$$\displaystyle (4-5) $$
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$$\displaystyle f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} \ ... $$
$$\displaystyle (4-6) $$
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$$\therefore {e}^{x} = \sum_{k=0}^{\infty} \frac{x^k}{k!} $$
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Solution of 2)

Using Taylor series and Maclaurin series

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$$\displaystyle f(\underline{x}) :=  \underbrace{{e}^{\underline{A}}}_{\underline{A} \ : \ n \times n \ Matrix} $$
$$\displaystyle (4-7) $$
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<Background Knowledge> - Exponential Matrix, Identity Matrix

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$$\displaystyle f(\underline{0}) = \underbrace{\underline{\mathit  I }}_{n \times n \ Unit \ mat.}, \ f'(\underline{x}) = {e}^{\underline{A}}, \ f'(\underline{0}) = \underline{\mathit  I }, \ ... $$
$$\displaystyle (4-8) $$
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$$\displaystyle f(\underline{x}) = 1 + \underline{A} + \frac{\underline{A}^2}{2!} + \frac{\underline{A}^3}{3!} \ ... $$
$$\displaystyle (4-9) $$
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$$\therefore {e}^{\underline{A}} = \sum_{k=0}^{\infty} \frac{\underline{A}^k}{k!} $$
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Author and Proof-reader
[Author] Oh, Sang Min

[Proof-reader]

= Problem 5 - Generalized to SC-L1-ODE-VC = From Meeting 14, p. 14-2

Given
Dimension of matrix

$$\displaystyle \underline{x} \ : \ n \times \ 1 $$

$$\displaystyle \underline{A} \ : \ n \times \ n $$

$$\displaystyle \underline{B} \ : \ n \times \ m $$

$$\displaystyle \underline{U} \ : \ m \times \ 1 $$

Find
Generalized (5-3) to SC-L1-ODE-VC

Solve
 SC-L1-ODE-CC can be generalized to SC-L1-ODE-CC as same as L1-ODE-CC is generalized to L1-ODE-VC

Using (5-1) ~ (5-3)

 SC-L1-ODE-VC

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$$ \underline{x}(t)= {e}^{\int_{{t}_{0}}^{t} \underline{A} (\tau) d\tau}\cdot \underline{x}({t}_{0}) + \int_{{t}_{0}}^{t}{e}^{\int_{\tau}^{t} \underline{A} (s) ds} \cdot \underline{B}\cdot \underline{U}(\tau)d\tau $$
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Dimension of matrix

$$\displaystyle \underline{x} \ : \ n \times \ 1 $$

$$\displaystyle \underline{A} \ : \ n \times \ n $$

$$\displaystyle \underline{B} \ : \ n \times \ m $$

$$\displaystyle \underline{U} \ : \ m \times \ 1 $$

Author and Proof-reader
[Author] Oh, Sang Min

[Proof-reader]

= Problem 6 - Obtaining SC-L1-ODE-CC with int. factor method = From Meeting 15, p. 15-1

Author and Proof-reader
[Author]

[Proof-reader]

= Problem 7 - Application SC-L1-ODE-CC about rolling control of rocket = From Meeting 15, p. 15-1

Given

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$$\displaystyle \delta $$  =   aileron angle(deflection)

$$\displaystyle \phi $$ =   roll angle

$$\displaystyle \omega $$ =  roll angular velocity

$$\displaystyle Q $$ =   aileron efflectiveness

$$\displaystyle \tau $$ = roll time constant

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Find
Put (7-1) ~ (7-3) in form of (7-4)

Solve

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$$
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\begin{pmatrix} \dot\phi\\ \dot\omega\\ \dot\delta \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & \frac{-1}{\tau} & \frac{Q}{\tau}\\ 0 & 0 & 0 \end{pmatrix}

\begin{pmatrix} \phi\\ \omega\\ \delta \end{pmatrix} + \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} u $$

$$ \underline{A}= \begin{pmatrix} 0 & 1 & 0 \\ 0 & \frac{-1}{\tau} & \frac{Q}{\tau}\\ 0 & 0 & 0 \end{pmatrix}

, \ \underline{B}= \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}

$$
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Author and Proof-reader
[Author] Oh, Sang Min

[Proof-reader]

= Problem 8 =

We can rewrite the eqn(8.1) as (8.2).

We are familiar with this equation, as we learned already. Total derivative - Egm6321.f10_HW1_prob#1_team6

As $$\begin{align} \frac{d}{dx}h(x,y)=0 \end{align}$$, we know that $$\begin{align} h(x,y)=f(y) \end{align}$$ only. It means $$\begin{align} {h}_{x}=0 \end{align}$$

Hence, eqn(8.2) becomes,

There are two possible solutions.

1) $$\begin{align} P=y'=0 \end{align}$$

2) $$\begin{align} {h}_{y}=0 \end{align}$$

If 1) were satisfied, whole problems became zero, which is trivial. We can conclude that 2) is the solution.

As $$\begin{align} {h}_{x}=0 \end{align}$$ and $$\begin{align} {h}_{y}=0 \end{align}$$,

= Problem 9 =

= Problem 10 =

= References =

title: User talk:Egm6321.f10.team5.oh/hw3