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Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor [email protected] Course website:...
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![Page 1: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/1.jpg)
Analytical mechanicsFall 2015S. Barraza-Lopez. Instructor
Course website:http://comp.uark.edu/~sbarraza/AM.html
![Page 2: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/2.jpg)
Syllabus (available at http://comp.uark.edu/~sbarraza/AM.html)
• Chapters 1 to 12. Marion and Thornton, Fifth Edition.
• Three cumulative tests: -Monday September 28 (50 minutes)-Monday November 2 (50 minutes)-Monday December 14 (2 hours)
• About 14 homeworks to be assigned on Monday, and due Monday of next week
![Page 3: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/3.jpg)
Grading Policy:
• Each exam: 20% of your grade• HW: 35% of your grade
• Assistance: 5%
Grading scores:86-100: A 71-85: B 56-70: C 40-55: D
![Page 4: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/4.jpg)
HW 1 assigned (12 problems: quick review of vector analysis
and matrix alegebra)
• Available at:http://comp.uark.edu/~sbarraza/AM.html
Due on Friday August 28.
![Page 5: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/5.jpg)
Vectors
• A scalar is a single quantity (a number or a single-valued function), but a vector is a collection of numbers or/and functions.
Notation:A vector will usually be represented by a bold letter: (e.g., r), by a regular letter with a “vector sign” on top (e.g., Ā).
(this vector has three entries)
![Page 6: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/6.jpg)
Dot product and vector length
• Given two vectors a and b with the same number of entries, we define a “dot product” as:
• Similarly, we define the length of a vector v as .
![Page 7: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/7.jpg)
A vector with n-entries
![Page 8: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/8.jpg)
“Unit” vectors
• A unit vector is usually written with a “hat” symbol, and it is defined as follows:
• An important example of unit vectors are those that indicate orthogonal vectors…
![Page 9: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/9.jpg)
“Unit” vectors
• A unit vector is usually written with a “hat” symbol, and it is defined as follows:
• An important example of unit vectors are those that indicate orthogonal vectors…
![Page 10: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/10.jpg)
“Unit” vectors
• So in three-dimensions, these special unit vectors are:
• Question: express r in terms of i, j, and k.
![Page 11: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/11.jpg)
Vector algebra
• r=x1i+x2j+x3k, is an example of vector algebra.
What is ar where a is a scalar? What is ar when a=0?
A+b = b+A? (why? Or why not?)
![Page 12: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/12.jpg)
Towards matrix algebra: Rotation of coordinates
Galileo once stated that the laws of mechanics should not depend on the choice you make of your coordinate
axes (this is known as Galilean relativity).
This means that the (right-handed) set of vectors on this figure are not “special:” others sets of vectors will
describe mechanical phenomena just as well.
![Page 13: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/13.jpg)
![Page 14: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/14.jpg)
So we have that k’=k (the axis that is kept
unchanged is the axis of rotation)
We also see that i’, j’ are linear combinations of i, j.
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![Page 16: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/16.jpg)
![Page 17: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/17.jpg)
• We will address the conditions this matrix must satisfy in order to be a rotation matrix next time.
![Page 18: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/18.jpg)
As a way to conclude:
• Most difficulties faced will be related to learning the notation.
• Please make a few notes with the definitions introduced today.
• Always seek clarification, if in doubt.
![Page 19: Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor sbarraza@uark.edu Course website: sbarraza/AM.html.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfc71a28abf838ca79b4/html5/thumbnails/19.jpg)
Next time: Basic calculus on vector quantities