Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations please visit my website at
http://www.solohermelin.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations please visit my website at
http://www.solohermelin.com.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
We explain a methodology to compute families of homo/heteroclinic connections between periodic orbits. We show the relation of some homoclinic connections with resonant transitions in the RTBP
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
We explain a methodology to compute families of homo/heteroclinic connections between periodic orbits. We show the relation of some homoclinic connections with resonant transitions in the RTBP
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Gravitational field and potential, escape velocity, universal gravitational l...lovizabasharat
What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples
Elasticity, Plasticity and elastic plastic analysisJAGARANCHAKMA2
It is actually the basis of structural engineering to study elasticity and plasticity analysis. So people who are also studying in various fields of structure and need to analyze finite element analysis also need to study this basis.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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3. Problem 1:
A Smooth straight thin tube
revolves with uniform angular velocity ‘𝜔′ in a
vertical plane about one extremity which is fixed; if
at zero time the tube be horizontal and a particle
insides it beat a distance ‘a’ from the fixed end,
and be moving with velocity V along the tube,
show that the distance at time ‘t’ is
a cos ℎ 𝜔𝑡 +
𝑉
𝜔
−
𝑔
2𝜔2
sin ℎ 𝜔𝑡 +
𝑔
2𝜔2
sin ℎ 𝜔𝑡
5. Let at time t, P be the position of the particle of mass m on the tube OB. The forces
acting at P are (i) its weight mg vertically downwards and (ii)normal reaction R
perpendicular to OB
Let P be (𝑟, 𝜃)
Angular Velocity= 𝜃 =
𝑑𝜃
𝑑𝑡
= 𝜔
Integrating,
𝜃 = 𝜔𝑡 + 𝐴
Initially when t=0,𝜃 = 0
𝜃 = 𝜔𝑡 -------(1)
Resolving along the radius vector OB
m( 𝑟 − 𝑟 𝜃2) = −𝑚𝑔 cos (90 𝑜 − 𝜃) = −𝑚𝑔 sin 𝜃
𝑟 − 𝑟𝜔2 = −𝑔𝑠𝑖𝑛𝜃 = −𝑔𝑠𝑖𝑛𝜔𝑡 (using (1))
𝐷2 − 𝜔2 𝑟 = −𝑔𝑠𝑖𝑛𝜔𝑡 ---------(2) Where D=
𝑑
𝑑𝑡
The complementary function Y is found such that
(𝐷2
− 𝜔2
)𝑌 = 0
6. The solution of this differential equation is
𝑦 = 𝐴𝑒 𝜔𝑡 + 𝐵𝑒−𝜔𝑡---------(3)
Where A and B are constants. The particular integral u of the equation (2) is
given by
𝐷2 − 𝜔2 𝑢 = −𝑔𝑠𝑖𝑛𝜔𝑡
𝑢 = −
𝑔
𝐷2−𝜔2 𝑠𝑖𝑛𝜔𝑡 = −
𝑔
𝜔2−𝜔2 𝑠𝑖𝑛𝜔𝑡 =
𝑔
2𝜔2 𝑠𝑖𝑛𝜔𝑡 -----------(4)
Hence the general of (2) is
𝑟 = 𝑌 + 𝑢 = 𝐴𝑒 𝜔𝑡 + 𝐵𝑒−𝜔𝑡+
𝑔
2𝜔2 𝑠𝑖𝑛𝜔𝑡 ------------(5)
The initial conditions are : when t=0,r=a and 𝑟 = 𝑉
Hence (5) gives A+B=a --------(6)
Differentiating (5)
𝑟 = 𝐴𝜔𝑒 𝜔𝑡
− 𝐵𝜔𝑒−𝜔𝑡
+
𝑔
2𝜔
cos 𝜔𝑡---------(7)
8. Problem 2:
Find the law of force
towards the pole under which the
curve 𝑟 𝑛
= 𝑎 𝑛
cos 𝑛𝜃 can be
described.
9. Solution
Given curve, 𝑟 𝑛
= 𝑎 𝑛
cos 𝑛𝜃
Since 𝑟 =
1
𝑢
, the equations is
un 𝑎 𝑛 cos 𝑛𝜃 = 1 ------(1)
Taking log on both sides
𝑛 log 𝑢 + 𝑛 log 𝑎 + log cos 𝑛𝜃 = 0-----(2)
Differentiating (2) with respect to 𝜃 ,
𝑛.
1
𝑢
𝑑𝑢
𝑑𝜃
−
𝑛𝑠𝑖𝑛 𝑛𝜃
cos 𝑛𝜃
= 0
(ie)
𝑑𝑢
𝑑𝜃
= 𝑢 tan 𝑛𝜃-------(3)
Differentiating (3) with respect to 𝜃𝑑2
𝑢
𝑑𝜃2
= 𝑢 𝑛 sec2 𝑛𝜃 + tan 𝑛𝜃 .
𝑑𝑢
𝑑𝜃
𝑢 +
𝑑2 𝑢
𝑑𝜃2 = 𝑢 + 𝑛𝑢 sec2
𝑛𝜃 + 𝑢 tan2
𝑛𝜃
= 𝑛𝑢 sec2
𝑛𝜃 + 𝑢(1 + tan2
𝑛𝜃)
10. = 𝑛𝑢 𝑠𝑒𝑐2 𝑛𝜃 + 𝑢 𝑠𝑒𝑐2 𝑛𝜃
= 𝑛 + 1 𝑢 𝑠𝑒𝑐2
𝑛𝜃
= 𝑛 + 1 𝑢. 𝑢2𝑛 𝑎2𝑛 using (1) to substitute for sec2 𝑛𝜃
= 𝑛 + 1 𝑎2𝑛 𝑢2𝑛+1
P∝
1
𝑟2𝑛+3 which means that the central acceleration varies inversely as the(2n+3) rd
power of the distance.
∎
11. Problem 3:
Find the law of force to an
internal point under which a
body will describe a circle.
12. Solution :
From the pedal equation of the circle for general position of the pole is 𝑐2
=
𝑟2 + 𝑎2 − 2𝑎𝑝 -----(1)
Differentiating with respect to r,
0 = 2𝑟 − 2𝑎
𝑑𝑝
𝑑𝑟
(i.e)
𝑑𝑝
𝑑𝑟
=
𝑟
𝑎
Now the central acceleration
𝑃 =
ℎ2
𝑝3
𝑑𝑝
𝑑𝑟
substituting for p from (1) ∎
13. Problem 4:
A particle moves in
an ellipse under a force which is
always directed towards its focus. Find
the law of force, the velocity at any
point of the path and its periodic time.
14. Solution
The polar equation to the ellipse is
𝑙
𝑟
= 1 + 𝑒𝑐𝑜𝑠𝜃 -----(1)
where e is the eccentricity and 𝑙 is the semi latus rectum.
From (1) 𝑢 =
1
𝑟
=
1+𝑒 cos 𝜃
𝑙
Hence
𝑑𝑢
𝑑𝜃
= −
𝑒𝑠𝑖𝑛𝜃
𝑙
and
𝑑2 𝑢
𝑑𝜃2 = −
𝑒𝑐𝑜𝑠𝜃
𝑙
𝑢 +
𝑑2
𝑢
𝑑𝜃2
=
1 + 𝑒 cos 𝜃
𝑙
−
𝑒 cos 𝜃
𝑙
=
1
𝑙
We know that
𝑃
ℎ2 𝑢2
= 𝑢 +
𝑑2 𝑢
𝑑𝜃2
=
1
𝑙
i.e The force varies inversely as the square of the distance from the pole.
1
𝑝2 = 𝑢2
+
𝑑𝑢
𝑑𝜃
2
=
1 + 𝑒𝑐𝑜𝑠𝜃
𝑙
2
+
𝑒𝑠𝑖𝑛𝜃
𝑙
=
1 + 2𝑒𝑐𝑜𝑠𝜃 + 𝑒2
𝑙2
Hence 𝑣2
=
ℎ2
𝑝2 =
ℎ2(!+2𝑒𝑐𝑜𝑠𝜃+𝑒2)
𝑙2
15. =
𝜇𝑙
𝑙2 (1 + 𝑒2 + 2
𝑙
𝑟
− 1 ) substituting for 𝑒𝑐𝑜𝑠𝜃 from (1)
=
𝜇
𝑙
𝑒2
+
2𝑙
𝑟
− 1
=
𝜇
𝑙
(
2𝑙
𝑟
− 1 − e2
)
= 𝜇[
2
𝑟
−
1−𝑒2
𝑙
)------(2)
Now if a and b are the semi axes of ellipse. We know that
𝑙 =
𝑏2
𝑎
=
𝑎2(1 − 𝑒2)
𝑎
= 𝑎(1 − 𝑒2
)
𝑣2 = 𝜇
2
𝑟
−
1
𝑎
, giving the velocity v.
Areal velocity in the orbit =
1
2
ℎ and this is constant.
The total area of the ellipse = 𝜋𝑎𝑏.
Periodic Time T=
𝜋𝑎𝑏
(
1
2
ℎ)
=
2𝜋𝑎𝑏
ℎ
=
2𝜋𝑎𝑏
√𝜇𝑙
where 𝜇 =
ℎ2
𝑙
=
2𝜋𝑎𝑏
𝜇.𝑏
. √𝑎, since 𝑙 =
𝑏2
𝑎
=
2𝜋
√𝜇
. 𝑎3/2 ∎
16. Problem 5:
A particle moves in a
curve under a central attraction so
that its velocity at any point is equal to
that in a circle at the same distance
and under the same attraction. Show
that the path is an equiangular spiral
and that the law of force is that of the
inverse cube.
17. Solution
Let the central acceleration be P. If v is the velocity in a circle at a distance r under the
normal acceleration P, then
𝑣2
𝑟
= 𝑃
i.e 𝑣2
= 𝑃𝑟 -------(1)
Since v is also the velocity in the central orbit,
ℎ = 𝑝𝑣 or 𝑣 =
ℎ
𝑝
putting this is (1),
ℎ2
𝑝2 = 𝑃𝑟 ------(2)
We know that, 𝑃 =
ℎ2
𝑃3 .
𝑑𝑝
𝑑𝑟
----------(3)
Substituting (3) in (2)
ℎ2
𝑝2 =
ℎ2
𝑝3 .
𝑑𝑝
𝑑𝑟
. 𝑟
(i.e)
𝑑𝑝
𝑝
= 𝐴
Substituting this in (3),
𝑃 =
ℎ2
𝑝3 . 𝐴 =
𝐴ℎ2
𝐴3 𝑟3 using (4)
=
ℎ2
𝐴2 (
1
𝑟3) (ie)𝑃 ∝
1
𝑟3 ∎