This document contains 5 problems related to motion under central forces. Problem 1 involves a particle inside a rotating tube and derives an expression for the distance of the particle over time. Problem 2 finds the law of force for a curve defined by r^n = a^n cos(nθ). Problem 3 finds the law of force for circular motion. Problem 4 finds the law of force, velocity, and period for elliptical motion under a central force directed toward the focus. Problem 5 shows that motion under a central force where velocity equals circular motion at the same distance results in an inverse cube law of force and an equiangular spiral path.
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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
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Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
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 ∎