Analytical Mechanics Fowles 7th Edition Chapter 2 Solutions
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Let p = line density of chain; let x be length of chain in overhang;
Let v be velocity of chain or dx/dt; L…more Answer to question on "Analytical Mechanics"
Let p = line density of chain; let x be length of chain in overhang;
Let v be velocity of chain or dx/dt; Let m be mass of chain in overhang at a particular time; acceleration is dv/dt Let ^ be an index of a quantity raised to a power. g is acceleration due to gravity
In this question we have an equation of motion applied to a variable mass.
Initially m=p(b+x); Force=d/dt(mv) = {dm/dt.v + m. dv/dt}
F=p(b+x).g
P(b+x).g= d/dt{p(b+x).v} Hence: (b+x).g={v.v + (b+x).dv/dt}
Now dv/dt=d/dx(1/2 v^2) where v^2 = v squared etc.
(b+x).g = 2.(v^2) + (b+x). d/dx(v^2)
(b+x) d/dx(v^2) + 2.v^2 = 2(b+x).g
d/(v^2) + 2.(v^2)/(b+x) = 2.g Put y=v^2 for brevity.
dy/dx + 2.y/(b+x) = 2.g This is a first order Ordinary Differential Equation.
The required integrating factor is e raised to the integral ( w.r.t. x)of 2/(b+x)
This is (b+x)^2. Hence:
dy/dx.(b+x)^2 + 2(b+x).y = (b+x)^2. multiplied by 2g. Hence:
d/dx{y.(b+x)^2} = (b+x)^2 multiplied by 2g
Therefore: y.(b+x)^2 = { (b+x)^3} divided by 3 multiplied by2g + C
where C is a constant of integration.
Now when x=0; y=0 This implies C = -b^3 multiplied by2g /3
Therefore y.(b+x)^2 = (b+x)^3 mult by2g/3 -2b^3g/3
Now 0
Therefore y when x =(a-b) is (i.e. chain leaves bench)
y= 1/(a^2). {a^3 .2g/3 - 2b^3 mult by g/3} { y=v^2}
Therefore v^2 =1/(a^2 - 2g/3{a^3-b^3}
Hence v^2 = 2g/3(a^2) . times {a^3-b^3} OR
v=speed =Square root of above expression.
Bill Oliver # 957335
(The denominator is 3 times"a" squared, not" a" cubed)(less) (hide spoiler)]
Community Reviews
Newton's Laws of Motion
Harmonic motion
Kepler's Laws
Rigid bodies
Lagrangian Mechanics
The authors use real world examples such as falling raindrops to illustrate various concepts. The problems at the end of each chapter are quite challenging and interesting. There are also some numerical problems in the book, that should be solved with a comput
Mechanics was one of the first (if not the first) courses I followed in University. "Analytical Mechanics" is a brilliant junior level book that covers: Newton's Laws of Motion
Harmonic motion
Kepler's Laws
Rigid bodies
Lagrangian Mechanics
The authors use real world examples such as falling raindrops to illustrate various concepts. The problems at the end of each chapter are quite challenging and interesting. There are also some numerical problems in the book, that should be solved with a computer. In my opinion there could have been more of those. The study group, that I was part of, ignored the "computer" problems completely.
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John Calvin
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His work was so clear concise and well written , that it emphatically induced me to lean towards physics. When I became an assistant professor of physics , I used his style of conceptual presentation to help my students to grasp the subtle theoretical concepts of mechanic When I was an undergraduate physics student we used Grant R Fowles Analytical mechanics text; prior to the introduction to this work I was caught up in the delimima as weather a would get a physics major or an engineering major.
His work was so clear concise and well written , that it emphatically induced me to lean towards physics. When I became an assistant professor of physics , I used his style of conceptual presentation to help my students to grasp the subtle theoretical concepts of mechanics. ...more
I found the text book useful for provision of additional examples when I was studying Applied Maths 2
during the maths degree course. The tutor recommended it.
Also useful were the texts:
"Mechanics" by R.C. Smith and P. Smith published by Wiley originally in 1968 This had a useful chapter on orbits.
"Theory of Mechanics" by K.E. Bullen - Science Press -originally in 1965. (Bill Oliver) #957335
I found the text book useful for provision of additional examples when I was studying Applied Maths 2
during the maths degree course. The tutor recommended it.
Also useful were the texts:
"Mechanics" by R.C. Smith and P. Smith published by Wiley originally in 1968 This had a useful chapter on orbits.
"Theory of Mechanics" by K.E. Bullen - Science Press -originally in 1965. (Bill Oliver) #957335
...more
It provides a decent amount of mathematical rigor without being too overwhelming. The minimum background necessary for an adequate understanding of the material is multivariable calculus. It's helpful if you've had a semester of ODE, linear algebra, and complex analysis, but it's not cruci
We used this book in my junior-level classical mechanics course. My initial impression is that it is a great introduction to mechanics for students who have studied calculus-based physics at the freshman level.It provides a decent amount of mathematical rigor without being too overwhelming. The minimum background necessary for an adequate understanding of the material is multivariable calculus. It's helpful if you've had a semester of ODE, linear algebra, and complex analysis, but it's not crucial since the book contains several appendices that help with the extra, specific maths needed. I personally hadn't taken linear algebra or complex analysis when I took this course.
The general topics covered include
1. a review of vectors and vector concepts,
2. newtonian mechanics in one, two, and three dimensions,
3. non-inertial reference systems,
4. gravitation and central forces,
5. dynamics of systems of particles,
6. rigid-body dynamics in two and three dimensions,
7. Lagrangian and Hamiltonian mechanics, and
8. oscillating systems with arbitrary degrees of freedom
in that order.
Every chapter begins with a thoughtful quote to prime your brain juices with insightful questions and conceptual descriptions, and whenever possible, each chapter provides a historical context of the physics to be analyzed. Less frequently (but just as insightful) the authors sometimes go off on a bit of a tangent in certain sections to discuss what might be classified as philosophy of physics and/or philosophy of science. For those who (like me) enjoy more than just the physics itself, you'll appreciate these little nuggets of wisdom scattered throughout the text. They were a joy to read.
Every chapter ends with section of problems, which itself is further divided into two subsections: (1) analytical problems meant to be solved by hand and (2) numerical problems to be solved with software like MATLAB, Mathematica, Maple, etc. The problems ranged in difficulty from both ends of the spectrum with no indicator of how challenging it is intended to be. Overall, they are helpful exercises needed to solidify the physics learned from reading the chapter. Unfortunately, only a small selection of problems from each chapter has a solution in the back of the the book.
The actual meat (aka the physics) of each chapter is usually presented in a relatively straightforward manner. In general, the sections begin with derivations and explanations followed by examples. This pattern is followed throughout the entirety of the text. Several chapters also contain sections dealing with numerical problem solving with Mathematica. The only downside I found was that it sometimes feels like it does too much of the teaching through examples. What I mean is that for a lot of equations it only shows you how to apply them in very specific example scenarios. But what do I know? Maybe this is the best pedagogical approach to teaching undergraduate level mechanics.
My final take on this book is that it provides a good, gentle introduction into the world of classical mechanics beyond the freshman level. It is not too dumbed down considering the relatively low level of math it demands; yet it does not try to go beyond what a typical undergraduate student is capable of. I would recommend this book as an introductory text for classical mechanics.
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Analytical Mechanics Fowles 7th Edition Chapter 2 Solutions
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