Physics 262: Mechanics

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Physics 262 Mechanics
This course covers most of classical mechanics in
great depth
Topics include numeric calculation, basics of
Newtonian dynamics, oscillatory motion,
nonlinearity and chaos, central forces and
orbits, scattering, and motion in noninertial
reference frames.
Topics omitted Relativity (Phy 215)
Lagrangians, Hamiltonians and anything that would
be done better with those (solid bodies, normal
modes, etc) (Phy 337),
First Week Course overview, dimensional
analysis, approximations, limits, and numeric
solutions. Read the handouts by Friday Make sure
your computer has Matlab working by Friday Phy
266 students, make sure your computer has Maple
working. Homework (problems 6 and 7) are due
Tuesday (due to MLK day) at 4pm. Recommend
problems 5, 14, 15
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Materials
Textbook Analytical Mechanics, Fowles and
Cassidy additional readings will be handed out
as needed.
Other materials igtclicker bring to class
daily Matlab contact the help desk ASAP if
you have trouble installing/accessing matlab.
Bring to class when listed on the course
calendar Webpage http//bob.olin.wfu.edu/web,
and click on teaching and Phy 262 Spring
2009. The course schedule with topics and due
dates is there!
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Office Hours
When would you like to have office hours?

• Tuesday 2-3 pm
• Wednesday 230-330pm
• Friday 200-300pm
• Friday 300-400pm

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Assignments and Grading
The schedule is online any specific mention
there over-rides general deadlines
Reading Assignments Daily, with in-class
clicker-based quizzes worth 1-2 points each
Homework Due every Monday at 4pm in the graders
mailbox in Olin 100. Late assignments will be ½
off if turned in within 48 hrs of the deadline.
Each required problem is worth 10 points.
Midterm 03/06/09. 100 points, no make-ups or
alternatives.
Final 05/07/09, remaining points to make 1000
for the course. Covers the whole semester. No
make-ups or alternatives. Failure will may result
in failing the course.
There is a set of minimum cutoffs on the course
policies.
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Homework
Homework is due weekly. There will be required
and recommended problems. The required problems
will be turned in and graded.
Standards Every problem turned in must be
worked out in full detail with full work. The
work must be coherent, logically, understandable
and complete. Do NOT use Maple for symbolic
manipulations. It is your obligation to prove
that you understand the workings of your
homework the grader assumes nothing. Correct
answers with little work are worthless Correct
answers with partial work are worth partial
credit.
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Homework Expectations
Your homework is expected to be your work. You
may work in groups, however, the final answer
must be yours. You may not copy answers from
other sources.
Question Copying solutions from sources found
on the web is A) Okay and expected. B) Not
expected, but tolerated C) Cheating and will be
referred to the Deans office
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Problem Solving A few key steps

• Diagram and label the physical setup!
• This is the most important step in many kinematic
  problems.
• Makes sure you understand the physical situation
• Write down what you know and what you are trying
  to find
• Lets you figure out if something is missing
• Use equations (Fma, conservation laws, etc) to
  connect things you know and dont know
• Solve things symbolically first
• Check out dimensions
• Check limiting cases
• If you need a number, solve numerically
• Check if the answer is reasonable
• Units are often an issue!

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Dimensional analysis
Your answer needs to have the right powers of
mass, length and time (or equivalently kg, m, s).
Otherwise, your answer is nonsensical. You can
often figure out forms for expressions and/or
find Where errors have been made by checking
dimensions. Example Waves moving in a fluid,
what sorts of expressions are valid for the wave
speed. The mechanical properties of a fluid are
its density and bulk modulus (F/A) and we want
speed. So lets work this out.
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Problem 1

• Lets suppose you have a vibrating water drop
  whose frequency of oscillation depends on its
  radius, R, density, r, and surface tension, s.
• What is the dependence?
• sqrt(s/(r R3)
• sqrt(r R3/s)
• s/R3
• R3/s
• s/r

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Problem 2

• Lets suppose you have a vibrating star whose
  frequency of oscillation depends on its radius,
  R, density, r, and Newtons Gravitational
  constant, G
• What is the dependence?
• sqrt(G/(r R3)
• sqrt(r R3/G)
• sqrt(GR3)
• sqrt(Gr)
• G/r

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Questions to think about
Why did we have s in one problem and G in the
other? Why could we use polynomials ? When
could more complex functions be used?
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Approximations
Approximations are ubiquitous in physics. Doing
physics well is usually about finding the write
approximations First approximation,
non-relativistic Newtonian mechanics. Within
this set of approximations, the most common
approximation is to assume that something is
small. What can we do if things are small?

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Example Approximations and Limiting cases
Lets suppose a ball is dropped from rest from a
height, h. Assume that the drag force on the ball
is linear in velocity, F-mav . We will
eventually figure out the velocity and position
as a function of time as v(t)(-g/a)(1-e-at) and
y(t)-(g/a)(t-(1/a)(1-e-at)) What should I do to
check that this is a reasonable result (assuming
that I have already checked dimensions)?
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Example Approximations and Limiting cases
Check to make sure I have the physically correct
behavior at short and long time. At short time,
what is e-at ? v(t)(-g/a)(1-e-at) and
y(t)h-(g/a)(t-(1/a)(1-e-at))
Does this make sense?
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Example Approximations and Limiting cases
Check to make sure I have the physically correct
behavior at short and long time. At long time,
what is e-at ? v(t)(-g/a)(1-e-at) and
y(t)h-(g/a)(t-(1/a)(1-e-at))
Does this make sense?
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Solving differential equations numerically
Most equations we deal with in physics are
differential equations. These equations cannot
always be solved analytically, so we need to
solve them numerically. How? By using a Taylor
series . Simplest approach (Eulers) v(tDt)v(t
)(dv/dt)Dt x(tDt)x(t)(dx/dt)Dt
Or v(tDt)v(t)a(t)Dt x(tDt)x(t)v(t)Dt
Explicitly assuming that higher order terms
can be ignored Whats the truncation error
here?
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Solving differential equations numerically
Look closely, v(tDt)v(t)a(t)Dt
x(tDt)x(t)v(t)Dt This is a generic
expression, the potential underlying the system
does not appear! Also, you need to start
somewhere, at t0. There is an explicit
dependence on initial conditions. Lets look at
an example in matlab. Take our your computers
now. (constant and harmonic)
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Solving differential equations numerically
Eulers algorithim is actually quite poor,
although probably good enough for most of what we
will do. Simple improvement Euler-Cromer
v(tDt)v(t)a(t)Dt x(tDt)x(t)v(tDt )Dt
Slightly more complex (simplest one actually
used) v(t0.5Dt)v(t-0.5Dt)a(t)Dt
x(tDt)x(t)v(t0.5Dt )Dt Can you do this
one?