Analytical Mechanics Problem Set #1

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```					 Physics 330                                                                            Fall (3) 2006
Analytical Mechanics: Problem Set #1
Math Methods and Newtonian Mechanics
r             r    r dpr     r
∂r         1 ∂r              dv r dm
hi =     ; ei =
ˆ           ; F=    =m    +v
∂ui        hi ∂ui      dt    dt    dt

Due: Sunday Dec. 3 by 1 pm
€
Reading assignment: for Wednesday,€ 1.9-1.16 (vectors and coordinate systems)
€
for Thursday,   2.1-2.4 (equations of motion from Newton's 2nd law)
for Friday,     2.4-2.7 (retarding forces and conservation laws)
9.11      (rockets)

Overview: We will begin our study of analytic mechanics with a brief review of vectors and
associated math methods. We then launch into the Newtonian approach to classical mechanics.
The "Newtonian program" is summarized in Newton's second law F=ma (or, more correctly, as
given above). A determination of the forces gives the accelerations; velocities and positions are
then computed via integration. Newton's laws completely describe all of the phenomena of
classical mechanics and, at this point in your careers, I'm sure you all are intimately familiar with
them. What you may not be aware of is that the Newtonian approach is neither the most
convenient nor the most general formalism. We will be learning other quite elegant formulations
of classical mechanics in this course. However, there are still many cases where good ole F=ma
remains the method of choice. Note that all formulations of mechanics, from the pedestrian to
the polished, lead to the same equations of motion. Unfortunately, these differential equations
are often intractable and thus one must resort to perturbation techniques or numerical methods.
You should get used to turning to the computer as a tool in solving problems for this class. One
can only get so far with analytic techniques and the "real" world is a complicated place! But
never fear, I won't deny you the delight of elegant analytical solutions. Note however that even
for these, it is often quite instructive to make plots of your final results to get a better feeling for
the physical behavior described by your equations.

In-Class Problems:
Wednesday       1.9 (vector manipulations) [SD]
1.10 (velocity and acceleration for an elliptic orbit) [PD]
1.40 (analysis of a surface z(x,y)) [RW]
1.41 (perpendicular vectors) [NT]
Thursday        2.14 (shooting uphill) [SD and PD]
2.32 (blocks on an inclined plane) [RW and NT]
Friday          2.12 (particle in drag) [PD and SD]
2.25 (block on a track) [NT and RW]

(over for problem assignment)
Problem assignment:

1.25 (acceleration in spherical coordinates)
1.26 (velocity and acceleration for constrained motion)
2.13   (particle subject to a weird resistive force)
2.17   (home run hitter)
2.18   (home runs are harder to hit with air friction ... use Newton, submit plots)
2.23   (complicated force, but elementary integrals, plot your results)
2.28   (the superball and the marble)
2.43   (determine the potentials, plot U(x)/k)
9.58 (rocket launch)                                                        L

A.1 Peg and the pendulum: A pendulum with mass m and length L is                            d
released from rest from a horizontal position. When the pendulum
attains a vertical orientation, it encounters a peg, located a distance d
below the pivot point. This peg causes the pendulum to move along a
peg
circular path (of radius L–d) as shown in the figure. Find the
minimum distance d such that the pendulum will swing completely                                 m
around the circle.

Bonus: 2.42 (stability of a cube on a cylinder)

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