PHYS 3202 Classical Mechanics II - Quiz #2
Please write your name.
Conceptual (circle ALL that apply - there may be
one or more correct items in each problem)
1. (4 points) The Hamiltonian has the same dimension as
A. the product of someones mass and heig
PHYS 3202 Classical Mechanics II - Quiz #1 Solutions
1.
2.
3.
4.
5.
6.
BD (correction: problem 1 has 5 points; 45 points in total for quiz 1)
No
AC
BCE
C
BC
Problem 7: The potential energy is
r
U (r) =
Gm2
r2
dr =
Gm2
.
r
In the center of mass frame the
PHYS 3202 Classical Mechanics II - Quiz #1
Please write your name.
Conceptual (circle ALL that apply - there may be
one or more correct items in each problem)
1. (4 points) Several stars are interacting gravitationally, and have nonzero total
momentum. If
Curvature Vector: from Newtons Second Law to Beam
Deection and to Relativity
Shina Tan
1
Curvature Vector
Any point r on a curve can be parameterized according to the length s from some starting
point on the curve to point r. We dene
=
s
dr
= r (s).
ds
(1
Solution to Eq.(6.36) of Thornton & Marion, 5th Edition
Shina Tan
Many problems appear dicult at rst sight. If you attack them directly, some of them
will yield beautiful solutions, so that in hindsight they do not seem hard.
In the problem of minimizing
A Corollary of Eulers Equations
Shina Tan
1
The Corollary
From Eulers equations
f
d f
= 0,
yi dx yi
we can derive an important Corollary:
d
dx
i = 1, 2, , n,
n
f
yi
i=1
f
yi
=
f
.
x
(1)
In particular, if the function f (y1 , y1 ; y2 , y2 ; ; yn , yn ; x)
What Are Tensors?
Shina Tan
Geometry in Euclids time had nothing to do with coordinates. A point is just a point,
not a set of numbers (coordinates). A surface is just a surface, not an equation. A circle is
a perfect beautiful you know what curve, not th
Tensor exercise - Solutions
Problem 1: Although
dI
dt
= 0 in general, show that
Solution: Since I =
3
i=1 Ii ei ei ,
dI
=
dt
dI
= 0.
dt
and Ii = 0, we have
(Ii ei ei + Ii ei ei + Ii ei ei )
i
(Ii ei ei + Ii ei ei )
=
i
Ii ( ei )ei + Ii ei ( ei ).
=
i
Now
Pre-class exercise (not graded)
Problem 1: Although
dI
dt
= 0 in general, show that
dI
= 0.
dt
Hints: the inertia tensor takes the form
3
I=
Ii ei ei .
i=1
The principal moments of inertia I1 , I2 , I3 remain constant in time, since the rigid
body has a
Finding the period of some oscillation
Suppose that we obtain the following equation of motion for a physical system:
= 2 sin .
(1)
If (t) oscillates with a small amplitude, we have sin , and the solution is simple:
(t) = A sin(t + ), with the amplitude
PHYS 3202 Classical Mechanics II - Homework #6
Due at 12:05pm, Friday March 11, 2011 in the class
In homework, quiz, and nal exam, please show intermediate steps of
your calculations. Whenever appropriate, you may draw diagram(s).
Problem 1 (10 points): F
PHYS 3202 Classical Mechanics II - Homework #5
Due at 12:05pm, Friday March 4, 2011 in the class
In homework, quizzes, and nal exam, please show intermediate steps
of your calculations. Whenever appropriate, you may draw diagram(s).
Problem 1 (10 points):
PHYS 3202 Classical Mechanics II - Homework #4
Due at 12:05pm, Friday Feb 25, 2011 in the class
In homework, quizzes, and nal exam, please show intermediate steps
of your calculations. Whenever appropriate, you may draw diagram(s).
Do Problems 7.5, 7.7, 7
PHYS 3202 Classical Mechanics II - Homework #4 Hints and
Clarications
Problem 7.5: A is positive, and the additional force is attractive toward the
origin.
Problem 7.7: Try to simplify your expression for the kinetic energy rst. Depending on your choice o
PHYS 3202 Classical Mechanics II - Homework #3
Due at 12:05pm, Friday Feb 11, 2011 in the class
In homework, quizzes, and nal exam, please show intermediate steps
of your calculations. Whenever appropriate, you may draw diagram(s).
If a problem is dicult,
PHYS 3202 Classical Mechanics II - Homework #3
Additional Hints
Problem 1. Part 1- The circle is
r = constant,
z = another constant
in the cylindrical coordinate system.
See Appendix F.2 of Thornton and Marion, 5th Edition for information about the
cylind
PHYS 3202 Classical Mechanics II - Homework #2
Due at 12:05pm, Friday Feb 4, 2011 in the class
In homework, quizzes, and nal exam, please show intermediate steps
of your calculations. Whenever appropriate, you may draw diagram(s).
Problem 1 (10 points): T