Notes for Lecture 19
Inertia Tensor, Non-linear Physics
19.1
Inertia tensor
Before going on to the non-linear physics, let us look at some typical inertia tensors
and their derivations.
1
19.1. INERTI
Greens function method
Phys. 105, UCSC, F2011
Greens function method for SHO
This is an optional topic the method of Greens function for solving the SHO
problem highly recommended for your reading, if
Notes for Lecture 6
Small Oscillations
We consider small oscillations around a stable equilibrium point. We assume that
the second derivative of the potential is non-zero. The resulting motion is a si
Notes for Lecture 5
Conservation principles and 1D
motions
5.1
Conservation theorems
It is not possible to overemphasize the importance of the principle of conservation. In
classical mechanics, these
Notes for Lecture 4
Lorentz force
4.1
Example 2.10 of text
Motion of a charge in a uniform B eld.
B = B0 z . (This is a more common convention. The textbook chooses the y
direction.) The Lorentz force
Notes for Lecture 3
Perturbation, air resistance (cont.)
3.1
Perturbation
As a general strategy to solve a real world problem, the perturbation theory is very
important. Please read Appendix A (and yo
Problems for rigid body
Phys. 105, UCSC, F2011
Problems for your study (especially the rst two problems). NOT due.
Problem 1 Consider a disk with a small hole cut out from it. It is a thin disk, whose
Homework 8
Phys. 105, UCSC, F2011
Due Nov. 29, Tuesday Dec. 1, Thursday.
70 points total. 30 points for the extra credit (but you are advised to understand all
problems).
Problem 1 (20 points) Find th
Notes for Lecture 7
Driven SHO
7.1
Particular solution, complementary function
Suppose that a force, F (t), drives a SHO.
m + bx + kx = F (t)
x
2
x + 2 x + 0 x = f (t)
Lx = f (t)
def
f (t) = F (t)/m
d
Notes for Lecture 9
Symmetry and conservation
Conservation principles are very fundamental in physics. Most conservation principles
that we know arise from symmetry.
9.1
Symmetry principles
Symmetry i
Notes for Lecture 18
Coupled Oscillators and Rigid
Body
Let us look at couple more examples about coupled oscillators. Then, we cover very
basic stu on rigid body.
18.1
Coupled oscillator example 2
Th
Notes for Lecture 17
Coupled Oscillators, cont.
You should read the previous lecture note carefully, if you have not.
Essentially, the problem of coupled oscillators is to reduce a problem that seems
Notes for Lecture 16
Coupled Oscillators
We discuss one of the most important general topics of mechanics, the problem of
coupled oscillators. We will approach this problem using a simple example, as
Notes for Lecture 15
Scattering
Here, we describe two particle collisions and then introduce the concept of the scattering crossection. The terms collision and scattering are used interchangeably
in t
Notes for Lecture 14
Kepler problem and many body
problem
14.1
Elliptical orbits
14.1.1
Turning points and angular momentum conservation
Let us come back to the discussion of elliptical orbits, applic
Notes for Lecture 13
Central force and the Kepler
problem
Central force problems are important. All planet motion problems are of this kind.
Also, crude but useful classical mechanics models of atoms
Notes for Lecture 11
Eective potential, gravity
11.1
Eective potential
Let us consider a problem for which the Lagrangian is not explicitly dependent on
time. So, the Hamiltonian is conserved. Suppose
Homework 7
Phys. 105, UCSC, F2011
Due Nov. 17, Thursday.
Problem 1 (20 points) Consider a cylindrical shell with uniform density and innite
length. The cylinder parallel to the z direction. Mass exist
Homework 6
Phys. 105, UCSC, F2011
Due Nov. 10, Thursday.
Problem 1 (10 points) Mechanical similarity.
(a) Show that scaling the Lagrangian
L (q, q, t) = CL(q, q, t)
where C is a non-zero constant, doe
Homework 5
Phys. 105, UCSC, F2011
Due Nov. 1, Tuesday.
Problem 1 (5 points) A general property of the Lagrangian formalism. Consider a
Lagrangian L, and another Lagrangian L , which is related to the
Notes for Lecture 1
Newtons laws
In general, my notes are not complete. You should read both my notes and the
textbook.
Chapter 1 materials will be spread around and parts of them will be taken up
dur
Problems for rigid body and collisions
Phys. 105, UCSC, F2010
Problems for your study. NOT due.
Problem 1 Consider a disk with a small hole cut out from it. It is a thin disk, whose
thickness is negli
Homework 7
Phys. 105, UCSC, F2010
Due Nov. 23, Tuesday.
Problem 1 (20 points) Consider the example that was studied in Lecture 14 and
summarized in pages 9 and 10 of LN 14.
(a) Assume that the initial
Homework 6
Phys. 105, UCSC, F2010
Due Nov. 16, Tuesday.
Problem 1 (30 points) Consider problem 5 of the pre-midterm. It involves three
masses: m (mass on string, pulled by gravity), M (block on a fric
Homework 5
Phys. 105, UCSC, F2010
Due Nov. 2, Tuesday.
Problem 1 (10 points) Fermats principle and the PoLA. Let us consider a
photon, the quantum of light, with the angular frequency . While the stud
Homework 4
Phys. 105, UCSC, F2010
Due Oct. 26, Tuesday.
Problem 1 (10 points) [Reading this problem is as important/easy as doing this
problem.] Here, we shall consider a subtle but important point ab
Homework 3
Phys. 105, UCSC, F2010
Due Oct. 19, Tuesday.
Problem 1 (10 points) Near an unstable equilibrium point, x0 , of a one dimensional
motion, the potential energy can be written as U (x) = U0 1