AA 242A Homework 1 Solution
September 30, 2014
1
1
Problem 1
Consider the vector A = Ax i + Ay j + Az k . Find the components A1 , A2 ,
A3 of the vector A in a skewed coordinate system whose axes have directions
specied by the following unit vector triad:

AA 242A Homework 3 Solution
October 22, 2014
1
1
Problem 1
Diagram of the mass in the tube, spinning with angular velocity . The
normal force, Fn and frictional force, Ff are shown as well.
Let us rst form the acceleration of the particle m:
= r
p
r
=

Homework 9: # 8.19, 8.24, 8.25
Michael Good
Nov 2, 2004
8.19
The point of suspension of a simple pendulum of length l and mass m is constrained to move on a parabola z = ax2 in the vertical plane. Derive a Hamiltonian governing the motion of the pendulum

Phys 7221, Fall 2006: Homework # 6
Gabriela Gonzlez a October 29, 2006
Problem 3-7
In the laboratory system, the scattering angle of the incident particle is , and that of the initially stationary target particle, which recoils, is ; see Fig. 3.24, or Fig

Example: Falling Chain
AA 242A
A chain of uniform density of length b and total mass M is suspended at its ends from adjacent
points A and B. At time t = 0, the end at point B is released. Find the tension in the chain at
point A after the free end has fa

Example: Simple Pendulum
AA 242A
A simple pendulum on Earth with point mass m on a massless wire of length l is attached to a
point O and is allowed to swing in a plane. An additional force is applied in the direction normal
to the pendulums motion.
(a) W

AA 242A
TOPIC 6: Linear and Angular Momentum
Linear Momentum (p)
IF A PARTICLE IS FREE, THEN LINEAR MOMENTUM IS CONSERVED.
THIS IS A VECTOR EQUATION AND APPLIES TO EACH COMPONENT OF
LINEAR MOMENTUM.
Linear momentum is written as follows: !

Phys 7221 Homework # 8
Gabriela Gonzlez a November 15, 2006
Derivation 5-6: Torque free symmetric top
In a torque free, symmetric top, with Ix = Iy = I , the angular velocity vector in body coordinates with axes along the principal axes, is given by x = 0

Phys 7221 Hwk # 7
Gabriela Gonzlez a November 6, 2006
Derivation 4-4
Show that if A is a real 3x3 antisymmetric matrix, then the matrices 1 A are nonsingular, and the matrix B = (1 + A)(1 A)1 is orthogonal. If A = A, then A has only three independent comp

Homework 4: # 2.18, 2.21, 3.13, 3.14, 3.20
Michael Good
Sept 20, 2004
2.18 A point mass is constrained to move on a massless hoop of radius a xed
in a vertical plane that rotates about its vertical symmetry axis with constant
angular speed . Obtain the La

Homework 1: # 1.21, 2.7, 2.12
Michael Good
Sept 3, 2004
1.21. Two mass points of mass m1 and m2 are connected by a string passing
through a hole in a smooth table so that m1 rests on the table surface and
m2 hangs suspended. Assuming m2 moves only in a ve

Homework 7: # 4.22, 5.15, 5.21, 5.23, Foucault
pendulum
Michael Good
Oct 9, 2004
4.22
A projectile is red horizontally along Earths surface. Show that to a rst
approximation the angular deviation from the direction of re resulting from
the Coriolis eect v

Homework 5: # 3.31, 3.32, 3.7a
Michael Good
Sept 27, 2004
3.7a Show that the angle of recoil of the target particle relative to the incident
1
direction of the scattered particle is simply = 2 ( ).
Answer:
It helps to draw a gure for this problem. I dont

Phys 7221 Homework #2
Gabriela Gonzlez a September 18, 2006
1. Derivation 1-9: Gauge transformations for electromagnetic potential If two Lagrangians dier by a total derivative of a function of coordinates and time, they lead to the same equation of motio

Phys 7221, Fall 2006: Homework # 5
Gabriela Gonzlez a October 1, 2006
Prob 3-11: Collapse of an orbital system
Consider two particles falling into each other due to gravitational forces, starting from rest at a distance a. The system has zero angular mome

Phys 7221 Homework #3
Gabriela Gonzlez a September 27, 2006
1. Derivation 2-4: Geodesics on a spherical surface Points on a sphere of radius R are determined by two angular coordinates, an azimuthal angle and a polar angle : r = x + y + z k = R(sin cos +

Phys 7221, Fall 2006: Homework # 4
Gabriela Gonzlez a October 2, 2006
Problem 3-10: A comet striking a planet
A planet has a very eccentric orbit about the Sun, with eccentricity e = 1 with 1. When the planet is at the greatest distance from the Sun (aphe

Phys 7221 Hwk #9: Small Oscillations
Gabriela Gonzlez a December 5, 2006
Prob 6-4: Double Pendulum
We follow the conventions for angles in Figure 1.4 (notice that 1 is counterclockwise, and 2 is clockwise!). We set up a coordinate system with the origin a

AA 242A
TOPIC 7: Linear and Angular Impulse
Linear Impulse (F)
Change in linear momentum of a particle during a given time interval is equal to the total
impulse of the external forces acting on the particle over the same interval, or
!
" F

AA 242A
TOPIC 2: Rate of Change of Vectors
A = A
Cartesian:
r = xx + yy + zz; v = xx + yy + zz
Cylindrical:
r = rr + zz; v = rr + rr + zz
= z
r = r = z r =
= = r
v = rr + r + zz
z
r

c 1999 Society for Industrial and Applied Mathematics
SIAM REVIEW
Vol. 41, No. 4, pp. 637676
Centroidal Voronoi Tessellations:
Applications and Algorithms
Qiang Du
Vance Faber
Max Gunzburger
Abstract. A centroidal Voronoi tessellation is a Voronoi tessell

AA 242A Homework 5
Assigned: October 21, 2014
Due: October 30. 2014
1. A pendulum of mass m is attached to a cart of mass M on a spring. Take the rest length of
the spring to be zero.
a. Find EOMs by Lagrange method and discuss the physical significance a

AA 242A Homework 8
Assigned: November 20, 2014
Due: December 5, 2014
1. Greenwood 7-36
A disk of mass m and radius r can roll without slipping on a rod of mass m and length
l = 4r which is pivoted at one end.
a.) Using x and as generalized coordinates, ob

AA 242A Homework 7
Assigned: November 11, 2014
Due: November 20, 2014
1. In aircraft flight dynamics, control and stability is typically analyzed using a standard
set of four frames. For short duration simulations, one typically makes a flat Earth
approxi