Physics 326 Homework #4
due Friday, 1 pm HARD DEADLINE
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without explanati
Phys 326 Discussion 4 Weakly Coupled Oscillators, Part II
Last week, you calculated the general solution for the coupled-oscillator demo from class. The system
consisted of a support-spring K whose top end was attached to a xed point and whose bottom end
Physics 326 Homework #12
Due FRIDAY, 1 pm AFTER MIDTERM 2
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without expla
Physics 326 Homework #14
HARD DEADLINE : due Thursday, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without exp
Physics 326 Homework #13
due Friday, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without explanation will not
Physics 326 Homework #8
due in course homework box by FRIDAY, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given witho
Physics 326 Homework #9
due in course homework box by FRIDAY, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given witho
Physics 326 Homework #10
due in course homework box by FRIDAY, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given with
Physics 326 Homework #7
due in course homework box by FRIDAY, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given witho
Physics 326 Homework #3
due in course homework box by Fri 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without e
Physics 326 Homework #2
due in course homework box by Fri, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without
Physics 326 Homework #6
due FRIDAY, 1 pm, No Work = No Points
Problem Pair 0 : Theyre not for points on this homework, but please do not forget to work through Discussion
6 Problem 2 (which no one got to) and the Homework 4 Extra Problem (the pre-exam pr
Physics 326 Homework #1
due in course homework box by Fri, 1 pm
All solutions must clearly show the steps and/or reasoning you used to arrive at your result. You will lose points
for poorly written solutions or incorrect reasoning. Answers given without
x
U 2 U
~ 2 ~
U
x
Thus, we expect the boundary layer thickness to scale with
Furthermore, since we assumed that <x, we have:
x for a laminar flow over a flat plate.
1 Re x 1
(8.5)
x
Ux
The assumption that <x is the same as saying that the Reynolds number
2.
Boundary Layer Equations
To determine the boundary layer velocity profile for a given flow, we need to go back to the governing
equations of fluid motion. Lets consider steady, incompressible, laminar flow over a sharp-edged flat plate
as shown in the
3.
momentum thickness, M or :
The momentum thickness, M, is the thickness of a stagnant layer that has the same momentum deficit,
relative to the outer flow, as the actual boundary layer profile. The concept is similar to that for the
displacement thickne
Boundary Layer Thickness Definitions
Before continuing further, we should define what we mean by the thickness of a boundary layer. There
are three commonly used definitions:
1.
(99%) boundary layer thickness, :
This thickness definition is the most commo
Phys 326 Discussion 3 Weakly Coupled Oscillator Demo
In this discussion, we will analyze the behavior of the demo we viewed in todays lecture: two equal masses on
identical springs were suspended from either side of a horizontal bar that was attached to a
Phys 326 Discussion 2 Critical Damping & More Complex-ity
Problem 1 : Critical Damping, Part 1
Checkpoints1
In our study of the damped linear oscillator, with equation of motion (EOM) and general solution
2
+ 2 x + 0 x = 0
x
2
x(t) = Re A1e + t + A2
! 3 !
v = " (v ! ri )! ri
i =1
!
!
dl
dl path = !du
du
!
!
! # !l !l &
dA = %
"
du dv
$ !u !v (
'
!
!
!
# !l !l & !l
dV = %
"
)
du dv dw
$ !u !v ( !w
'
!f
!dxi
i =1 !xi
n
df (x1 ,., xn ) = "
Taylor
"
f (x) = #
n=0
0
f (n) (x0 )
(x ! x0 )n
n!
sin
1st ord
Phys 325 Discussion 15 Introduction to Hamiltonian Mechanics
The Hamiltonian formulation of mechanics is a modied version of Lagrangian mechanics. At its heart, the
Lagrange-to-Hamilton transition is a change of variables. Consider a system with n degrees
Phys 326 Discussion 14 Rotational Trajectories; Eulers Equations
Problem 1 : Time-Dependence of Vectors Under Constant Rotation
Checkpoints 1
In Phys 325 we proved this crucial relation:
dB
= B for any constant vector B that is xed in a body rotating
Phys 326 Discussion 12 The Inertia Tensor
In lecture, we met the inertia tensor I describing the rotational properties of a rigid body around a point:
L = I
where
I ij = dm ij r 2 ri rj
(
)
Lets develop our intuition about this new object, I, by exploring
Phys 326 Discussion 11 GR, SR, & The Global Positioning System
In lecture we went through an introduction to General Relativity (GR) = Einsteins geometric theory of gravity.
GR has proved extremely successful in describing 100 years worth of experimental
Phys 326 Discussion 13 Symmetries and Principal Axes
Today we will study some important special cases that can greatly simplify our work with inertia tensors: when
an object has certain symmetries, we can guess the principal axes in advance. This is hugel
Phys 326 Discussion 10 Rutherford Scattering & Repulsive Kepler Trajectories
Our Kepler formula set is now updated to include two things: (1) the possibility of repulsive 1/r2 forces with
negative force-constants (2) relations needed for scattering proble
Phys 326 Discussion 8 Central Force Orbits
Here is our growing collection of useful formulae concerning the two-body central force problem. Remember
from our study of particle collections in 325: unscripted CAPITAL letters denote
TOTALs for the system if
Phys 326 Discussion 9 Transfer Orbits ; Scattering
Here is our complete set of useful formulae concerning the two-body central force problem:
mm
m
m
Coordinates & Reduced Mass : r1 = R + 2 r ,
r2 = R 1 r ,
= 1 2
M
M
M
2
2
L
L
Centrifugal force &