Exercise 2.6
2.6.1
Impossibility of Oscillation
Explain this paradox: a simple harmonic oscillation
is a system that oscillates in one
dimension (along the xaxis). But the text says that onedimensional systems cannot oscillate.
Solution:
The equation
is
4.5.3 (Excitable systems) Suppose you stimulate a neuron by injecting it with a pulse of current. If the
stimulus is small, nothing dramatic happens: the neuron increases its membrane potential slightly, and
then relaxes back to its resting potential. How
Solving Equations on the Computer:
2.8.2 Sketch the slope field for the following differential equations. Then "integrate" the equation
manually by drawing trajectories that are everywhere parallel to the local slope.
a) =
b) = 1 2
c) = 1 4(1 )
d) = sin
Nonlinear Dynamics
Some exercises and solutions
S. Strogatz Nonlinear dynamics and chaos
Dominik Zobel
[email protected]
Please note: The following exercises should but mustnt be correct.
If you are convinced to have found an error, feel free to
Egye 332 probset 2
4.1.2 For each of following vector fields, find and classify all the fixed points, and sketch the
phase portrait on the circle
=1+2 cos
0=1+ 2cos
1
=cos
2
2 4
,
Fixed points: =
3 3
d
2
=
Stability check:
= 1.73, stable point
dt
3
d
4
Lecture 4
Introduction to Databases
Database Management for
Strategic Advantage
We live in the Information Age
Information used to make us more
productive and competitive
Database can be found everywhere: POS in
Supermarket, Banks, SMS logs, Call logs,
Electric Field &
Electric Forces
Objectives
Describe the electric field due to a point charge quantitatively and
qualitatively
Establish the relationship between the electric field & the electric
force on a test charge
Predict the trajectory of a massi
ATQ: 216, 217
What is the equation relating the
torque of an electric dipole with
its dipole moment and a uniform
electric field ?
2. Draw the electric field lines for a
dipole separated by a distance d.
Use a maximum of 5 field lines for
each charge.
1
Energy Reservoir Model (ERM) for a
Heat Engine Heat engines absorb heat from a
source at a relatively high temperature,
perform some mechanical work , and
discard some heat at a lower
temperature.
For an engine that undergoes a cyclic
process:
= 0
=
=
4.3.6 Draw the phase portrait as function of the control parameter . Classify the bifurcations that occur
as varies, and find all the bifurcation values of .
= + sin + cos 2
= + sin + cos 2 = + sin + 1 2 sin2
For   < 2, fixed points satisfy:
sin =
1
4.3.9 (Alternative derivation of scaling law) For systems close to a saddlenode bifurcation, the scaling
1
law Tbottleneck ~ ( 2 ) can also be derived as follows.
a) Suppose that x has a characteristic scale ( ), where a is unknown for now. Then = ,
wher
5.3.6 (Romeo the robot) Nothing could ever change the way Romeo feels about Juliet = 0 , = +
. Does Juliet end up loving him or hating him?
In matrix form:
=[
0
0
]
= 0 0 = [ ]
0
= = [ ]
1
There exists a line attractor along 0 . R stays constant.
b
a
b
Egye 332 probset 2
4.1.2 For each of following vector fields, find and classify all the fixed points, and sketch the
phase portrait on the circle
Fixed points:
Stability check:
= 1.73, stable point
= 1.73, unstable point
4.1.8
a)
, hence, for every poin
Exercise 2.1
A Geometric Way of Thinking
In the next three exercises, interpret
2.1
as a flow on the line.
Find all the fixed points on the flow.
Solution:
Fixed points:
x = n where n
2.1.2
At which points x does the flow have greatest velocity to the r
2.3
Solve
Population Growth
(
) analytically for an arbitrary initial condition
.
a) Separate variables and integrate using partial fractions.
Solution:
(
)
(
(
)
)
Therefore,


(
)
At t = 0, N(0) = N0. Solving for C,
(
)
Solving for N,
( (
)
( (
)
(
)
2.2 Fixed Points and Stability
Analyze the following equations graphically. In each case, sketch the vector field on the real line, find all
the fixed points, classify their stability, and sketch the graph of x(t) for different initial conditions. Then
tr
Electric Field Calculations
Objective
Evaluate the electric field at a point in space due
arbitrary charge distributions
to a system of
Approach: Continuous charge distribution
2.
Define a small charge element, find its electric field. Write
electric fie
Microscopic Interpretation of
Entropy
Entropy (Microscopic Interpretation)
Entropy (quantitative measure of disorder) is related
to probability.
State of high order have low probability
State of low order have high probability
Some terms to remember:
M
Carnot Cycle
Second Law of Thermodynamics
What is the most efficient engine?
How efficient can an engine be
given two heat reservoirs at
temperatures TH and TC?
Answered in 1824 by Sadi Carnot
by developing a hypothetical,
idealized heat engine that has
EEE 33 Homework 5
Due: 23 September, 5 PM
Problem 1
Problem 2
The height of the waveform is 1. Assume that the waveform value prior to t = 2 is 1.
Problem 3
Assume that the form of the steadystate response is K exp(2t).
Problem 4
Problem 5
Problem 6
Pr
EEE 33 Homework 3
Due 3 September 2015, 5 PM
Problem 1
Problem 2
Write the mesh equations for the following circuit:
Problem 3
Write the nodal equations for the following circuit:
NATIONAL INSTITUTE OF PHYSICS
U.P. DILIMAN CAMPUS
PROBLEM SET NO. 3 (Gauss Law)
Date Due : 01 September 2015 (Tuesday)
1.
The electric field components in the Figure are Ex = 0, Ey =
by1/2 (b = 8830 N/Cm1/2) and Ez = 0. Find :
A ) the electric flux throu
Chapter 35
Study this chapter then answer the uploaded
Recit 17 in UVLE. Print a copy and encircle the
letter of the best answer.
Submission due: Nov 26 (Thurs)
Nov 26 (Thurs): Inclass recit 18
(Topic: Diffraction)
Physics 72 Bareza AY 1516 1st sem
1
#11
1.76
! 10of
m."
Determine
a time
1.6
10#2 s? the kinetic energy of the electron.
* 24.
unstrained
spring has
a length
0.32
m and a
* 48. AnThe
drawinghorizontal
shows a positive
point
chargeof!q
1, a second point
m mmh Two tiny spheres have the same m
EEE 33 Homework 4
Due 10 September 2015
Problem 1
Problem 2
Problem 3. Find v(t) for t < 0 and t > 0.
Problem 4. Find i(t) for t < 0 and t > 0.
Problem 5
Problem 6. In pro
NATIONAL INSTITUTE OF PHYSICS
U.P. DILIMAN CAMPUS
PROBLEM SET NO. 2 (Electric Force and Electric Field)
Date Due : 18 August 2015 (Tuesday)
1.
An insulating rod of length L has charge q
uniformly distributed along its length.
A ) What is the linear charge
EEE 33 Homework 6
Due 30 September 2015
Problem 1 (treat the two items as one problem)
Problem 2
Problem 3
Write the form of the solution to the following differential equations:
a.
b.
d4 v
d3 v
d2 v
dv
+7
+33
+97 +102 v=0
4
3
2
dt
dt
dt
dt
6
5
4
d i
d i
Chapter 34
Geometric Optics
Physics 72 Bareza AY 1516 1st sem
1
Chapter 34
Geometric Optics
341 Reflection and Refraction at a Plane Surface
342 Reflection at a Spherical Surface
343 Refraction at a Spherical Surface
344 Thin Lens
Physics 72 Bare
EEE 33 Homework 2
Due: 27 August 2015
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5. Find the effective inductance seen at terminals ab.
Problem 6
Problem 7