Motivating Applications
Medical Treatment Strategies:
Ultrasound to break up kidney stones
Acoustic barriers for EM treatment
Ultrasonic dental tools
Motivating Applications
SAW (Surface Acoustic Wave) Devices:
Used in filters, oscillators and
transfo
Chapter 13 Simple Harmonic Motion
v = A2 x 2
(13.15)
Acceleration as a function of time
a = 2A cos t
(13.17)
Angular frequency of a spring
=
Velocity as a function of displacement
k
m
f= 1
2
Frequency in simple harmonic motion
(13.20)
k
m
T = 2 m
k
Period
Chapter 13 Simple Harmonic Motion
is fired at an angle of 62.50, the range of the projectile will be 1.50 m. The mass of
the ball is 25.2 g.
*38. In the simple pendulum shown in the diagram, find the tension in the
string when the height of the pendulum i
Acoustic Models
It's easy to play any musical instrument: all you have to do is touch the right key
at the right time and the instrument will play itself, J.S. Bach
Motivating Applications
Jet Noise Reduction:
Employ chevrons to improve
mixing and decrea
102. (b) The requirement is to calculate the
degree of
operating leverage of the company. The
formula for degree of
operating leverage is
DOL = Percent change in operating income
Percent change in unit volume
In this case, the percent change in operating
LINEAR PROGRAMMING
CASE STUDIES AND SOLUTIONS
Vassilis Kostoglou
E-mail: vkostogl@it.teithe.gr
URL: www.it.teithe.gr/~vkostogl
1
LINEAR PROGRAMMING CASE STUDIES AND SOLUTIONS
Case study 1
Portfolio selection
Select a portfolio package from a set of altern
Explanations
1. (c) The requirement is to identify the
preferred method
of calculating the expected return for an
investment that is
expected to be held for a long period of time.
Answer (c) is
correct because the geometric average depicts
the compound
an
LINEAR PROGRAMMING:
MODEL FORMULATION AND
GRAPHICAL SOLUTION
2-1
Chapter Topics
Model Formulation
A Maximization Model Example
G hi l S
Graphical
Solutions
l ti
off Li
Linear P
Programming
i M
Models
d l
A Minimization Model Example
Irregular Types of Lin
SSC_R Seminar 1 & II
(Saturday, August 13, 2016)
Here is an excerpt from the Educational Plan of the Order of Augustinian, Recollects. This is to
be realized also in the College of Law of San Sebastian College - Recoletos Read, reflect and
find out
1. Whi
Anti-Drugs Campaign
Since the new administration of the government sat in power, the
Philippines has taken steps on war against drugs. While it is understandable
that the drug syndicate be loathed and, naturally, be subject to
extermination in the society
A. Total Project Cost(Pre-operating expense)
Year
2016
Cost of processing Licenses &
Permits
Php10,000
Installation Cost of
Electricity
20,000
Installation cost of water
supply
10,000
Other pre-operating expenses
Total pre-operating expenses
60,000
Php100
Chapter 13 Simple Harmonic Motion
acceleration of gravity of 9.80 m/s2 on the earth. (This is an example of Einsteins
principle of equivalence in general relativity.) An oscillating pendulum of measured
length l can be placed in an elevator and the period
Chapter 13 Simple Harmonic Motion
When the oscillating mass reaches x = A, the kinetic energy becomes zero since
1 kA2 = 1 kA2 + 1 mv2
2
2
2
2
2
1 mv = 1 kA 1 kA2 = 0
2
2
2
= KE = 0
As the oscillation continues there is a constant interchange of energy be
Chapter 13 Simple Harmonic Motion
that the restoring force is pulling the mass back toward its equilibrium position. At
the equilibrium position x = 0, a = 0, and F = 0, but v has its maximum velocity of
A toward the left. As x goes to negative values, th
Examples of Sound Pressure and Sound Pressure Levels
Sound Source
Sound Pressure Level
Pa
Theoretical limit for
undistorted sound
RMS Sound Pressure
dB re 20 microPA
101,325
191
1883 Krakatoa eruption
Approx 180 at 100 miles
Stun grenades
170-180
Rocket l
Models for Sound Waves
Nonlinear System:
Assumption: Sound is perturbations of pressure and density about static or
slowly time-varying state.
Static Quantities:
Models for Sound Waves
Dynamic Perturbations:
Strategy: Linearize about steady, silent case
E
Wave Equation in Potential
Initial Assumption:
Wave Equation in Potential
WLOG:
Wave Equation in Potential
Nonconstant Density:
Wave Equation in Potential: See book, pages 236-237
Linear Wave Equations for Sound
Pressure:
Velocity:
Potential:
Music
Pitch:
Music
Note: Going up a fifth (e.g., C and G) causes frequencies to align every third
interval for the G - e.g., First two notes of Star Wars theme.
Reason: Equal tempering which spaces all 12 notes equally
Each semitone is thus
C sharp thus has a freque
Solving the Harmonic
Oscillator Equation
Morgan Root
NCSU Department of Math
Spring-Mass System
Consider a mass attached to a wall by means of a
spring. Define y=0 to be the equilibrium position of
the block. y(t) will be a measure of the displacement
fro
Alternate Method:
The Characteristic Equation
Again we take 0 =
k
m
and look at the system :
2
&(t ) + 0 y (t ) = 0
y
Assume an exponential solution, y (t ) = e rt .
&
y (t ) = re rt and &(t ) = r 2 e rt
y
Subbing this into the equation we have :
2
r 2 e
Chapter 13 Simple Harmonic Motion
We are to admit no more causes of natural things
than such as are both true and sufficient to explain
their appearances. Isaac Newton
13.1 Introduction to Periodic Motion
Periodic motion is any motion that repeats itself
Chapter 13 Simple Harmonic Motion
the simple harmonic motion of a body m moving under the influence of an elastic
restoring force, as shown in figure 13.2.
The position of P on the x-axis and hence the position of the mass m is
described in terms of the p
Extrajudicial Killings
Extrajudicial killing is the killing of a person, by a governmental
authorities, without the prior sanction of a judicial proceeding. Normally,
the punishment of an execution can only be determined through a proper
trial. However, i