Physics
Math Basics Instructor: Padmini Arora
The Math Basics
Order of Operations:
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
Examples
2 + 5-3 2+5 X3 (2+5) X3
(22+5) X3 important exponents: 10%
103
4 = 17 = 21 = 27 = 1 X 102 = 100 = 1 X 103 = 1000 = 1 X 10-2 = .01 = 1X10-3 = .001
10-2
10-3
***Remember: PE MD AS
Other Important exponents:
22 =
2X2=4
=
2X2X2=8
2X2 X 2 X 2 = 16
3 X3=9
= 3 X 3 X 3 = 27
34 =
3X3X3 X 3 = 81
..
.
Relationship in terms
Unrelated: When the there is no relation between terms
Ex: Test score is unrelated to the student's seat number
Directly Related: When one item increases, the other increases
Ex: Skill is directly related directly proportional to practice
Inversely Related: When one item increases, the other decreases
Ex: When Temperature goes down, the sweater sales go up
Reciprocal Relationship: Multiplication of reciprocal numbers equal one
Units of Measurement
All Numerical values must have corresponding unit
Ex: Runner finished 2 miles race in 20 minutes
Conversions:
Ex: How many minutes are there in one hour
**Notes:
✦ Unit is very important to consider with the numerical value
✦ Percent is unitless
✦ Increase by a factor: Multiply by that number
✦ Decrease by a factor: Divide by that number
Metric System - Powers of Ten
Symbol
1 + Exponents
109 106
М.
103
102 101
Prefix
giga mega kilo hecto deca deci centi milli micro --- Meaning billion million
thousand hundred ten tenth hundredth thousandth millionth billionth
10-1 10-2 10-3 10-6 10-9
DE 3 0 2 0
nano
Scientific Notation
• Scientific Notation: Shorthand representation of very big or very small number:
Examples:
1,000 =
103
2,000=
2X 1,000 = 2X 103
.001 =
10-3
.002=
2x.001 = 2x 10-3
Complimentary Units
Pairs
Prefixes
Abbreviations
billions & billionth
giga & nano
G&n
millions & millionths
mega & micro
M&u
thousands & thousandth
kilo & milli
K&m
hundreds & hundredths
hecto & centi
h&c
.
tens &tenths
deca & deci
da & d
The Graph
X-axis (Horizontal axis): runs side to side
Y-axis (Vertical axis): runs up & down
Review Questions
• Are the following pairs inversely related directly related
or unrelated?
1) Cholesterol level & longevity
2) Smoking & the likelihood of cardiovascular disease
*3) Years employed & the days of vacation earned per year
4) IQ & shoe size
5)Calorie intake & weight
6) Hours spent exercising & weight
7) Alcohol intake & sobriety
8) What is reciprocal of 20?
Waves
Sound Waves
Sound is mechanical wave:
The particles move back & fourth
Sound is longitudional wave:
The wave travel in straight line
Biologic effect:The effects of sound wave on medium. Acoustic propagation properties:
Effects of medium on sound wave ***Sound cannot travel through vacuum
Transverse waves
Particles displace perpendicular to the direction of propagation of the wave
Ex: Wave on a string & ripple on a pond
Longitudinal waves
Particles displace parallel to the propagation of the wave
Ex: Sound wave in the air & seismic p-wave created during earthquakes & explosions
In-Phase & Out of Phase Waves
In-phase :Peaks at same time
Out of phase: Peaks at different time
Interference
***Combination of 2 or more than 2 waves produce interference -- original waves lose
their identity
• In-phase waves go through interference called constructive interference [resultant
wave is bigger] while out of phase waves undergo destructive interference [resultant
wave is smaller]
7
Constructive interference is Different Frequencies
• With different frequency waves, both constructive and
destructive interference occur
Acoustic Variables
The alteration in these 4 variables generate sound wave:
• A) Pressure : Force per unit area
Unit : pascals (Pa),
• B) Density : Mass per unit volume
Unit: kg/cm3
• C) Distance: Measure of length
Unit : cm, feet, mile
• D) Temperature : Degree of hotness or coldness of a
body or environment Unit : °C or OF
Acoustic Variables
* Rhythmic movement of any of the four variable generate sound wave [acoustic wave]
* If there is movement of any other variable then the wave is not a sound wave
*** Know:
1) 4 variables of sound wave
2) their definition
3) their units
.
Acoustic Parameters
• Acoustic parameters are the parameters required to describe any sound wave
• Seven Acoustic parameters:
1) Period 2) Frequency 3) Amplitude 4) Power 5) Intensity 6) Wavelength 7)
Propagation speed
Review
• Define sound wave.
• Define biologic effect.
. What do waves transfer from one place to another?
• Name three acoustic variables.
• Unit of all three variables.
• Name seven acoustic variables.
• What in in phase & out of phase waves?
• When do two waves have constructive & destructive
interface?
Sound Waves
Acoustic Parameters
• Acoustic parameters are the parameters required to describe any sound wave:
7 Acoustic parameters:
1) Period 2) Frequency 3) Amplitude 4) Power 5) Intensity 6) Wavelength 7)
Propagation speed
Frequency
Definition: Number of cycles per second
Formula: # of cycles/seconds
Unit: hertz, Hz, 1/ second, or per second Determined by: sound source
Adjustable: Sonographers cannot change this value
Typical Values: 2 MHz to 10 MHz
Frequency
Sound wave classification based on frequencies:
Infrasonic: Frequency < 20 Hz
Audible: Frequency between 20 Hz To 20 kHz
ultrasonic: Frequency > 20 KHz.
***Frequency is very important in ultrasound because it effects the resolution &
penetration
Period
Definition: Time it takes for one cycle to occur -- it’s the reciprocal of frequency
Formula: Period = 1/ frequency Period X frequency = 1
Unit: seconds, minutes or any other unit of time
Determined by: source [US system & transducer]
Adjustable: Sonographers cannot change this value
Typical Values: 0.1 to 0.5 microseconds [us]
Frequency & Period
There is inverse relation in frequency and period Frequency = 1/Period
If FP
If F p1
ex: if p=1 f= 1/1 = 1
if p= 2 => f= 42 (p= double, f = half)
Always use complimentary units with period and frequency Ex: seconds and hertz,
microseconds with megahertz
"Bigness" Parameter
These three "Bigness" Parameters describe the strength of sound wave
.1) Amplitude: Maximum variation that occurs in an acoustic variable [Maximum value normal value]
2) Power: Rate of energy transfer
3) Intensity: rate at which energy passes through a unit area
Amplitude
Definition: difference between maximum & undisturbed value
Unit: unit of any acoustic value [Pa, g/cm3, cm, °C]
Determined by: sound source
Adjustable: Sonographers can change this value
Typical Values: 1 MHz to 3 MHz
1) Amplitude: difference between middle value to maximum value
2) Peak to Peak amplitude: difference between minimum value & maximum value
Power
Definition: rate of energy transfer
Formula: power amplitude? in other words power is proportional to amplitude square
Unit: watts
Determined by: source [US system & transducer]
Adjustable: Initial power can be changed
Typical Values: 4 to 90 mw [milliwatts]
Intensity
Definition: rate at which energy passes through a unit area
Formula: intensity = power /area
intensity co power
|
intensity o amplitude
Unit: W/cm2
Determined by: source [US system & transducer]
Adjustable: Initial intensity can be changed
Typical Values: 0.01 to 300 W/cm2
*** Intensity of power & amplitude 1
Wavelength
Definition: Distance or length of one complete cycle
Formula: Wavelength = propagation speed/ frequency
(n = c/f)
Unit: Any of distance [m, mm, cm]
Determined by: Source & Medium
*** Only parameter determined by both [source & medium]
Adjustable: Sonographers cannot change this value
Typical Values: 0.15 to 0.8 mm
Wavelength & Frequency
A = c/f
***Iff or iff
A4
f [frequency] is inversely related to the A [wavelength]
If C= 1.54 mm/us so A (mm)= 1.54 mm/us /f MHz
If frequency is 2 MHz then A = 1.54/2 = 0.77 mm
Propagation Speed
Definition: distance traveled by the sound wave in 1
second
Formula: Propagation speed (c)=distance (d)/time (t)
Propagation speed (c)= frequency (f) Xwavelength (A)
Unit: m/s, mm/us
Determined by: medium only
Adjustable: sonographer cannot change the speed
Typical Values: 500 m/s to 4000 m/s
*** Speed of sound in soft tissue: 1.54 mm/us or 1540 m/s
Propagation Speed
Speed of sound in biologic tissues:
Tissue type
Lung
Fat
Speed (m/s)
500
1,450 soft
tissue
1,540 Liver
1,560 Muscle
1,600 Tendon
1,700 Bone
3,500 In other media:
Water
Metal
Air
330
1,480
2,000 to 7,000
Propagation Speed
Characteristics of medium that effects speed of sound:
1) Stiffness: Objects ability to resist compression
Propogation speed is directly related to
stiffness
Stiffness → Propagation Speed 1
2) Density: Relative weight of a material
Propagation Speed is inversely proportional to density
Density 1 → Propagation Speed
***Opposite of stiffness is compressibility & elasticity
Rarefactions & Compressions
• Compression: When the molecules of the medium are squeezed together
• Rarefaction: When the molecules of the medium are stretched apart
Review Questions
• Define the following acoustic parameters & give there units:
1) frequency
2) period
3) Wavelength
4) Amplitude
5) Intensity
6) Power
7) propagation speed
Which is the only parameter depending on both the media & the source?
Which parameters cannot be changed by the operator?
Formulas/Units
Chapter 1: Know conversions
Chapter 2: pressure = Force/area [Pa, mmHg, N/cm2]
density = mass/volume [kg/cm3] distance [cm, mm, feet, mile] temperature [°C, °F]
Chapter 3: Period = 1/frequency [seconds, minutes]
Frequency = 1/period [Hz, per second] Amplitude [unit of any acoustic value] power =
power ou amplitude? [watts] Intensity = power /area, intensity -- power → intensity
amplitude Wavelength = propagation speed/frequency [any unit of length] Propagation
speed = distance/time [m/s, mm/us] Stiffness 1 → Speed 1, Density 1 → Speed 10