physics applied to radiology chapter 3 fundamentals of physics
TRANSCRIPT
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Physics Applied to Radiology
Chapter 3Chapter 3
Fundamentals of PhysicsFundamentals of Physics
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Physics natural science deals with matter and energy
defines & characterizes interactions between matter and energy
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Matter a physical substance characteristics of all matter
occupies space has mass
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Energy capacity for doing work
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Math exact vs. approximate numbers
exact -- defined or counted approximate -- measured
examples your height # of chairs in room # of seconds in a minute # seconds to run 100 m dash
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# of digits in a value when... leading & trailing zeros are ignored
trailing 0 may be designated as significant the decimal place is disregarded
How many significant figures?Value: significant figures 3.47 0.039 206.1 5.90
Significant Figures
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# of digits in a value when... leading & trailing zeros are ignored
trailing 0 may be designated as significant the decimal place is disregarded
How many significant figures?Value: significant figures 3.47 3 0.039 2 206.1 4 5.90 2
Significant Figures
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Accuracy vs. Precision accuracy -- # of significant figures
3.47 is more accurate than 0.039
precision -- decimal position of the last significant figure
0.039 is more precise than 3.47
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Example Describe the accuracy and precision of the
following information. 2.5 cm metal sheet with a .025 cm coat of paint
accuracy is same for both (2 sig. fig.) precision is > for paint (1/1000 vs. 1/10)
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Rounded Numbers all approximate # are rounded last digit of approx. number is rounded last sig. fig. of an approx. # is never an
accurate # error of last number is ½ of the last
digit's place value (if place value is .1 then error = .05)
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Rounded Number example:
if a measured value = 32.63
error is .005 (½ of .01)
actual # is between
32.635 (32.63 + .005)
32.625 (32.63 - .005)
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Rounding Rules round at the end of the total calculation
do not round after each step in complex calculations
when - or + use least precise #(same # of decimal places)
when x or ÷ use least accurate #(same # of sig. figures)
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Rounding Example 173.28.062793.57+ 66.296 241.1287
241.1 # decimal places = to least precise value
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Rounding Example 22.4832
x 30.51
75.762432
75.76# significant figures =to least accurate number
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Numerical Relationships direct linear
as x y (or vice versa) example formula y = k x expressed as proportion y x example: x y (for y = 5x) 1 5
2 103 15
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Numerical Relationships direct exponential
direct square (or other exponent) as x y by an exponential value(or vice
versa)
example formula y = k x2
expressed as proportion y x2
example: x y (for y = 5x2) 1 5
2 203 45
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Numerical Relationships (cont.) indirect
as x y example formula x y = constant
expressed as proportion y 1/x example: x y (for xy = 100)
1 100
2 50
4 25
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Numerical Relationships (cont.) indirect exponential
inverse square (or other exponent) as x y by an exponential value(or vice
versa)
example formula y x2 = constant expressed as proportion y 1/ x2
example: x y (for x2y = 100) 1 100
2 254 6.25
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Graphs used to display
relationships between 2 variables Y-axis (dependent)
measured value
X-axis (independent) controlled value
x-axisy-
axis
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Graphic Relationships ( on linear graph paper)
slope (left to right)
direct = ascending
indirect = descending
shape linear = straight
exponential = curved
X
Ym
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Evaluating Graphed Information identify variables describe shape & slope of line correlate information to theory
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Example #1 Relationship of mA to Intensity
0102030405060708090
100
0 100 200 300 400 500 600
mA
Exp
os
ure
(m
R)
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Example #1 (evaluated) Relationship of mA to Intensity
variables independent = mA dependent = Exposure
shape & slope slope = ascending (=direct) shape = straight line (=linear)
correlate to theory mA has a direct linear relationship to exposure;
as mA increases exposure increases in a similar fashion; the graph demonstrates that if you double the mA (200 to 400) you also double the exposure (30 mR to 60 mR)
0
20
40
60
80
100
0 100 200 300 400 500 600
mA
Ex
po
sure
(m
R)
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Example #2 Relationship of the # days before exam to
amount of study time
012345
0 1 2 3 4 5 6
Days before exam
Stu
dy T
ime
(HR
S)
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Quantities & Units quantity = measurable property
quantity definition (what is measured) length distance between two points
mass amount of matter (not weight)
time duration of an event
unit = standard used to express a measurementquantity unit other units length meter
mass kilogram
time second
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Unit SystemsSystem length mass time
English foot slug (pound) second
metric SI** meter kilogram second
** also ampere, Kelvin, mole, candela
metric MKS meter kilogram second
metric CGS centimeter gram second
Do not mix unit systems when doing calculations!!
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Converting Units convert 3825 seconds to hours
identify conversion factor(s) neededfactors needed: 60 sec = 1 min & 60 min = 1 hour
arrange factors in logical progressionFor seconds hours
sec min/sec hour/min
set up calculation
60min
1hour
60sec
1minsec3825 hour1.0625 hour1.063
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Dimensional Prefixes Bushong, table 2-3 (pg 23)
used with metric unit systems modifiers used with unit a power of 10 to express the magnitude prefix symbol factor numerical equivalent
tera- T 1012 1 000 000 000 000 giga- G 109 1 000 000 000 mega- M 106 1 000 000 kilo- k 103 1 000 centi- c 10-2 .01 milli m 10-3 .001 micro- 10-6 .000 001 nano- n 10-9 .000 000 001 pico- p 10-12 .000 000 000 001
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Rules for Using Prefixes To use a prefix divide by prefix value &
include the prefix with the unit
kmmm kmm 45104500045000 3
lmlml mll 85.10850850 3
To remove a prefix multiply by prefix value & delete prefix notation from the unit
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Base Quantities & Units (SI) describes a fundamental property of matter cannot be broken down further quantity SI unit definition for quantity
length meter distance between two points
mass kilogram amount of matter (not weight)
time second duration of an event
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Derived Quantities & Units properties which arrived at by combining base
quantities
quantity units definition for quantity
area m x m m2 surface measure
volume m x m x m m3 capacity
velocity m/s m/s distance traveled per unit time
acceleration m/s/s m/s2 rate of change of velocity
ms-2
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Derived Quantities with Named Units quantities with complex SI units
quantity units definition
frequency Hertz Hz # of ?? per second
force Newton N "push or pull"
energy Joule J ability to do work
absorbed dose Gray Gy radiation energy
deposited (rad) in matter
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Solving Problems1. Determine unknown quantity2. Identify known quantities3. Select an equation (fits known & unknown quantities)
4. Set up numerical values in equation same unit or unit system
5. Solve for the unknown write answer with magnitude & units raw answer vs. answer in significant figures
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Mechanics study of motion & forces motion = change in position or orientation
types of motion translation
one place to another
rotation around axis of object's mass
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Measuring Quantities in Mechanics all have magnitude & unit
scalar vs. vector quantities
Scalar -- magnitude & unit
Vector -- magnitude, unit & direction
run 2 kmvs
run 2 km east
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Vector Addition/Subtraction requires use of graphs, trigonometry or
special mathematical rules to solve example:
F1
F2F1 + F2 =Net force
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Quantities in Mechanics speed
rate at which an object covers distance rate
indicates a relationship between 2 quantities $/hour exams/tech # of people/sq. mile
speed = distance/time
speed is a scalar quantity
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Speed (cont.)v =
d t
d in mt in sv = m/s
same at all times
total distancetotal time
General Formula:
Variations:
instantaneous uniform average
dist
ance
time
v at 1 point in time
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Speed Example An e- travels the 6.0 cm distance between the anode &
the cathode in .25 ns. What is the e- speed? [Assume 0 in 6.0 is significant]
v = ?? 6.0 cm = distance .25 ns = time
v = d/t (units: m/s need to convert)
6.0 cm = 6.0 x 10-2 m .25 ns = .25x10-9s
= 6 x 10-2 m / .25x10-9s
= 2.40000 x 108 m/s (raw answer)
= 2.4 x 108 m/s (sig. fig. answer)
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Velocity speed + the direction of the motion vector quantity
A boat is traveling east at 15 km/hr and must pass through a current that is moving northeast at 10 km/hr. What will be the true velocity of the boat?
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Acceleration rate of change of velocity with time
if velocity changes there is acceleration includes: v v direction formula:
v = vf - vi
units v in m/s t in s a = m/s2
a = v t
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Acceleration Example A car is traveling at 48 m/s. After 12 seconds
it is traveling at 32 m/s. What is the car’s acceleration?a = ? 48 m/s = vi 12 s = t 32 m/s = vf
a = v / t
v = vf - vi = 32m/s - 48 m/s = -16 m/s
a = -16m/s / 12 s = -1.3333333333 m/s2
= -1.3 m/s2 [ -sign designates slowing down]
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Application of v and a in Radiology
KE (motion) of e- used to produce x rays controlling the v of e- enables the control of the
photon energies Brems photons are produced when e-
undergo a -a close to the nucleus of an atom
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Newton's Laws of Motion
1. Inertia
2. Force
3. Recoil
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Newton's First Law defined -- in notes inertia: resistance to a in motion
property of all matter mass = a measure of inertia
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InertiaSemi-trailer truck
large mass large inertia
Bicycle small mass small inertia
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Newton's 2nd Law (Force) Force
anything that can object's motion Fundamental forces
Nuclear forces "strong" & "weak"
Gravitational force Electromagnetic force
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Mechanical Force push or pull vector quantity
net force = vector sum of all forces
push on box + friction from floor
equilibrium -- net force = 0
Vector sum
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2nd Law (Force) defined -- in notes formula for the quantity “force”
force = mass x acceleration
F = m x aa =
v t
kg ms2
Newton N
units kg x m/s2
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Example Problem for 2nd LawWhat is the net force needed to accelerate a 5.1 kg laundry cart to 3.2 m/s2?
F =?? 5.1 kg = mass 3.2 m/s2 = acceleration
F = m a
= 5.1 kg x 3.2 m/s2
= 16.32 kg m/s2
= 16 N
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Example 2:A net force of 275 N is applied to a 110 kilogram mobile unit. What is the unit's acceleration?
acceleration =?? 275 N = F 110 kg = mass
F = m a
a = F/m
= 275[kg m/s2] / 110kg
= 2.5 m/s2
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Example 3An object experiences a net force of 376N. After 2 seconds the change in the object's velocity 15m/s. What is the object's mass?
mass =?? 376 N = F 2 s = t 15 m/s = v
F = m a m = F/a
a = v/t
= 15 m/s / 2 s = 7.5 m/s2
m = 376 [kg m/s2] / 7.5 m/s2
= 50.13333333333 kg = 50 kg
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Weight adaptation of Newton's 2nd law weight = force caused by the pull of gravitation
weight massgravitational force inertia of the object
varies with gravity always constant
unit = N [pound] unit = kg [slug]
when g is a constant then weight proportional mass
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Weight (cont.) formula for quantity “weight”
modified from force formulaF = m x a
Wt. = m x g gearth = 9.8m/s2
kg ms2
Newton N
units kg x m/s2
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Weight ProblemWhat is the weight (on earth) of a 42 kg person?
Wt. = ?? 42 kg = mass [9.8m/s2 = gravity]
Wt. = m x g
= 42 kg x 9.8m/s2
= 411.6 kg m/s2
= 410 N
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Weight Problem #2What is the mass of a 2287N mobile x-ray unit?
mass = ?? 2287N = Wt [9.8m/s2 = gravity]
Wt. = m x g
m = Wt./g
= 2287N / 9.8m/s2
= 233.3673469388 kg
= 233.4 kg
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3rd Law (Recoil) Defined -- in notes
no single force in nature all forces act in pairs
action vs. reaction
formula
FAB = -FBA
A
B
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Momentum (Linear) measures the amount of motion of an object tendency of an object to go in straight line
when at a constant velocity formula
p = m x v units
= kg x m/s= kg m
s
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Momentum vs. Mass (Inertia)p = m x v
p m
m = pm = p
Direct proportional relationship
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Momentum vs. Velocityp = m x v
p v
50 km/hr
v = p
100 km/hr
v = pDirect proportional relationship
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Momentum ProblemWhat is the momentum of a 8.8 kg cart that has a speed of 1.24 m/s?
p = ?? 8.8 kg = mass 1.24 m/s = velocity
p = m x v
= 8.8 kg x 1.24 m/s
= 10.912 kg m/s
= 11 kg m/s
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Momentum Problem #2What is the speed of a 3.5x104 kg car that has a momentum of 1.4x105 kg m/s?
velocity = ?? 3.5x104 kg = mass 1.4x105 kg m/s = momentum
p = m x v
v = p / m
= 1.4x105 kg m/s / 3.5x104 kg
= 4.0 x 100 m/s
= 4.0 m/s
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Conservation Laws Statements about quantities which remain
the same under specified conditions. Most Notable Conservation Laws
Conservation of Energy Conservation of Matter Conservation of Linear Momentum
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Conservation of Linear Momentum
momentum after a collision will equal momentum before collision
results in a redistribution momentum among the objects
p1 = p2
m1v1 = m2v2
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Example
m1v1= 1kg m/s mv = 0
mv = 0 m2v2= 1kg m/s
before collision
collision occurs
after collision
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Example #2
m1v1= 5kg m/s mv = 0
m2v2= 5kg m/s
before collision
collision occurs
after collisionm2 = mA + mB v2 = vA + vB
A B
A B
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Work defined -- in notes
measures the change a force has on an object's position or motion
If there is NO change in position or motion, NO mechanical work is done.
F
d
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Work (cont.) formula
Work = force x distance W = F x d
units = N x m=kg m
s2 x m
kg m2
s2 = Joule J=
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ExampleHow much mechanical work is done to lift a 12 kg mass 8.2 m off of the floor if a force of 130 N is applied?work = ?? 12 kg = mass 8.2 m = distance 130 N = force
W = F x d
= 130 N x 8.2 m
= 1066 N m
= 1100 J (1.1 kJ)
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Example #2 A 162 N force is used to move a 45 kg box 32 m.
What is the work that is done moving the box?work = ?? 162 N = force 45 kg = mass 32 m = distance
W = F x d
= 162 N x 32 m
= 5184 N m
= 5200 J or 5.2 kJ
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Energy property of matter enables matter to perform work broad categories
Kinetic Energy: due to motion Potential Energy: due to position in a force field Rest Energy: due to mass
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Kinetic Energy work done by the motion of an object
translation, rotation, or vibration formula
KE = ½ mass x velocity squared = ½ m v2
units = kg x [m/s]2
kg m2
s2 = Joule J=
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ExampleFind the kinetic energy of a 450 kg mobile unit moving at 6 m/s.
kinetic energy = ?? 450 kg = mass 6 m/s = velocity
KE = ½ m v2
= ½ x 450 kg x [6 m/s]2
= 8100 kg m2 /s2
= 8000 J or 8 kJ
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Potential Energy capacity to do work because of the object's
position in a force field fields
nuclear electromagnetic gravitational
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Gravitational Potential Energy barbell with PE formula
PEg = mass x gravity x height
= m x g x h
units= kg x m/s2 x m
=
hg
m
kg m2
s2
= Joule J
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ExampleHow much energy does a 460 kg mobile unit possess when it is stationed on the 3rd floor of the hospital? (42m above ground)PE = ?? 460 kg = mass 42 m = height [9.8 m/s2 = gravity]
Peg = m x g x h
= 460 kg x 9.8 m/s2 x 42 m
= 189 336 kg m2 /s2
= 190 000 J or 1.9x105 J or 190 kJ
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Rest Mass Energy energy due to mass Einstein's Theory formula (variation of KE formula)
Em = mass x speed of light squared
= m c2 [c = 3x108 m/s] units = kg x [m/s]2
kg m2
s2 = Joule J=
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ExampleWhat is the energy equivalent of a 2.2 kg object? Em = ?? 2.2 kg = mass [3x108 m/s = speed of light]
Em = m c2
= 2.2 kg x [3x108 m/s ]2
= 1.98 x 1017 kg m2 /s2
= 2.0 x 1017 J [trailing 0 is significant]
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Conservation Of Energy (Matter)
Energy is neither created nor destroyed but can be interchanged
(Matter is neither created nor destroyed but can be interchanged)
Because mass has rest energy, conservation of matter & energy can be combined
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Power Rate at which work is done
Faster work = more power Rate at which energy changes
Large E = more power
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Power (cont.) formula
power = work / time or energy / time
P = W / t or E / t units = J / s
kg m2
s3 = Watt W=
kg m2
s2 = s
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ExampleHow much power is used when an 80N force moves a box 15 m during a 12 s period of time?
(hint: solve for work first)
P = ?? 80 N = force 15 m = distance 12 s = time
P = W/t & W = Fd
P = (F d) / t
= (80 N x 15 m) / 12 s
= 100 Nm/s
= 100 W
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Heat energy internal kinetic energy of matter
from the random motion of molecules or atoms KE & PE of molecules heat E in matter moves from area of higher E in
object to area of lower internal E
Unit -- Calorie (a form of the joule) amount of heat required to raise one gram of water
one degree Celsius.
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Heat Transfer movement of heat energy from the hotter to
cooler object (or portion of object) 3 methods of transfer
conduction convection radiation
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conduction primary means in solid objects classification of matter by heat transfer
conductors--rapid transfer insulator--very slow to transfer
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convection primary means in gasses and liquids convection current--continuing rise of
heated g/l and sinking of cool g/l
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radiation transfer without the use of a medium
(i.e. no solid, liquid or gas) occurs in a vacuum
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Heat Radiation term “radiation” may simply refer to heat
energy and not the transfer of heat infra-red radiation, part of EM spectrum, is
heat energy
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Effects of Heat Transfer change in physical state of matter
solidliquidgas
melt boil change in temperature
measure of the average KE of an object relative measure of sensible heat or cold
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Temperature ScalesScales Boil (steam) Freeze (ice) No
KE
Fahrenheit 212° 32° -460°
Celsius 100° 0° -273°
Kelvin (SI) 373 273 0
1K = 1°C = 1.8°FConversion formulae
°F = 32 + (1.8 °C)
°C = (°F - 32) 1.8
K = °C + 273