(column
0)
(col.1)
In the spaces below you
should draw diagrams to illustrate the meaning of each respective proportion..
(You may tape on extra paper as needed or write on back.)
(col.2)
Name of the newly
Defined concept.
(Please realize that the new defined concept of the proportion is necessarily
a
Constant !!! )------
Please realize that the definition step (ş) is the creation of a new concept which we “experience” and know “intuitively” but without the ş step we are at loss to clearly discuss or handle quantitatively. (col. 3,4,5)
(col.6)
Special Notes concerning what specific conditions must be true in the indicated proportion. Study examples given.
When we see
“ µ ”
proportion in
the data , then
Definition
“ ş ” will be pre-sent and
Then
“ = ”
gives the
equation.
Examples for you to follow
Vertical. (x,y) + (X,Y)
m = Slope of graph. This gives a general equation for proportion: y = mx (unitsvert. / unitshoriz)
vertical µ horizontal
y µ x
m ş y/x=Dy/Dx
where b=0
y = m x
Line must be straight and through (o,o). Column four is always “left over right”. This is a good way to remember what to do!
$
$
P = Price of an item you purchase
( $ per gal )
dollars µ # items
$ µ n
P ş $ / n
$ = P n
In column five is always put constant first after the “ = ” .
m
______
______ Short Rope
M
_____________
_____________
LongRope
H = “Heavy-ness” of a rope or cable.
( kg/m )
mass µ length
m µ l
H ş m / l
m = H l
The rope or cable must be "Uni-form". i.e. Everywhere the same.
Common proportions for first semester physics. These should be easy to fill out
d = Density of a material.
( )Put Units
( ) µ ( )
( ) µ ( )
( ) ş ( )/( )
( )=( )( )
The sample must be
homogeneous. (i.e. the
same everywhere).
( ) Put Units
distance µ time
µ
ş
=
Must be a constant rate of motion (vel. is a constant).
( ) Put Units
change vel. µ time
(v –v0) µ t
ş
=
Must be a constant rate of “speed-up” (accel. is const).
( Joule/sec ) Put Units
Work µ time
What must be condition of car engine?
____ = Spring Constant.
( ) As above.
Force µ stretch
Spring must....
____ = Coefficient of Static Friction.
( ) As above.
____ = Coefficient of Kinetic Friction.
( ) "
( ) ; )
Work Out µ Work In
Diagrams,
Maps, and Building Plans: Rules for Space we live in.
(Picture on a Diagram) µ (to World )
S.F. = Scale Factor for all scale diagrams. (Similar to concept of magnification. ( ) " "
(l in World ) µ
( l in Diagrm)
(l= Length)
In scale diagrams angles are same as world, but sides are all proportionally smaller.
Trigonometry
(Sides Small Rt. D)
µ
(Sides Large Rt.
D)
Angle J
H
h
o O
Definition of
sine of angle J
o = side opposite
h = hypotenuse ( )
(side opposite)
µ (hypotenuse)
µ
What is same? My Answer:
The space we live in every-where accurately follows Euclid’s Plane Geometry and standard trigonometry !!
This means no near-by Black Holes and no “curved space”. Under these conditions the Commutative Property of Vectors is experimentally true and independent of scale.
Definition of
cosine of Angle J
( )
Definition of tangent of angle J =Dy/Dx =rise/run =slope = grade of a road = so-called “gradient”.
Geometry of Circles.
Above have shown example diagrams. Here and below you are to show your own diagrams.
(Sm. Circle) µ (Lg. Circle)
Definition of p
( )
Sm. Wedge µ Lg. Wedge
J = Angle measured in radians.
( )
arc distance µ radius
s µ r
What is same?
Circle moving particle
w = Angular velocity. Here fol-low (by analogy) velocity above.
a = Angular acceleration. Here follow (by analogy) with acceleration above.
s,v,a,J, v,a
are variables.
From geometry of a circle the definitions above relate linear to angular as follows: (ignore units)
s µ J
=
=
What's constant ? __ . For small angles, J @ sin J follows from the definitions.
v µ w
=
=
a µ a
=
=
Dynamics
Newton’s Law where
mass is a constant.
( )
=
ş
Here Col. 4 does not define m but Col. 5 defines force!
“Rotary Newton’s Law” where I= moment of Inertia = constant.
( )
ş
=
Here Col. 4 defines moment of inertia.
Gravity & Astronomy
(g at surface of earth) = _______________
(g at 2 earth radius above
earth surface) =___________.
g = Gravitational Field Strength A new way to look at an old equation. ( )
gravity force µ its
on any object mass at any position.
g ş FG / m˘
FG = g m˘ (familiar equation)
So g = 9.8m/s = "Gravitational Field Strength" at surface of earth and is less far from earth.
May omit sketch, unless you have good idea.
FG µ ( )
FG µ ( )
FGµ 1/ ( )
G = Newton’s Universal Gravity Constant.
( )
( ) ş ( )( )
( )
Why name “Universal”?
Kepler’s Third Law gives a proportion (T & r) and a constant ratio given as symbols by New-ton which is may be called Kepler's Const.
What things are constant? Look at Newton’s symbols!
Look in book only when done.
Put any other proportions here. More space on back.
THINK!!!
(cycles/sec, revolutions/sec,
waves/sec, events/sec, etc )
# events µ time
( )
Force on µ area of
a sign sign
The __________of the wind must __________________.
The constant is a property of a material. See handbooks.
Surface Tension
( )
Force liquid µ length
surface edge
Surface Tension (a 2nd equivalent definition)
( )
Energy liq. µ area
surface surface
May omit sketch, unless you have good idea.
D l µ D T
D l µ l
= Coefficient of Thermal
Expansion of a Material.
( )
Ditto above.
DV µ DT
DV µ V
= Coefficient of Thermal
Expansion of a Liquid.
( )
Ditto above.
DH µ DT
DH µ m
cp =
( )
= Heat of combustion or
other chemical reaction
( )
heat µ mass involved
= latent heat to(melt/fuse)
or (vaporize/condense)
( )
heat µ mass involved
Force held by a rod of given area.
F = Force (holding ability)
A = Area of Cross-section.
Force µ Area
holding
stress ş F/A
l = Length of object being
stretched. D l = length stretch
D l µ l
Strain ş
Force on rod. omit sketch
D l µ Force (applied or
holding ability)
D l µ length of object
D l µ 1/ Area x-sec obj.
E = Elastic Modulus
(may ignore units)
( ) µ
( ) ( )
( )
(1/Eş
( )__________ ( _)( _)
( )
Spring Equation is F= k Dx.
E is defined to give similar eqn: ( F / A) = E (D l / l ). And from definitions (stress) = E (strain )
("pressure") = E (“ % stretch”)
Atoms and Gasses.
= (Avogadro’s Number)
( )
(# atoms) µ (# moles)
N µ n
sketch a graph for each law
Charles's P µ ( )
Gay-Lus. P µ ( )
Boyle’s P µ 1/ ( )
= Ideal Gas Constant
( )
( ) µ
( )( )
( )
( ) ş
( )__________ ( _)( _)
( )
k = Boltzman’s Constant
( )
energy per µ temp.
atom/deg. of gas
of freedom
P = percent ş # per 100 ş
# for each 100 items. Example: ( # $ per 100 $ )
(Interest) µ(Principle/100)
P ş
S = Entropy
Q = Reversible Heat Flow
( )
Q µ (Temperature)
Ef µ (Area )
Ef µ (time )
I = Intensity of light or sound
Ef = Energy (Flow) of light or sound. (Joules/ sq. meter/sec)
I ş ( )( )
Put any other proportions you can think of here.
Additio Summary
The proportionality constant is the relationship between the two variables. Example: Density is the constant relationship between mass & volume in homogeneous substance like steel. Density is an example of an "intensive quanity", i.e. how much "concentration".
The two var-iables above are called “extensive variables”.
Calculus is used to change the ratio constant (above) to a variable, in which case the defined concept is now called an “intensive variable”.
The proportions listed on this table are only those where the constant ratio defines a new concept. There are many other proportions in physics where where for a variety of practical reasons the constant ratio happens not to be given special name. Here are some examples:
Work ş (Fconst.) s OR Torque ş F (xconst. ) OR KE = 1/2mv2 OR PE = mgh OR Velocity Wave = f l .