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properties and units
. . . A. Object Properties
Items that are Material Objects posess Properties
WE are the active subject which investigates the passive object's properties
Some properties are Qualities. Here are 3 main kinds:
. . . existent (posessed by that item) or not ... the item is living, or is not ... weather outside is raining, or is not
difficult items & properties, for us to decide existence about,
are usually clarified by us more precisely defining the property
... ambiguity is almost never due to the item's actual properties being "fuzzy".
. . . sortable into categories
dichotomies have only 2 categories ... humans are male or female (approximately!)
trichotomies have 3 categories ... animal, vegetable, mineral (an antique view)
MU course grades need to be sorted into 5 categories A,B,C,D,F
6 "primary" colors: Red, Orange, Yellow, Green, Blue, Violet
. . . rankable in a consistent order
two statements A > B, and B > C, imply that A > C even more than A > B.
rankable properties always have an "inverse" property that ranks oppositely
... if getting a C is "better" than getting a D, then getting a D is "worse" than getting a C
Some properties are Quantities. Quantities must be additive:
. . . there is a physical procedure to add one item's quantity to that same quantity posessed by another item
each quantity-property's procedure is specific to that quantity
... same procedure to add that property for every item
. . . the math procedure to add properties is "addition" (symbol "+")
which adds the numerical results from two measurement procedures.
measurement procedures compare an item's property to that property posessed by a standard item.
. . . measurement counts the number of standard items one needs to add, have the same property as the measured item.
The standard item is called the unit, so any measurement result is a number of units.
"of" here (as usual) means the number is "multiplied by" that unit item's property.
So the unit after the number carries the info about which property is meant,
... which implies the addition procedure ... not just which of alternative standards was used.
It is the properties, not the items they belong to, that science makes claims about !
The natural language to use, to relate one quantity to another, is mathematics (mostly arithmetic and geometry).
the subject in a math sentence is the left side of the equal sign (or other relationship sign)
. . . question scenarios might provide info about several different quantities in various situations;
write a new math sentence for each piece of information that's given in the problem
. . . in (the last line of) your answer to a question, it should be the unknown which the question asked about
if you don't know what is being asked for, you won't recognize it ... so you won't stop!
. . . include adjectives - traditionally as subscripts after each noun's (property's) symbol
enough to distinguish all the quantities for each item and for each situation (instance, or instant)
the predicate in a math sentence is almost always "equals"
if they want "greater" or "lesser", re-word their question to ask about that extreme limit which IS equality
... inequalities are so much harder to work with, re-wording is always worthwhile.
. . . statements about Nature should really have predicate "causes" or "is caused by"
... (less often, "is spread among" or "is mitigated by", etc.)
notice that "causes" and "is caused by" are an anti-symmetric pair
I might write mass in gravity causes Force : mg ⇒ F ... this implies F ⇐ mg ... NOT F ⇒ mg
the object in a math sentence is the right-hand side of the equation
. . . compound subject or compound object ... conjunctions like addition and/or subtraction
. . . prepositions ... try to use descriptive prepositions, to avoid misleading yourself or others
don't say "speeding up" for an item becoming faster downward or southward
don't say "pushing on" some item if you're actually pushing upward from underneath it
. . . B. Quantity Measurement , Addition & Subtraction - Basis of Measurement Systems
Some Properties are easier to measure than others.
We select a set of several properties, to form a Measurement System based on them.
Width, Length, Height
. . . these "directed" properties are added by juxtaposing a new unit object beside, in front of, or above the others
the table's Width is measured by how many 1 foot wide rulers fit beside each other
touching "tail-to-tip" ... think of each ruler as an arrow pointing from the "0" end to the "12" end.
each ruler must be oriented so that their own 1 foot width points along the table's width!
. . . Most directed properties are vectors, which behave in a similar way to W,L,H
the table's Length is measured by how many 1 foot long rulers fit along its Length.
it is really helpful that the same 1 foot wide ruler can be re-oriented to become a 1 foot long ruler
... (without turning it, my 1 foot wide ruler is only 1½ inches long.)
. . . that count will be the table's Width compared to the 1-foot ruler's Width.
the 1-foot ruler's property has been chosen as the unit for this measurement.
Unit conversion: we might have (instead) chosen to use the 1-inch marks Width as the measuring unit
. . . there are 12 inch-widths in each 1-foot width, so the 1-inch number-count will be 12× as many, for the same table.
our front table is 6 foot wide ... = 6 foot wide × (12 inch) / (1 foot) = 72 inch wide ... notice that the foot/foot = 1 (units cancel)
. . . most folks use the same ruler, turned upright, to measure Height.
... (architects in ancient Egypt measured W and L using a ruler that was 3.14 times as long as their Height ruler.)
. . . If we use the same ruler (turned as needed) for all these directed measures,
adding Length to Width yields a bent path from the original corner to the opposite diagonal corner.
= adding Width to Length ... different path, but same ending corner.
=> think of W, L, H as arrows that are added (next)tail-to-(prior)tip.
If W is | L , and they are both | H , then
Pythagorus' theorem relates the diagonal measure to the Leg measurements.
=> d ² = W ² + L ² + H ² . . . ( W | L | H )
Copy-Paper box (11" × 17" × 8½") has 22" diagonal
SI "System International" & metric prefixes
the SI unit for length is the meter, abbreviated m .
originally there were to be 10,000,000m from Earth's Equator to North Pole
(but a war delayed the surveyors for years, and precise N.Pole is hard to find)
. . . careful people specify that they are using the "mks" system
because technically the "cgs" system is within the SI.
cgs uses centimeters as its basis, rather than meters ... "centi" means per 100, so 1 cm = 0.01 meter
. . . SAE has decided to define their "inch" now, as exactly 25.4 millimeters ... so 1 foot (=12") is 304.8 mm ⇐ If you don't know how to get this, ask in Sci.159!
(SAE is the abbreviation for Society of Automotive Engineers ... US Bureau of Standards has stopped using SAE units)
. . . "milli" means per thousand, so 1 mm = 0.001 meter ... you will eventually learn prefixes for these factors of 1/1000
milli (1/1000) ; micro (1/1000 000) ; nano (1/1000 000 000) ; pico (1/1000 000 000 000) ; femto (1/1000 000 000 000 000)
index finger ~80 mm long ... fine-tip line ~500 μm wide ... sugar molecule ~2 nm long ... atom ~200 pm diameter ... atom's nucleus ~ 5 fm diameter
... ignore centi, so it makes sense ... the rest are unusually small → [ ... atto (1/10-18) zepto (1/10-21) ; yocto (1/10-24) ]
. . . "kilo" means thousand, so 1 km = 1000 meters. You will eventually learn prefixes for these factors of 1000:
kilo (1000) ; Mega (1000 000) ; Giga (1000 000 000) ; Tera (1000 000 000 000) ...
6 city blocks ~ km ... ~1½ Mm from WV to FL ... 150 Gm from Earth to Sun ... ~3 Tm from Sun to Uranus
... the rest are unusually big → [ ... Peta (1015) ; Exa (1018 ; Zetta (1021) ; Yotta (1024) ]
nearest star ~10 Pm ... galaxy center ~250 Em ... galaxy dia ~1 Zm ... Andromeda ~25 Ym
Notice that Width and Length and Height are not useful properties for liquid samples nor gaseous samples.
These properties are not intrinsic to these samples, but rather depend on environmental conditions
... for example, water sample Height depends on the Area of the beaker that contains the water.
The key feature is that a liquid sample has a constant Volume, but acquires its shape from the container.
=> Volume = Area × Height ... use the average Area if the container is cone-shaped !
Gas samples don't even have a fixed Volume, but occupy the entire Volume of their container.
. . . A. Big Idea #0 . . . about how the Universe Works: Mass is conserved
. . . an undirected (scalar) intrinsic object property.
depends on nothing except how much matter the object contains.
additive, so it is a quantity. Summing a lot of individual mass gives the total mass:
=> Σ mi = mtotal . . . capital greek Sigma means sum the individual item properties.
SI unit for mass is the kilo-gram , abbreviated kg ... = 1000 grams (the cgs unit)
originally there were to be 1000kg (= 1 "metric tonne") in 1 cubit meter of water
. . . mass is conserved ... this means that the total mass amount is always the same
"mass can not be created, nor destroyed"
if there is less water mass in the beaker now, than there used to be earlier,
it is because some water mass left the beaker (moved out, through the open top's Area).
Weight
. . . directed downward (vector), object-in-environment (product) property.
"US Customary unit" system chose weight rather than mass as its measurement.
. . . Earth gravity pulls an object's mass property downward (masses attract other masses)
so weight depends on object mass and on the local gravity intensity:
=> Weight = mg . . . gEarth surface ≈ 32 pounds/slug (↓) = 9.8 Newtons/kg (↓)
. . . Weight (a Force) is usually measured by having adjusting a different Force to (exactly) cancel it;
in most Force-scales, Weight deforms a spring until the spring's Force (↑) cancels mg (↓) .
the spring deformation (usually stretch or compression Length change) is calibrated in pounds or Newtons.
... "calibrate" means to hang a "legal" 1-lb weight from it, and mark that deformation as 1 lb ; then do 2-lb, etc.
. . . "Good springs" apply Force proportional to their stretch or compression Length change:
=> Fspring = − kstiffness ΔL . . . stiff springs push hard even with small deformations.
bathroom-scale springs deform very little (~1mm), so that ΔH is tricky to measure with much precision;
modern scales measure the deformation electrically (strain⇒resistance), rather than with mechanically (levers & gears).
Total mass is constant, so by itself it is somewhat boring property.
. . . Where the mass is becomes the big deal, because the mass property can move,
carried by the object which posesses it.
. . . mass moment is the product when an object's mass is multiplied by the object's location.
mass moment is another quantity ... that is, its sum is meaningful.
=> Σ miri = mtotal rcenter . . . that place is the "center of the mass" .
. . D. Comparison Ratio : One Quantity Divided by a Different Quantity
A Quantity per Volume = that Quantity Density
. . . Everybody expects bigger things to contain more mass (and weigh more, cost more, pollute more, etc)
these quantities are extensive ; their values depend on the object's size-extention.
. . . So the ratio (big block mass)/(big block Volume) might be the same as (small block mass)/(small block volume)
that ratio yields an intensive material (or environment) property .
=> ρ = m / Vol ... Archimedes' ratio ... called the mass density.
. . . a species' population density, a fuel's Energy density, gasoline's cost density, molecule density (mol/L) ... all describe "concentration".
A Quantity per Mass = Specific that Quantity
. . . Everybody expects that more massive things will occupy more space (and weigh more, and cost more, ...)
very similar to having more members.
. . . the ratio (big bag price)/(big bag mass) should be quite similar to (≈) (small bag price)/(small bag mass)
that ratio is called "specific price" (though most shoppers just refer to it by its units : $/lb or ¢/oz )
. . . Why does diesel fuel cost more per gallon? ... all hydrocarbons have nearly the same specific Energy (mass is more relevant than volume)
=> Volume / mass = 1/ρ ... would be called a material's "specific Volume".
many interesting properties really depend on the number of items - "per capita" or "per connection" would be appropriate
- so "specific heat capacities" decrease (essentially as 1/A) along the periodic table.
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