Topic 1 homework practice (not for grading) . . . careful of ± signs here!
Topic 1 Summary :
To measure an artifact means to compare one aspect of it to the same aspect of a "Unit Artifact" ;
which procedure is used determines which quantity (aspect) is being measured (e.g, height, weight, Volume...);
that property of the artifact is divided by the same property of the Unit.
... a quantity is reported : multiply the count by the Unit ... the Unit contains all the info about the procedure.
Units are more important than the number ; the property (name) is more important than the units
... greatest importance is whether it is an instantaneous condition , or an ongoing process
You should draw a sketch of each situation , label relevant quantities on the sketch.
. . . the ones you "know" values for , and the ones still "unknown" => especially the answer subject !
All of your statements should begin with a subject , before the predicate "is" ( = ) , then the object
. . . the "climax statement" should begin with the answer subject (the "desired unknown")
perform the same computations on the units that you do on the numbers
. . . you'll learn more if you de-brief yourself after the answer :
does it seem to be the "right size" to make sense? why is it so big or so small?
Mass : m ; is an intrinsic property of an object ; mass exhibits weight depending on its environment condition local gravity field strength
but an object has mass ; mass is its "unconditional" condition . NOT directional , measured in [kilograms] = [kg] .
explanation depends on mass ; cause-and-effect statements require quantities which do include mass
Location: r , or x , or y ; measured in [meters] , abbreviated [m].
condition vector , draw as originating at (arrow tail is at) the origin (observer) ; location is relative to the origin
... location is important in description , but not in explanation
If you want to find locations of things relative to object A , just subtract object A's location from every origin-based location
. . . notice how this makes object A's location = 0 , so it is at the "origin" relative to itself.
=> Length L ; is the "difference in location" between the two ends ... ( δx ) ... at the same time
. . . verify that the length relative to object A is the same as the length relative to the original observer
=> distance is the total length along the path ... each small step-length added tail-to-tip
Area: A ; is the average Length in one direction × entire Width (perpendicular to L)
. . . notice the adjectives in the statement above . . . usually these are written as subscripts after the symbol .
Volume: V ; is the average Area (cross-section) × entire height (perp. to each cross-sect.surface)
Any quantity per Volume is called that quantity's density . . . abbreviation symbol ρ "rho"
. . . if the spacing between individual items is essentially the same in different regions ("homogeneous") then the Number of items will depend on how much Volume is considered
but the Number density will be the same regardless of the size of region sampled .
. . . so the number density, and the mass density, is a property of the material , rather than just that object .
typical mass densities, in [kg/m³] : (at room Temperature, except for ice)
H2 0.083 . . . He 0.168 . . . NH3 0.717 . . . H2O vapor 0.749 . . . air 1.2 . . . O2 1.331 . . . CO2 1.842
solid water 920 . . . liquid water 1000 . . . clay 1500 . . . calcium 1650 . . . sandstone 2200 . . . limestone 2600
aluminum 2700 . . . granite 2700 . . . basalt 3200 . . . malachite 3850 . . . hematite 5150 . . . iron 7870
nickle 8900 . . . copper 8960 . . . silver 1050 . . . lead 1135 . . . mercury 1355 . . . gold 1932 . . . platinum 2145
. . . how compact a population is, is usually expressed by "population surface Area density" = N/Area instead of N/Volume
Time : a parameter that increases at a constant rate ...
... time of day might be descriptive, but is not important in explanation
=> the physically meaningful time is the duration (interval) , measured by events that repeat identically , in [seconds] = [s] .
"Change" (abbreviation : Δ ) implies that something (say, quantity Q ) is different at a later instant
. . . (things which can change have a "starting condition" and an "ending condition")
=> Qearlier + ΔQ = Qlater . . . time-ordered story-line
Duration is "change in time" between "start time" and "finish time" ... ( Δt ) ... at the same place
duration is the process of time changing ; other conditions might also change meanwhile.
(it is non-sense to refer to any process "at an instant" ; nothing happens unless the two instants are separated)
... if some quantity (say, Rc.o.m.) endures for a while, then there is an average value of the quantity within that duration.
A quantity per time duration is that quantity's rate
... frequency ... f ... is an "event" rate , so Time Duration from one event to the next is : Δt = 1/f
... speed is the travel distance rate ; velocity is the displacement rate .
Important Statement #1 about the Universe :
mass is a "conserved quantity" ... this means that the total mass is the same at every time.
=> Σmtime 1 = Σmtime 2 . . . Greek capital Sigma " Σ " means : Sum up all of the ____ ;
. . . Σmt_1 = Mtotal,t_1 = mobject_a,t1 + mobject_b,t1 + mobject_c,t1 ...
If the mass inside some Volume has changed, this implies that this same amount of mass has crossed the Volume's boundary (surface Area)
A quantity per mass is often called the "specific quantity" ... it is meaningful because mass is conserved .
the "mass-weighted sum" of locations : ma·ra + mb·rb + mc·rc ... = Msystem·Rsystem
=> center-of-mass location Rystem is the place which the entire set of objects seems to be centered .
mass is usually measured by comparing gravity's pull to the unknown with gravity's pull to a unit mass [kilogam].
. . . this procedure is called balancing the unknown mass ... balance is achieved when the mass-center is vertical below the knife-edge pivot.
modern mechanical balances have masses (0.100 kg & 0.010 kg) that can be slid along a ruler
displacement : the "change in location" (of an object) during a duration , ( Δr ) , is called "displacement" ... it is a process .
=> an interval of time is the total of all the durations along the path ... in each different condition ... it is a process .
velocity : vaverage = Δr / Δt ... the velocity is averaged all thru that duration
. . . but if the duration is short enough , the RATIO becomes constant therein (so won't depend on the duration)
=> so we can consider velocity to be a condition - an object has a velocity at every instant .
. . . the displacement process Δr occurs when average velocity vavg endures thru a duration process Δt .
=> rbefore + vavg.during·Δt = rafter .
If some object's velocity is supposed to be referenced to a (moving) object A ("object's v relative to A"),
we use the relative final location (xobj,f − xA,f) and its relative starting position (xobj,i − xA,i)
. . . to get its relative displacement Δxobj,rel.to.A = Δxobj,rel.to.0 − ΔxA,rel.to.0 ...
. . . and its velocity relative to A is vobj,rel.to.A = vobj,rel.to.0 − vA,rel.to.0 since A and 0 agree on Δt .
The physically important feature in motion includes velocity but also how much substance has that velocity :
. . . momentum , abbreviated p , can be computed as mass · velocity .
momentum specifies the "quantity of motion" within an object (impetus concept of John Philoponos, 512 AD)
. . . is NOT an intrinsic property of the object , but will depend on its condition (speed & direction of motion)
it is NOT intended to be a "non-material substance" injected between the object's atoms ,
but rather is a vector (directional) quantity within each atom (or subatomic piece of matter).
Because an object can have starting velocity, ending velocity, average velocity, change in velocity ... all different,
the object can have starting, ending, average, and change in momentum , all being different
=> momentum is a conditional object property - because mass and velocity both describe an object at any instant.
. . . "momentum at any instant" is meaningful if the object's velocity does not change in "jumps" (that is, if v(t) is continuous , which means r(t) is smooth)
momentum specifies the "quantity of motion" within an object (impetus concept of John Philoponos, 512 AD)
Important Statement #2 about How the Universe Works (from Philiponos but usually called "Newton's 1st Law")
Total momentum is always the same (unless there's an unbalanced external cause that changes it).
. . . this "momentum conservation" statement is useful, to predict many outcomes ... remember that momentum is a vector (directional +/-) !
=> Σ pbefore = Σ pafter
a) the limit of average velocity , as Δt→0 , will exist as a single value .
. . . notice that the average location, and the average time, and the average velocity, are all in the middle of the interval
this symmetric derivative works essentially the same as the one-sided derivative usually used in math courses.
=> v(t) = dr(t)/dt , as a derivative .
b) the velocity of a system's center-of-mass is constant (always the same)
. . . we get to choose which objects we include in "the system" (just include same ones later!)
c) if some process occurs that does cause momentum to change , called an Impulse ,
. . . then that Impulse must have occupied a (non-zero) time duration (... so expect the Impulse rate to be important)
d) we can choose any (constant) velocity to observe from
. . . only relative velocities are physically important.
Forces that are applied to the object must be added directionally (as vectors) so that they cancel each other, for momentum to be constant.
. . . gravity pulls each mass downward, toward Earth's center ... proportional to the mass amount ... proportional to gravity
=> Fgrav = m g (↓) . . . g on Earth surface ≈ 9.8 N/kg
. . . cords, ropes, strings, threads pull their ends inward against their length (toward their center) , sometimes called Tension Force T
=> FTension = (in) , just strong enough to maintain their length (i.e, keep from breaking) ... until they break as their Tensile Strength is reached
. . . hard surfaces push directly outward (through their contact Area against the other object) , sometimes called Normal Surface Force N
=> Fsurface,N = (out) , just strong enough to maintain their shape (i.e, keep object from sinking into it) ... until they break as their Compressive Strength is reached
more about Forces in later Topics
conditions (instantaneous) processes
t, m(t), r(t), v(t), p(t) . . . Δt, Δm, Δr
Can we put together a physically meaningful process that would be done by an object (and expected to be non-zero), yet?
Action , usually abbreviated S , is done while the object carries its momentum for a ways .
=> S can be computed by p·Δr . . if momentum is constant thru the displacement . . .
. . . since any negative displacement can be done only if the average momentum is negative,
the Action done is always positive (or zero) . . . measured in [kg·m²/s]
. . . if the momentum is not constant (time-wise), it can't be the same at every location within the displacement
then Action is approximated by a continuous sequence of little "partial actions" (p(x)·dx ), added together (an integration)
Examples done in 211 class
- mass of air in the lecture room:
m = ρ V ; V = L×W×H ; we had to average the height Havg ≈ ½ (Hmax + Hmin) => V ≈ 300[m³]
using the mass density of air (presuming typical Pressure) ρ ≈ 1.28[kg/m³] , we got m ≈ 384[kg] ... ~5 people's mass, for 75[kg] people.
- center-of-mass for Sun + Earth + Jupiter "system"
Σ(mi Xi) = Msystem Xsystem
(2 000 000 E24[kg])·0[m] + 6 E24[kg]·150[Gm] + 1900 E24[kg]·778[Gm] . . .
- Quantity = average rate * total duration
average extraction rate during the "good ole days"
was about half of the maximum rate ...
Q = ½·9[Mbarrel/day] ·55[yr]·365[day/yr] = 900,000[Mbarrel] = 900[Gbarrel]
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