 |
|
Phy.201 Links:
» Syllabus
» 201 home
- - - Unit 1 - - -
» topic 1 (I&1a&2a&5a)
» topic 2 (1b&2b&5b)
here! ⇒
» topic 3 (3&4a)
» topic 4 (4b&7)
- - - Unit 2 - - -
» topic 5 (6&4b)
» topic 6 (8)
» topic 7 (19)
» topic 8 (20)
- - - Unit 3 - - -
» topic 9 (10&11)
» topic10 (12&9a)
» topic11 (13&9b)
- Off-Campus Sites -
» MasteringPhysics
(online hw login)
» Blackboard
(So.Chas.Access)
» physicsforums
(human.hw.help)
» hyperphysics
(detailed ebook)
» Minds-On-Physics
(decent practice like hw)
("continue" below red words)
|
|
|
|
College Physics I (PHY.201 - §201, 2018 Spring => CRN 4575)
Class Meets in : Science 276 ... Mon & Wed & Fri 8:00 - 8:50 (am)
My office: Science 159 (below ramp to 3rd Ave) e-mail : foltzc @ marshall.edu phone : (304) 696-2519
... Topic 2 hw is up, due Thursday Feb.01 (midnite)
... Plan for Quiz 2 to be Fri.Feb.02 ... here is my solutions key as jpeg image
Physics 1 Topic 2: Topic 1 Reminders :
Units are more important than the number ; the "thing that it is" is more important than the units
... even more important is whether it is an instantaneous condition , or an ongoing process
... maybe most important is whether the thing describes an object , or the environment the object is in .
You should diagram each situation , label relevant quantities on the sketch , in symbols
. . . the ones you "know" values for , and the ones still "unknown" => especially the answer subject !
All of your statements should begin with a subject symbol , before the predicate "=" , then the object.
. . . the symbol "=" can be read as "is" ; but you should recognize (and try to use) more precise meanings
including : "is defined as" , "causes" or "produces" , "comes from" or "is caused by" , "is determined by" ...
. . . (did you notice that none of this list is reflexive? that none are symmetric? ... which ones are transitive?)
The first statement in your answer should be a general statement , in symbols , about the situation or scenario
. . . we try to use meaningful symbols for nouns - often abbreviations for a phrase
we routinely modify nouns with adjectives and prepositions ; these are written as subscripts .
. . . the "climax statement" should begin with the desired unknown as its subject . . . (earlier ones don't need to)
. . . perform the same computations on the units that you do on the numbers
. . . you'll learn much more (with little more effort) if you de-brief yourself after getting the answer :
a) typical values at the scale of this scenario : is the answer the "right size"? why is it so big or so small?
b) improve your effectiveness : what did I get stuck on? what got me un-stuck? how to be stuck for less Δt ?
c) how physical quantities relate to each other: if given#n was 2× as big, result#a would be ____ × bigger ...
- - - choose an origin for time and location - - - to make important condition properties - - - zero - - -
If you start your stopwatch at the "right" instant (defining t=0) , you can make the initial time ti = 0
so that Δt (= tafter - tbefore) becomes just t .
. . . you might want to make x0 = 0 , or v0 = 0 (or at least known) , or ...
 |
3m cart collides with stationary 1m cart, and sticks:
... blue vector arrows are momentums (ptail @ c.o.m.) ; red vector arrows are Forces.
... (initial → is Force from 3m into 1m ; initial ← is Force from 1m into 3m)
initial Force pair (compression) transfers ½ the 3m's initial momentum into the 1m
later Force pair (⇄ tension) transfers ½ of that back to the 3m (¼ of pi), to stick them together
|
Topic 2 Summary : How things Change
. . . Big Idea #2 . . . about how the Universe Works:
Force being applied to an object for a time duration CAUSES the object's momentum to change.
... the Force is applied by the subject - distinct from object - as the object's external environment .
. . . While the Force is being applied , a process is occurring (called "Impulse") :
=> on objectFby subject Δt = Δp .
We can re-arrange the two changes into a ratio:
=> on objectFby subject = Δp/ Δt
Since the object's mass is constant, the momentum changes only because the velocity changes:
=> F = m Δv / Δt = m a . . .
This acceleration a has direction ... same direction is positive as for x , and v , and p .
. . . (by now you habitually indicate the "+'ve" direction on your sketch ...)
this a is the average value that exists for the duration Δt of the applied Force . . .
. . . connects the earlier condition (v1) with the later condition (v2) .
Since momentum , as a function of time has to be smooth through time (with no "corners") , the limit as Δt→0 exists , and
=> a(t) = dv(t)/dt , as a derivative .
. . . this means that the acceleration at is the slope of the vt graph .
. . . B. Forces are caused by location-dependent sources
While the Sum of Forces causes the (single) acceleration of a (single) object,
each individual source of Force is caused by conditions specific to that type of Force .
It turns out that the location of the object , relative to the source , is a key aspect for all types of Force .
. . . features which determine the strength and direction of a Force can be written as a function !
gravity Force: depends on the object's mass m - mass is a scalar intrinsic property of the object
(an extensive property ; two identical objects together have twice as much as either one)
. . . also on depends on the strength and direction of the gravity field at the place where the object is
9.84 N/kg (down) at Earth surface , 9.83 N/kg at 6.4 km below surface , 9.83 at 3.2 km altitude ; 1.6 N/kg on Moon
the gravity field at any place is an intensive property of the environment there .
=> Fgrav. = m g .
Tension : in rope , cord , string ... ideal (textbook) rope is perfectly flexible, does NOT change Length, and has no mass
. . . (although you need to apply a sideways force to flex a real rope , and they do have mass)
. . . they only pull , inward on both ends , equal strength , along the rope ; the pull is called Tension .
microscopically, the rope Tension is a pressure "stress" exerted thru its cross-sectional Area
= > FTension = parallel to rope .
spring : depends on the stiffness k of the spring , which we will treat as a scalar intrinsic property of a particular spring
. . . (although a real spring's stiffness does change with age or abuse)
. . . and also depends on the distance and direction that the spring has been stretched : s .
the spring pulls in the opposite direction from the stretch vector (or pushes opposite the compression vector)
. . . this is parallel to the length of the spring ! neglect the mass of ideal (textbook) springs
= > Fspring = - k s .
Pressure : depends on the pressure at the contact surface (a scalar intensive property)
. . . and also on the Area of the contact ... pressure Force is through (perpendicular to) the Area.
. . . one might argue that Pressure is more fundamental than Force
. . . P = F spread out over a surface that's | to the Force (Force pierces Area) .
=> Fpress. = P A .
Gravity : every mass in the universe pulls on every other mass in the universe . . . wait till Topic 3 !
. . . each source mass contributes to the gravity field g at each location.
the total gravitational influence emanating from source mass "M" ~ GM , toward the mass ,
with G = 6.67E−11[Nm²/kg²] = 66.7[m/s² · (Mkm)²/(1E30kg)] ,
or . . . G = 5.931E−3[m/s² · (au)²/Mass_Sun] <= astronomical units
. . . this influence spreads out, 2 ways in 3-d space, so the contribution to g is much weaker far from the source
the contribution to g from every source mass (reasonably nearby) must be added as a vector .
= > gby M = G M / r²M-to-point (toward M) . . . then use Fon m = mg ; NOTICE : m ≠ M !
It is the vector SUM of all Forces applied to an object by external subjects (at any instant)
. . . which causes the mass of that object to accelerate . . . or its momentum to change over time .
You write : ΣF|| = m a|| ... and : Σ F_|_ = m a_|_
. . . very often the component _|_ to the path (perp. v) will have zero acceleration (... in topic 3, we'll use a·r = −v·v to calculate it).
A ½kg cylinder hangs (stationary) from a spring's hook, stretching the spring 3 inches (45mm) ... a) how stiff is the spring?
object is hook ... ΣF = gravity Force down + spring Force up, cause hook mass to remain at height z = h ≈ 1½m above floor.
. . . so Δz =0 for all durations ; so vi =0 and vf =0 also ; so Δv =0, so a =0 ; so ma =0 .
. . . spring Force up is opposite gravity Force down ... (−ks-3) = (−)(mg)
. . . k = (0.5kg)(9.8N/kg) / (45mm) ⇒ 4.9N / 45mm = 0.109N/mm
b) if the hook and cylinder were pulled 2 inches farther down, then released from rest, what acceleration would result?
. . . ΣF = spring Force up + gravity Force down , causes m to a ... −ks-5 + mg
= (0.109N/mm)(75mm)(↑) + (0.5kg)(9.8N/kg)(↓) = 8.17N(↑) + 4.9N(↓) = 3.27N(↑).
⇒ a = ΣF/m = 3.27N/0.5kg = 6.53N/kg ( = 6.53 m/s²)
You get to choose which objects in the scene are included INSIDE your system
. . . it is often simplest to "hide" rope Tensions and some contact Forces inside the system
so that they are not included in the external Force Sum ... but the Total Mass inside gets accelerated
. . . later (when a is known) you can choose a smaller system , so the Tension pierces the system boundary
and will be included in the Sum of external Forces ... so that you can solve for it !
Atwood's Machine has a cord over a pulley; 1kg hanging from each side. at t=0 an extra 0.10kg will be added to the left side.
. . . if we choose "leftward along the rope" as positive direction, the only unbalanced Force is +0.98 N {= mg = (0.1 kg)(9.8 N/kg) leftward}.
Σ F = (Σ mi)(Δv / Δt) = (2.1 kg)(a)
=> a = 0.98 N / 2.1 kg = 0.467 m/s² . . . the cylinders should travel 1 meter distance (Δx ≈ ½ a t² =>) in t = sqrt(2·1m/0.467m/s²) = 2.07 s .
... verified in-class ... the momentum at floor impact was (0.98N)(2.07s)=2.03Ns , so its velocity at impact was (2.03kgm/s)/(2.1kg) = 0.966m/s.
... note that its average velocity was (vmin+vmax)/2 = (0+.966)/2 = 0.483m/s ... Δx = vavgΔt
. . C. Kinematic Equations . . . math withOUT cause-and-effect
(Reminder, definition of average velocity) Kinematic Equation #1 : vavg ≡ Δr / Δt
=> place story line : rstart + vavg·Δt = rend (acceleration is not explicit)
(definition of average acceleration) Kinematic Equation #2 : aavg ≡ Δv / Δt
=> motion story line: vstart + aavg·Δt = vend (displacement is not explicit)
IF acceleration is constant during the interval Δt , then : vavg = vstart + ½ Δv = vend − ½ Δv = ½ ( vstart + vend ) .
=> place story w/ accel : Kinematic Equation set #3 : has 3 different "wordings" mixing the info of #1 and #2 )
. . . (a) . . . rstart + vstart·Δt + ½ aavg (Δt)2 = rend (ending velocity is not explicit)
. . . (b) . . . rstart + vend·Δt − ½ aavg (Δt)2 = rend (starting velocity is not explicit)
. . . (_) . . . rstart + ½ (vend2 − vstart2) / a = rend (how to divide by a vector? Re-Write!)
. . . (c) . . . v2start + 2 aavg·Δr = v2end (time duration is not explicit)
CAUTION : avoid form "c" until you understand multiplication of vectors ... the directions of a and Δr are important !
caution : avoid form "c" until you find out how to identify v² on a diagram ... it discards direction of both velocities
- - - NOTICE - - - which quantity - - - does NOT explicitly show - - - in each equation above - - -
Solvable Kinematic scenarios can only have 2 __TWO___ of these quantities as unknown
. . . make sure the other 3 are __ON_YOUR_DIAGRAM__ . . .(4, if you have xi and xf , instead of Δx )
Your diagram should tell you whether you need to "sub-contract" to obtain a value from a preliminary scenario ...
Solve part (a) for a's unknown , by using whichever equation is _missing_ part b's unknown .
Read the equation as words ... appropriate to the scenario ...
... then , do symbolic algebra to isolate a's unknown . . . RE-READ YOUR RESULT as words
... when satisfied that the relationship is reasonable, plug numbers [ WITH UNITS ! ] and compute your result
. . . THINK ABOUT whether it's ABOUT RIGHT , size-and-unit-wise !
... finally , you can use any of the OTHER equations to calculate b's unknown
. . . even those which need to use a's _now_known_ .
Dynamic scenarios _imply_ a by telling F and m . . . or Δv , by telling Δp and m . . . or combinations with Δt
- - - - the KEY is - - - to TRANSLATE the STORY-LINE - - - to a SYMBOLIC STATEMENT - - -
. . D. Directional Addition . . . add vectors tail-to-tip
Quantities which have direction are called vectors. Vectors are drawn as arrows (with labels, of course).
... we will pattern our treatment of all object vectors on Displacement .
We will always use coordinates with axes that are perpendicular to each other ; then,
motion in any direction is independant of motion in all other directions
. . . an event condition relates these directions by its time
. . . algebraically, the key is to keep each component (x) separate from the others (y and z) !
then , a "big & scary" 2-dimensional scenario becomes 2 "small" 1-dimensional scenarios ... connected via time .
Vector representations : first choose an origin and coordinate system (showing each positive direction).
drawing : a location vector is drawn as an arrow with its tail at the origin, its tip at that location.
this represents a location as if it is a radially-pointing range at some angle from the coordinate axes ... hence the abbreviation r .
. . . vectors of other quantities can have their tail at the object being described,
but must point in the direction of the quantity (e.g, v in velocity's direction).
bold italic letters represent vector quantities !
component : The ordered set of the location coordinates ( x , y , z ) . . . = (right , forward , up) or (East , North , Zenith)
is the algebraic way to write this location vector (Cartesian component form)
. . . a velocity vector is written as the set of velocity components ( vx , vy , vz ) in the same order ... .
In almost every case, it is best to :
0) diagram the "Before" scenario ... include arrows (and labels!) for the vector condition quantities.
½) notice which vector quantities might be different in the "Meanwhile" and the "After" diagram
1) draw and label the important process quantities , from Before to After ,along the path taken.
2) choose a coordinate system ; often parallel (and perp.to) the path v or p ... except in free-fall .
3) split important vector quantities into (parallel , perpendicular) or ( horizontal , vertical ) components.
4) write how the each direction's vector quantities relate to one another ... or to TIME.
Addition : drawing : usually done on a small coordinate system , separate from the main diagram (so it doesn't get cluttered).
the first arrow tail is at the origin of the addition coordinate axes (x=0,y=0,z=0).
its tip points the same direction from its tail as it does in your sytory-book diagram ... to ( x1 , y1 , z1 ).
. . . the vector being added to the previous has its tail placed at the previous tip
its tip points the same direction from its tail as it does in your story-book diagram ... to (x1 + x2 , y1 + y2 , z1 + z2 ).
. . . vector arrows are appended tail-to-previous-tip in sequence, keeping each aligned with its arrow on the first diagram.
The Sum ("Result for the addition") of the vectors is the arrow from first tail (origin) to last tip .
algebraically : the Result of the addition is the list of components : (x1+x2+x3... , y1+y2+y3... , z1+z2+z3... ) .
Adding a vector to itself results in a vector twice as long , in the same direction ; i.e, v + v = 2 v ...
. . . the negative of a vector points in the opposite direction
... a vector added to its opposite ... Δx + ( − Δx ) ... makes the "zero length vector" ... = 0 = (0, 0, 0) .
Process quantities ... displacement (Δr) , boost (Δv) , Impulse (FΔt) , Action (p·Δx) ... add sequentially
make sure the durations being added DO NOT OVERLAP . . . change = change so , there is always one Δ in each term !
Condition vectors ... location , velocity , acceleration , momentum , Force ... are added at the instant of that condition
. . . it is the Sum of simultaneous Forces that cause the effected mass to accelerate (at that instant) : Σ F = m a .
|
