» 211 Syllabus

Topic Three - Vector Components

centripetal components : Ch.4 §5 , Ch.6 §3 - §4
elastic Force : Ch.6 §2
Gravitation Source : Ch.9 §1 - §4

elastic material : Ch.14 §4

elastic Force : ch.6 prob. 37,39,40,42,43,45 , 46,47,48

Gravity Field : ch.9 prob. 1,3,6,7,8,12 , 13,18 , 17 , 19,20,23,27,28,30,32,33,34

elastic moduli : ch.14 prob. 66,67,69,71,73,79,81 , 75,76

Pressure Force : ch.18 problems   topic 1 : 1,2,5,7,8 ;   topic 2 : 9 ;   topic 3 : 12,13,19,21,23 ;   topic 4 : 25 ;
        topic 5 : 11,29,31,35,38,39 , 44,45,47,49,52,55,56,61

Rules for Working on Homework for Grading:

Presenting someone else's work as if it was your own work
    is a violation of Marshall's Academic Dishonesty Policy.
This "work" includes the physical Work of pushing a pencil across a page, or typing on a keyboard
    and also to the artifact created by those actions -
    work includes the mental effort used in performing these activities,
    and also the thinking that was needed to create the response.

DO NOT COPY other people's homework solutions!

How to work with one or two study partners (or a tutor), and not copy :
                        (no, a group over 4 is not for learning)
  0.  draw your own diagram first ... then show ... reach a consensus diagram to answer
    (re-read & argue as needed)
  1.  now use your own thought process, to come up with an answer
    (get "unstuck" with "hints" of a few words - not "formulas")
  2.  do your own algebra, substitute your own values in, and do your own calculator entry.
  3.  then check with your study partner's answer (number and units)
  4.  have each describe their solution in words only (no math) ... (is their logic good?)
  5.  if your answers disagree , check each other's values for intermediate results

HomeWork 7 (classes 11,12,13) for grading , due Friday Sep.17 ... Monday's Quiz is about this ...

1. A 1.5[Mg] log rests on a 15% grade (that rises 15[m] for 100[m] horizontal distance) ;
     the surface is very slippery mud. A rope from the log is parallel to the slope and passes over a pulley, attaching to a 0.5[Mg] counterweight that hangs freely.
         a), b) find the acceleration of the log, and the Tension in the rope.
         c) Find the Normal (Pressure) Force by the muddy slope applied to the log.
         d) If the mud is NOT perfectly slippery, but has μ = 0.14 , use the friction Force to find the log's new acceleration.
2. Last week a 2[ton] device was removed from the Science building roof ; as it hung from the crane by a cable, a rope pulled sideways (+x) so the cable was 7° from vertical.
         a), b) Find the Tension in the cable, and the Tension in the rope.
         c) If the rope Tension suddenly doubled, what would be the initial acceleration for the device?
3. In the FirstMars Olympics (2112), a 0.8[kg] javelin, thrown with speed 30.5[m/s] at 41° above horizontal,
     hits the scoreboard with its velocity 17° below horizontal. Using g_mars = 3.72[N/kg] ,
         a) find both velocity components for initial condition , and for final condition
         b) find the time duration for javelin flight before hitting scoreboard (Mars!) . . . {hint: what change do we know? what made it change?}
         c) find the displacement components : Δx , and Δz
         d) If you had been given Δx and Δt instead of the initial velocity vector, outline a scheme to deduce v.
4. Wiley E. Coyote (20[kg]) lights his rocket-powered roller-skates just as Roadrunner passes him at 15[m/s] ; one rocket thrusts forward 60[N] and one thrusts upward 60[N] ... < oops, Wiley did not fasten rocket #2's orientation > .
         (a) How long (time) does it take for Wiley's speed to match Roadrunner's , on this level road?
         . . . (you should be able to verify that Wiley does stay in contact with the road ... what is F_N ? )
         (b) How far does Wiley need on level straight road before he catches up to Roadrunner? ... going how fast, by then?
     ... the road curves sharply moments (Δt≈0) before then ; Wiley goes straight instead, off a 400[m]-high-cliff, with rockets blazing.
         (c) How far from the cliff edge should we look , to see his puff of dust at the ground ? {show all steps!}
         (d) What would Wiley's velocity (vector components) be, just before he lands (if rockets keep blazing)?

HomeWork 6 (class 9) for grading , due Friday Sep.10 (serious attempt by class-time)

1. A 1.5[kg] lab cart on a ramp, inclined 22° from horizontal, is pushed (by hand)
     with a 10[N] Force up-wardly along (parallel to) the ramp .
         a) compute the Force caused by Earth gravity applied to the lab cart ... what direction is this?
         b) compute the two components of weight that are parallel to the ramp, and perpendicular to the ramp;
         c) calculate the total Force component along the ramp ... what is the cart's acceleration?
         d) does the cart accelerate into (or out from) the ramp surface? ... so, what's the (Normal) Pressure Force by the ramp?
2. a 0.058 [kg] tennis ball was moving −20[m/s] x + −2[m/s] z ; then Venus hits it with her racquet, which is tilted upward 8° , and squishes the ball 12[mm] before the ball's x-velocity component momentarily stops (soon after, it bounces off with positive x-velocity!)
         a) What was the ball's average acceleration x-component during this "squish" half of the collision?
         b) What was the average Force x-component during the "squish" ?
         c) If the racquet's Force is thru the surface (along A, at 8° upward) , what is the vertical Force component by the racquet?
         d) including the weight of the tennis ball, what is the total Force applied to the ball?
         e) If the same average Force is applied for the "un-squish" half of the collision, what would the ball's velocity components end as?
3. The 80[kg] hero drags a 102[kg] unconscious alien 12000[m] across the (horizontal) sand in 5 [hours],
     by pulling with 400[N] on parachute cord that is 37° from horizontal.
         a) compute the horizontal & vertical Force components applied to the alien by the cord ;
         b) compute the Force components applied to the alien by Earth's gravity
         c) calculate the two Force components applied to the alien by the sand {hint: what is the alien's accel?}
     . . . the component along the sand may be called friction.
         d) compare the Force component along surface (|| v) to Force component thru surface ( | v) ... [divide F_fr by F_N]
     . . . that ratio (called friction co-efficient , μ ) depends mostly on the surface condition (roughness)
4. . . . same hero-with-captured-alien scenario . . .
         a) use Newton#3 to write the horizontal & vertical Force components applied to the hero by the cord ;
         b) compute the Force components applied to the hero by Earth's gravity
         c) calculate the two Force components applied to the hero by the sand
     . . . notice that the hero's Friction-to-Normal ratio is less than the alien's (or else hero's feet would slip)

(to topic 1 summary)

USE UNITS all thru your answer ... don't just append them onto the final number !

DRAW a DIAGRAM for each scenario . . . label the diagram with SYMBOLS , as you read
. . . condition points , words like   "start" , "end" , "turn-around" . . . with vector arrows
. . . process spans as brackets or lines , words like   "average" , "moved" , "change" , "Δ" , "t=0→2" ...

Write statements as symbols first ; manipulate symbols before plugging numbers.
. . . each statement should have a SUBJECT . . .
. . . keep track of adjectives such as   "initial" , "average" , "final" , "stopped" . . . "change"

(to topic 2 summary)

subscripts for "total" or "system" , "part A" ...

keep track of the SOURCE for the external Forces applied to the (passive) object

Recognize whether you are predicting based on theory . . . reasoning from causes to their effects ... (what should it do?)
      or whether you are deducing from observation . . . (what does this imply about it?)

Get un-stuck (often) by wondering "why isn't it the same as it used to be?"

Topic 5 Summary :

While the Sum of Forces causes the (single) acceleration of an 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 !

spring Force : depends on the stiffness konstant k of the spring , which is a scalar property (almost) intrinsic to a particular spring
      . . . (although a real spring's stiffness does change with age or abuse)
. . . and also depends on the distance and direction which that end of the spring has been stretched : s .
      the spring's end pulls in the opposite direction from its stretch vector (or pushes opposite its compression vector)
      . . . this is parallel to the length of the spring !     neglect the mass of ideal (textbook) springs
=> F_spring = − k s .

Two springs hooked end-to-end in series (anchor...spring1...spring2...Force , as : |-v^v^v-·-^v^v^v→ )
. . . will stretch farther than either of them ...the stretch distances add , so s_total = s₁ + s₂ ,   but their Tensions are the same
=> 1 / k_series = 1 / k₁ + 1 / k₂ .
this implies that longer springs have smaller stiffness k than otherwise similar short springs .

If two springs are both connected to the anchor and to the external Force, in parallel, ( |‾‾‾→ ) , their Forces add
. . . their stretch distance is the same, but Forces add :
=> k_parallel = k₁ + k₂ .
this implies that thicker wire springs have larger stiffness k than otherwise similar thin-wire springs .

elastic modulus : even straight steel wire stretches a little bit if pulled with enough Tension ...
. . . the cause of the stretch is stress : Tension spread across the cross-sectional wire Area , in [N/m²] or [N/atom²]
      the effect of stress is strain : fractional length increase , ΔL /L , due to increased average atom-to-atom distance parallel to T
. . . the ratio : stress/strain is Young's modulus Y , which is a property of the material (not just the object)
      Y describes the Pressure that would cause its length to double (if it didn't break first)
=> P = Y ΔL/L , for Pressure in solids .

for fluids, which have no intrisic shape, we describe the change in Volume as they are compressed :
. . . each atom itself is a little bit compressible , with non-infinite elastic stiffness .
=> P = −B ΔV/V , where B is called the Bulk Modulus for the material.

Pressure : more intense Pressure at deeper depths within a non-accelerating fluid ;
. . . ΣF_z=m a_z = 0 => P_bottom·A_up = P_top·A_down + m_fluidg ;   m = ρ A×δh
      Pressure at bottom of fluid sample = Pressure at top of fluid sample + ρ g δh ...
=> δP = ρ g δh .

Pressure Units are [N/m²] ; this set of units is also called a "Pascal" , abbreviated Pa .
. . . but other pressure units are also used : besides the "psi" = pound-Force per square inch , there are also
      torr : pressure caused by 1[mm] deep mercury column in standard Earth gravity (9.81[N/kg) . . . and
      "inches of water" , P caused by 1" deep column of water in 9.81[N/kg] gravity . . . and
      "atmosphere" , the pressure caused by "standard" depth (to "sea level") dry Earth atmosphere in standard Earth gravity

. . . Force difference : F_{on bottom} − F_{on top} is called "Buoyant Force" by the fluid on object ;
=> δF = − ρ_fluid g V_object . . . upward, as it used to cancel the weight of fluid that isn't there anymore ... the "displaced fluid"
      typical Bouyant Force for condensed matter (solid or liquid) immersed in sparse matter (gas, plasma) is −0.001 × mg ;
      but condensed matter can float in other condensed matter ; and some gases float in other gases .
=> for many situations bouyant Forces can NOT be ignored .

Important statement #5 about how the Universe works:

Gravity : every mass in the universe pulls on every other mass in the universe

each source mass M_s contributes to the gravity field g at each location.
. . . the total gravitational influence emanating from source mass M_s ~ GM_s , pointed toward the source mass ,
      with G = 6.67E−11[Nm²/kg²] = 66.7[N/kg · (Mkm)²/(1E30kg)] ,
      or . . . G = 5.931E−3[N/kg · (au)²/M_Solar] <= astronomical units
. . . this influence spreads out, 2 ways in 3-d space, so the contribution to g is weaker far from the source ;
      so strength contributed goes as 1/"distance squared" = 1/r² from that source
      . . . { its contributed g piercing thru the whole surface of any surrounding shell is the same => g·A = 4π GM_s }
=> g_{by M} = G M / r²_M-to-point (toward M) . . . then use   F_{gravity, on m} = mg ; NOTICE : m ≠ M !

The contribution to g from every source mass (reasonably nearby) must be added as a vector
      to get the total   g   at that place (the field point) .
. . . so there IS only ONE gravity field vector at any location , after adding all the contributions ... (3 components, one vector)

Since the gravitational influence spreads with distance from the souce Mass ,
      the gravity field is more intense on the "near side" of an object than it is on the "far side" of the object.
. . . relative to the average field at the object , these "excess" and "deficit" Forces (per unit object mass) seem to pull outward
      Tidal Tension (per unit mass in the object) = g_near − g_average   and   g_average − g_near   .
=> Δg = g / (1 ± Δr/R)²
. . . this is almost the same on each side , and if the stressed object is small (compared with source Mass distance ; Δr/R << 1 )
=> Δg ≈ ( ^d/_dr{g(r)}|_R ) · Δr   . . . ≈ 2gΔr/R   .

For a "field point" anywhere inside a hollow shell of mass, the gravity field contributions from all the shell source Mass will cancel !
. . . only M located non-isotropically around the field point contribute to g .
=> gravity is weaker at the bottom of a deep hole into Earth ... only the "inside" Mass counts .