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Topic 2 (Rates - especially momentum & velocity)

Quiz 3 was thRs.Sep.26 ... here's my grading key jpg
Exam 1 was thRs.Oct.03 ... here's my grading key jpg page 1 ... page 2 ... page 3

the rate of momentum exchange is called Force . . . abbreviated F .
if there is no Force, or zero Force , the momentum is constant .
because Forces are directional , they can cancel each other
. . . if applied to the same object but in opposite directions.
      this is the usual way things stay near one another
an uncancelled Force , applied for a while , changes some object's momentum
=> F Δ t = Δ p   . . . (Newton's second Law , in cause => effect order)

being Pushed for a while , changes its motion   =>   F·Δt = Δp
. . . the motion (total) is momentum   p = mv
. . . the Push is called a Force ... weight (pull by gravity) is not the same as mass (amount of matter)

if something's velocity changes . . . then it accelerated
acceleration is the velocity change rate ... it is directional ...
      (you can not literally push yourself to go faster)
. . . We use the letter Δ , capital (greek) "Delta" , to mean "change in ____" something.
      so , a = Δv / Δt ,   where this Δt is the time interval that the velocity changed during.
=> a·Δt = Δv . . . accelerating for a while changes its velocity ... that's what acceleration does

What might cause acceleration? ... the same thing that causes momentum to change ... Force !
. . . must be an external Force , applied to our object (by an active subject)
. . . since mass doesn't change, F Δ t = Δ mv   becomes   F Δ t = m Δ v  
. . . Force causes mass to accelerate
=> F = m a   . . . famous equation called "Newton's 2nd Law" .

External Force causes a mass to accelerate   =>   Σ F = m·a . . . = m Δ v / Δ t

the mass is the mass in the system you're watching.
      inside the object (in contrast with "outside the system" or "outside the subject")
. . . it is the total mass in the system
      add up all the masses , if there's more than one in there.
. . . the mass of the system is what gets accelerated
      the acceleration of the center-of-mass ... is the average acceleration of the system.
. . . this is meaningful and not complicated, since mass does not depend on the environment
      Σ m   has the same value everywhere.

Force is exerted by something Outside the system       the active subject in the sentence describing what happened. . . . applied to the (passive) object inside the system       so each Force vector (arrow) must pierce the system boundary. . . . it is the total external Force , added directionally , that is important       net Force means what remains after they've all been summed ;       ... Forces tend to oppose one another , canceling . . . . represent each Force that is applied to the object as an arrow       pointing in the direction that the Force is pulling (or pushing)       and with length that shows how strong that Force is       ... decide which is positive and which is negative! . . . the special situation of zero net Force is called "equilibrium"       Forces cancel => acceleration is zero => velocity is constant

each Force has a nameable cause
      which depends on a particular property.
. . . gravity pulls any mass that is immersed in the gravity field downward
      around here, the gravity field has 9.8 N/kg intensity.
      ... each kilogram is pulled with a 9.8 Newton Force.
      ... (a few percent different at different places)
      on our Moon's surface, gravity's intensity averages 1.62 N/kg
      ... (a few percent different at different places)
. . . Surface Forces push outward, strong enough to keep the object from penetrating it
      (they do compress slightly, depending on how hard the surface is - spring-like!).
      the surface Area pushes straight out thru it (perpendicular = "Normal")
. . . a spring pulls inward depending on how far it has been stretched
      stiffer springs pull harder, for the same stretch distance.
      to stretch a spring, you need to pull outward on both ends
      ... this Tension makes the spring material deform
. . . compression springs (in click pens) push outward (stretch is negative)
      with strength that increases with how far they've been compressed
. . . Tension Forces pull inward, hard enough to keep from breaking
      (they do stretch slightly, depending on their stiffness - spring-like!).
      ... ropes and cords usually have much less mass than the object they are tied to, and stretch very little (neglect it)
      ... so Tension is essentially the same all along that rope.

most of the time, one Force (gravity's Force) pulls an object down onto the table
. . . the table deformed a wee bit as the object was placed onto it, until it pushes outward just strong enough
      that the surface Force upward (+) just cancels gravity's Force downward (−)
the table will deform much farther if the object is thrown onto its surface
. . . so the table pushes it up much harder than gravity pulls it down
      and the ball accelerates upward until the table stops pushing it so hard.

except for gravity's Force (and later electric and magnetic forces),
      these can all be identified by contact
      the subject pushing the object (which is being watched).
. . . each side of the contact surface pushes thru the contact Area, into the other side
      equal strength Forces, in opposite directions
. . . these pairs of Forces are applied to different things (object c.f. subject)
      so you do not add them to get a (zero) total Force.
. . . (if both are inside your system , neither pierces the system boundary
      so neither accelerates the system's c.o.m. ... make your system smaller!)

Friction Force often "Accidentally" Cancels another Force
friction Forces tend to resist sliding
      that is, they oppose the slide.
. . . if the pushing subject can only move so fast, then
      the pusher will often push at constant velocity for a while
      ... zero acceleration implies zero total ("net") Force.
      they started it moving by pushing stronger (so, +'ve net Force) for a little while, to give it some momentum (mv),
      ... but let it stop by pushing weaker than friction (so, −'ve net Force) until is slows to a stop.
      from very beginning to very end (stopped to stopped) is zero average net Force.
. . . wind resistance & water resistance Forces (both viscous and drag)
      are stronger for faster speed thru the fluid
      ... speed increases until the other (imposed) Force gradually becomes cancelled.
      bigger size objects encounter more resistance, if at the same speed
      ... because they hit more wind (or water) as they travel
      so fluffy objects achieve slower terminal (equilibrium) velocity.

Curved Path implies acceleration toward the Center of Curve

=> a_centrip.   =   v ² /r   . . .   centripetal = "center-pointing"

recall : velocity includes direction . . . represent as arrows . . . change in velocity . . . Δ v = vafter − vbefore       even though the speed is the same, Southward is not the same as Eastward!       the diagram is top view of an object moving clockwise around a circle . . . "subtract" means "add the opposite" . . . the "additive inverse" is the negative       ... (that's where negative numbers come from!)       the opposite of Eastward is Westward       so we should add Westward to Southward . . . . add vectors tail-to-tip, like road-trip "directions" or connect-the-numbers drawing game       the change in velocity during this ¼ circle is 1.41 |v| to the South-West       (check that by using Pythagoras: c² = a² + b² )       it needs to remove the original Eastwardness       as well as have all the Southwardness leftover. = Δv points toward the circle's center during that time span.

Large circles take a long time to travel Around
      Time for one complete go-round is called the (repeat time) Period ... usual abbreviation   T .
. . . total distance around is circumference   = 2 π r = π·diameter .
      from v = d / Δt ,   Δt = d / v
      ¼ circle will take   Δt = π r / 2 v . . . (<=right?)
. . . acceleration   a = Δ v / Δ t . . . substitute formulas to get
      a ≈ 1.41 |v| 2 v / π r
      . . . the 2·1.41/π = 0.9 is not exactly 1 here only because ¼ circle is very coarse to calculate Δv from
=> a_centripetal = v ² / r . . .

example: bucket of water overhead
suppose you try to swing a bucket filled with water all the way around in 1 second.
. . . using your shoulder as a pivot, the water is about 1 meter from that center.
speed ≈ d/Δt = 2 π r / T ≈ 6 (1m) / (1s) ≈ 6 m/s .
      (1 gal ~ 4 liter would have KE = ½ (4kg) (6m/s)² = 72 J   + PE = (4·10 N)(1m), so be careful!)
a = v² /r = (6m/s)² /(1m) = 36 m²/s²m = 36 m/s² .
. . . it does not need to go this fast to stay in the bucket, because gravity only provides 9.8 m/s²
      your shoulder will need to provide the other 26.2 m/s² acceleration.
      at ½ this speed,   a is ¼ as quick ... 9 m/s² ... water falls quicker than bucket
. . . or you can use a slower speed with a much shorter radius (notice: speed counts twice)
      ½ the speed (3m/s) with ¼ the radius again makes a = 36 m/s² .
      ... but takes ½ as long because the path is so short.

Force Applied 90° to velocity changes the object's path (deflects) , not its speed
Consider swinging a bucket of water around you, in a horizontal circle
      hold one end of rope above your head, the bucket on the other end moving at wist level.
. . . The rope Tension does no Work . . . so can not change the bucket's KE .
      change the speed of a bucket by pulling along the curved path
. . . caused Force (e.g. string) does not need to point to the center
      it is the net Force that causes acceleration
      so string + gravity will point inward to circle center (horizontal circle)
. . . "centripetal Force" IS NOT ANY FORCE'S NAME !
      name each Force by its cause ...
      even if centripetal acceleration is its effect .
. . . triangles of string Tension along the diagonal
      show that part cancels weight (Force by gravity downward)
      and some causes centripetal acceleration (horizontal).
      if you draw a Force triangle to scale you can verify each portion.

the Gravity Field is Caused by Mass ; g Intensity Decreases w/ Distance ² (as g spreads)

=> g = G M_subject / d ²   . . . G = 6.67 E−11 [ Nm²/kg² ] everywhere (Universal)

the property that gravity influences   (mass)   is the same property that causes it.
      but the cause M is the subject while the the influenced m is the object.
. . . the full formula for the Force that Earth's gravity applies to an object ... replace "g" with its formula above
=> F = m   G M / d ² .

Every Mass (even You) Causes a Gravity Field
. . . but only humungous masses have noticeable gravity .
example: 70 kg student ... what gravity do they cause at 1m distance
. . . find, on your calculator, the EE or Exp button
      I get 467E−11 [N/kg] = 4.67E−9 [N/kg]
      really small compared to Earth's 9.8 [N/kg].
=> but not zero!

Force Increases with Proximity ... 2× as Close , 4× as strong
. . . ½ as far means d = ½ m ... so d² = ¼ m² .
      4.67E−9 [Nm²/kg] / (¼ m²) = 18.68E−9 [N/kg]
      closeness counts twice!
. . . we should check the Sun's gravity out here at Earth distance:
      (66.7E−12) (2E30 [kg]) / (150E9 m )²
=> g_{by Sun} = 5.93E−3 N/kg . . . 0.006 N/kg ... compared to Earth's 9.8 N/kg here (we're close to it!)

Use the Average Distance to the Mass ... Center-to-Center
How to convince ourselves that this is the correct value of   G ? Try it for Earth!
. . . We're 6.37E6 meter from the middle of Earth's 5.98E24 kg
      g = (66.7E−12 ) ( 5.98E24 kg) / ( 6.37E6 m) ²
=> g_{by Earth} = 9.83 N/kg ... if you've punched it in right (use the EE button, not these 4 keystrokes "× 1 0 ^" )
. . . actually , this agreement is how we know Earth's mass ... in terms of laboratory-scale masses [kilogram]

very useful to use ratios if you know one g or F
      essentially using "G" in different units
. . . G = 9.8 Newton·EarthRadius²/(kg·EarthMass) . . . to find surface gravity for different planets.
. . . example: Moon's mass is 1/81 × Earth's mass ... implies weaker ... only 1/81 as strong ( M = 0.012345 EarthMass)
      but Moon's surface is only 1/3.66 × as far from its center as Earth's surface ( d = 0.273 EarthRadius)
      . . . that implies stronger by factor 3.66² = 13.4 ×
=> g_{on Moon} = g_{on Earth} ( 13.4 / 81 ) = (9.8 N/kg) × ( .012345 / .0273² ) = 1.62 N/kg . . . ~ 1/6 of Earth's.

. . . example: Sun's mass is 333,333 × Earth's mass ... implies intense gravity ...
      but Earth is 150 Million km from the Sun ... that is d = 150,000,000/6400 = 23,440 EarthRadius)
. . . g _{by Sun, here} = 9.8 Newton·EarthRadius²/(kg·EarthMass) × 333,333 EarthMass / (23440 EarthRadius)² = 0.00595 N/kg
=> G _{for planet orbits} = 0.00595 Newton·EarthOrbitRadius²/(kg·SunMass) . . . to find the Sun's gravity at different planet distances.
. . . the Earth orbit distance (from Sun) is called the "Astronomical Unit" , abbreviated AU . Used to describe solar-system sized distances

use the conceptual ratio technique above ... think of distance squared as an Area

Kepler #1 : Orbits are Ellipses (not circles) - Sun is at one Focus (not center) We all orbit the Sun counter-clockwise, as seen from Draco (looking down onto the North Pole) "North" means : if our motion turned a (right-hand) screw, it would move Northward along its axis . . . Earth's "orbit North" ... straight up from our orbit plane ... points toward the constellation Draco . . . Earth's "spin North" now points toward the star Polaris (pretty close, for the last thousand years)       spin North is tilted 23½° from orbit North, lately toward Taurus/Gemini       (but very slowly circles the N.ecliptic pole , every 26,000 yrs)       ... so Vega will be our North star in 12,000 years. an ellipse is a stretched-out circle . . . cut the circle's center in half (each half becomes a focus), move the two foci apart       the farther they're spread, the more eccentric the ellipse becomes. . . . not an oval (semicircles joined by straight lines) - a smooth curve! . . . pin string to the foci, pull tight with a pencil, trace around the ellipse. this orbit shows eccentricity of about 60% . . . the aphelion is 4× the perihelion . . . eccentricity is difference / sum : e = (ap - peri)/(ap + peri) ≈ (4-1)/(4+1) = 3/5       more like a comet orbit here . . . real planet orbits are only slightly eccentric ... 1.7% for Earth, 9% for Mars . . . we're closest to the Sun on Jan.03 or 04 , farthest on July 4 or 5 => gravity source Pulls from one focus   . . . planet's average location is toward the "empty" focus

Planet gains speed as it falls "toward" Sun . . . of course it misses, but it still gains speed as it gets closer (pulled by gravity).       it is fastest where it is closest to the Sun (peri helion , "around" Sun as in "peri"scope) . . . and loses speed as it "rises" away from (farther from) the Sun       it is slowest where they are farthest from the Sun (ap helion , "high above" Sun as in "apex") . . . until it starts falling inward again (gaining KE). Kepler #2 : sideways speed × radial distance = constant for one orbit . . . it is moving 4× as fast at the closest , than it is at the farthest.       (for orbits around Earth, these are called peri gee and ap ogee) . . . on the illustrator linked below, set the sweep time to 1/16 of an orbit ...       each sweep segment has the same Area (half as far moves twice as fast) => this is because Angular (Rotational) Momentum (d×mv) is Conserved (more later...)

Kepler #3 : Average Distance Cubed / Orbit Time Squared = ... Mass of Gravity Source Compare orbits to Earth's orbit, get Gravity Source compared to the Sun's Mass.       Kepler left the comparison in terms of base units distance and time . . . Newton's re-wording: a planet's acceleration a, is caused by gravity g       . . . v ² /r = GM / r ² . . . Leibniz re-wording: its Kinetic Energy is half-way out of its Potential-Energy hole       . . . ½mv² = ½ m(−GM/r)   ... but that's topic 3 . . . example : Mars' orbit distance 1.52× Earth's , Mars orbit time 1.88 yrs.       1.52 ³ / 1.88 ² = 3.52 / 3.53 ≈ 1 "Solar Mass".       trivial, since we know Mars orbits our Sun (but it's a check) . . . example : Jupiter's Europa : orbit distance 671,000 km (from Jup) = .671/150 = .00447 AU       orbit time 3.55 days = 3.55/365 = .00973 . . .       . . . (.00447)³ / (.00973)² = .0000000895/ .0000946 = .00095 => Mass of jupiter = .00095 × Mass of Sun (about 1/1000 ... 2E27 kg .

Kepler #3 is the way we figure out the masses of other stars ... and extrasolar planets !
here's a link to a pretty good illustrator for ellipse orbit vocab and Kepler #2.
. . . you need to move their planet from aphelion so you can see r₁ separate from r₂ .