Combustion Energy : ch.8 prob. 35,36,37,39(80%→heat),41,42,43 ; 44,45,47,48,49,51,53,101
Power in Transformations : ch.8 prob. 54,55,56,57,59,62,64,65 ; 67,69,71,72,76,77,78,85,87,89,96
Gravitational PE : ch.9 prob. 45,47,51,53,55 , 57,61,62,66 , 70,74
Energy in Orbits : ch.9 prob. 38,39 , 42,43,40,42,43 , 69,67,72,73,86
Energy in Collisions : ch.10 prob. 52,53,54 ; 57,68,69,71,75,72 ; 73 ;
Ch.11 prob. 1,5,9 ; 16,18,19,23,26,27,29,30,31,32 , 37,41,42 , 45,47,53,55,57,61,63,65,66
Topic 1 Reminders :
(to topic 1 summary)
USE UNITS all thru your answer ... don't just append them onto the final number !
Put symbols on a sketch, as you read . . . use meaningful phrases : "v_match" , "catch" , "ground" "turn-around"
. . . show process spans as brackets or vector arrows , with words like "average" , "losing" , "gaining" , "t=0→2" ...
Write statements as symbols first ... each statement has a SUBJECT ... manipulate symbols before plugging numbers.
. . . keep track of adjectives such as "initial" , "average" , "final" , "stopped" . . . "change"
Topic 2 Reminders :
(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 3 Reminders :
it's the Sum of external Forces that cause an object's mass to accelerate subscripts for "total" or "system" , "part A" ...
once you separate vectors into components, keep each component separate from the others!
ΣF = ma , for each component separately.
Topic 4 Reminders :
Name each Force by the source of that Force ... its cause ... a function of the environment.
spring Force : Fspring = − k s ... opposite the stretch vector .
Pressure Force : Fspring = P A generally ... in a fluid , δP = ρ g δh .
Gravity : every mass in the universe contributes to the gravity field g at any place of interest.
... add "nearby" contributions as vectors ... gby M = G M / r²M-to-point (toward M) . . . then use Fgravity, on m = mg ; NOTICE : m ≠ M !
Topic 5 Reminders :
Energy is conserved quantity : Σ Ebefore + Σ Wby F's not in PE list = Σ Eafter .
KE = ½ m v² = ½ p·v = ½ p²/m , in any of these equivalent forms .
Each Force either does Work during motion , or has an associated PE function ... not both .
=> Wby F = F·Δs .
=> − dU / dr = Fr , the component along that direction (r)
PEgravity,local = m g h , with h measured upward (z-direction) from a "zero-height reference" point
PEelastic = ½ k s² . . . stretch s must be measured from its relaxed length
PEPressure = P·V ; it is okay to measure from a non-zero reference pressure , but be consistent !
PEGravitation = − m GM / r .
There IS NO PE for Friction Force ... must be explicitly included as a "Work by Forces not in the PE list"
Topic 6 Summary :
the Gravitational Potential Energy of a planet , near the Sun , is negative .
As the planet-forming material fell inward from far away (where PE≈0) , to lower PE , it gained KE
. . . as the material lost some KE during collisions , it became trapped , unable to achieve the large distances that it once was at .
the amount of Energy an object needs , in order to achieve "infinite" distance , is called its Escape Energy
. . . the speed that corresponds to that much Kinetic Energy , which is the same for any mass , is the Escape Speed.
Because all source Masses contribute to the Gravitational Field g , at any location ,
the Gravitational PE of some field mass is NOT due to any single source Mass .
. . . instead, each source Mass Mi contributes to the total Gravitational Potential , called VGrav , at the place of interest .
notice that a "mass m" object at the place of interest has PE = m VGrav when it is there ,
but the place itself has Gravitational Potential (due to "space warp") , even if no mass is there .
=> VGrav(x,y,z) = Σ Vi . . . . . each contribution Vi = GMi /ri , depends on distance ri from its Mass to the place (x,y,z) .
In the early 1600's Kepler analyzed data for the direction to planets as seen from Earth (mostly Tycho Brahe's data)
=> Kepler #1 : each planet path is an ellipse , with the Sun at one focus of the ellipse
. . . each planet's ellipse is stretched along its own vector , and is tilted a few degrees relative to Earth's.
=> Kepler #2 : planets move faster where closer to the Sun ... move slower where farther from the Sun
. . . r × v | is constant for each planet ; v | is the velocity component perpendicular to r .
. . . since each planet's mass is constant, this implies that L = r × p = r p sin(φ) is constant (conserved).
the angular momentum vector L only includes the angular (or tangential) velocity component ... (not radial)
. . . since the velocity's tangential component is vφ = r ω = r (dφ/dt) , L = m r ² ω .
angular momentum is equal to rotational inertia (mr ²) times the angular velocity (ω) .
=> Kepler #3 : planets with smaller orbits move (on average) faster than planets with larger orbits ... T ² ~ R ³
. . . this works if T is in [years] , and R is in [a.u.]
here, the average distance from the Sun R , is half of the ellipse major axis
. . . R = ½ ( perihelion distance + aphelion distance ) .
if the orbit is around a planet , rather than the Sun , T ² = Morbited R ³ , with M in solar masses .
. . . we can interpret this as : (average) KEangular = ½ (4 π²) M / R ... and recognize G in these astronomical units .
An object can have non-zero KE even if its center-of-mass has zero velocity, if it is spinning.
. . . each little piece of mass is moving at different speed v = r ω , depending on its distance from the spin axis.
=> KErotation = ½ I ω ² = ½ ∑ m (r ω)² = L ² /(2 m r ²)
I is called the rotational Inertia , or the (second) moment of Inertia , for the object ... depends on shape!
I = ∑ m r ² , where r is the distance from the rotation axis to each point mass
. . . for real objects that are not made of point masses , there are two options :
a) integrate : I = ∫ ρ r ² dV over the entire volume of the object (write the Volume element dV carefully!)
b) add pieces : Itotal = ∑ (Ipart + mpart rpart² ) (using the parallel-axis theorem ).
... treat object as being made from a few parts , model each part as having regular geometric shape
... look up a rotational Inertia formula for each "part" (in a table) around its center-of-mass
... pay attention to the axis location (center, or edge) for that formula!
This means that Work can be done on an object , even if the object's center-of-mass doesn't move .
(Work is done by the Force application point moving parallel the Force)
. . . for a spinning object, the tangential Force component is the one which can do Work ; distance traveled is Δs = r Δθ .
=> WRotation = r × F Δθ = r F sin(φ) Δθ .
(to 
