Angular Kinematics : ch.12 prob. 1,3,5,7,9,11,15,13 ; 17,22,25 ; 27,29 ; 33,34,35,37,39,43,51,55,56,59,65,71
Torque Statics : ch.14 prob. 9,11,15,18,20,22,23,24,25,49,51,57,59,61,63,64 ; 33,37,39,45,46,47
L = r×p : (ch.9 prob. 38,39,41,43,44,67,69,72,73) ; Ch.13 prob. 47,49,51(L²/2mr²?) , 55,57,59,60 ; 61,62,63,64,66,67,69,73,74,75,76
Rotation Dynamics : ch.13 prob. 3,7,8,9,13,17,21,27,29,31,35,38,39,45 ; 79,81,102
practice writing these angular symbols on the homework problems, to reinforce the linear-angular analogies.
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 Reminders :
Some Energy forms are somewhat complicated to calculate, at a fundamental level, such as Chemical PE or Nuclear PE.
. . . they can be treated as experimentally-measured quantities which are "released" (transformed) during some conversion process (i.e, burning).
more material holds more Energy ; it is the PE density (PE/Volume) or the specific PE (PE/mass) that is measured.
efficiency is the fraction (or %) of the input Energy that is transformed into the "desired form".
. . . because total Energy is consserved, the sum of efficiencies in all output forms must be 100%.
Power is the rate that Energy is transformed (or transfered) into some other type (or to some other object)
=> P = ΔE / Δt , reported in [Watt] = [J/s] .
Because KE goes as v² = vx² + vy² + vz² , motion along each component adds in an independent manner .
. . . in particular, radial and angular motion components can be treated separately for objects in orbit
=> L = r × p is conserved then , so that KEangular = L²/2mr² is a useful formula .
Topic 7 Summary :
Even if the Sum of all Forces (applied to an object) is zero, the object's motion might be changing rotationally.
. . . a thing's spinning motion tends to stay the same <=> analogous to linear motion
. . . angular location, or angular position, angular orientation φ [radians] = (arc length)/r <=> x, y, or z .
in the direction that a right-hand screw would move (if right fingers curl along arc's arrow, thumb points in direction of φ)
. . . counter-clockwise arc has φ out of page . . . clockwise motion has Δφ into page .
. . . angular velocity ω [radian/sec] = Δφ / Δt = v/r is analogous to linear velocity
. . . angular acceleration α [radian/s²] = Δω / Δt = a/r is analogous to linear acceleration .
The analogy to any kinematic formula is obtained by directly replacing each linear quantity by its angular sibling :
φi + ωavg Δt = φf ;
Δφ = ωi Δt + ½ αavg (Δt)² ;
ωi + αavg Δt = ωf ;
ωi² + 2 αavg·Δφ = ωi² ; etc.
Rotational Dynamics statements are almost a straight-forward analogy to the familiar linear (translational) statements .
. . . the rotational analogy to Force is Torque τ [m×N] = r × F . . . "r cross F" . . . (direction via right-hand-rule)
Torque around a rotation axis depends on the distance from the axis (tail of r) to the point the Force is applied
. . . and only the component of F that is | to r provides torque
. . . or, use the entire Force at its lever-arm distance : the distance from the axis | to the Force's line of action
generally, τ [m N] = r × F = r F sin(φ) ; here, φ is the angle from the r-vector direction to the F-vector direction
=> r × F = τ . . . right-hand rule : right rist along r , fingers flip thru angle φ till point along F ; Thumb points ( | ) along Torque
is the pattern for any vector cross-product a × b .
. . . you should use the right-hand rule to verify that a × b = c implies that b × a = − c . . . <= ORDER MATTERS !
. . . if the total Torque is zero, the conserved rotational quantity is angular momentum L [m N s] = r × p . . . (cross product!)
of course the anguLar momentum is around the same rotation axis that has Στ =0 . . .
=> L = r_|_ p = r p_|_ = r p sin(φ) ... rite rist along r, pingers point to p , thumb shows L .
The "story-board" statement of how Nature behaves is just as valid in rotational form as in linear form :
"starting anguLar momentum is changed , by total Torque applied for a while , into the anguLar momentum afterward."
=> Σ Li + Σ τ Δt = Σ Lf
.
If you try to isolate the more easily-measured kinematic quantities , the analogies are not generally as useful
. . . the analogy to mass is rotational Inertia or "angular Inertia" or "(second) Moment of Inertia" :
=> I [kg m2] = Σ m r² .
... so it depends not only on the amount of mass , but also depends on how far from the axis the mass is
... because its velocity is v = − r × ω , more distant mass has more momentum
if the mass is spread out in some Volume, then the sum must become an integral of the mass density ρ
=> I = ∫ r² dm = ∫ r² ρ dV . . . over all the Volume that has non-zero mass density .
. . . the mass density is allowed to be different at different places : ρ(r) as a function of location .
. . . the differential Volume is dV = dx dy dz , in "rectangular" coordinates
in "spherical polar coordinates" dV = r cos θ dφ r dθ dr . . . measure θ from spin axis z , measure φ around the spin axis
if the mass density has rotational symmetry (same ρ at any φ) , then ∫ dφ = 2π . . . dV = 2πr cos θ r dθ dr .
if the mass is all in the x-y plane , θ = 0 so cos θ = 1 , and ∫ r dθ = t , the thickness of the sheet . . . dV = 2πr t dr .
... in that situation, ρ t = σ , called the surface density , is often given ... σ = m/Area [kg/m²]
. . . in any case, once the Inertia is known, the anguLar momentum is that Inertia I multiplied by the angular velocity vector ω :
=> L = I ω
If some mass's distance (from the axis) changes , it's essentially a "soft rotational collision" between the moving masses and the others.
. . . If you know the initial and final locations, you may compute the initial rotational Inertia and the different final Rotational Inertia
Ii ωi + Σ τ = If ωf
If the object does NOT change shape, and the rotation axis does not change,
then the moment of Inertia (around that axis) does NOT change ;
. . . then any change in angular momentum comes from change in angular velocity ω
. . . so the total external torque causes the object's rotational Inertia to have angular acceleration α :
=> ΣτA = I ΔωA / Δt = I αA ... where "A" is the axis , which the Inertia will rotate around ...
... a mass has tangential acceleration component aφ = − r × α , but radial acceleration component ar = − rω² .
because the different parts of the object do not have the same speed, there is an additional Kinetic Energy term :
=> KErotation = ½ I ω² calso be written KErot = ½ L ω , or KErot = L²/2I .
. . . this KE of mass around the rotation axis is added to the linear KE as if it was moving at the speed that the axis has .
this rotational KE is changed by rotational Work :
=> Wrot = τ ·Δθ . . . parallel components of Torque and angular displacement .
an implication of this is that an object's moment of inertia is smallest around its center-of-mass,
. . . more rotational inertia the farther the axis is from the c.o.m. ; if axis is distance d from c.o.m. :
=> Iaxis = Ic.o.m. + M d² .
You STILL get to choose which objects in the scene are included INSIDE your system
. . . use "bigger" systems if you intend to use angular momentum during a "rotational collision"
so that internal Forces do not produce external torques to Sum ... but the Total Inertia inside stays the same.
(to topic 8 summary)
