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Circular ThinkingCircular Thinking
Ken McGregorKen McGregor
Why?Why?
It’s a challenge.It’s a challenge. Can see & feel what’s going on. Can see & feel what’s going on.
Instinctive gut reaction that might be Instinctive gut reaction that might be right or wrong, but probably needs right or wrong, but probably needs explanation.explanation.
Allows for creative & critical thinking Allows for creative & critical thinking – perhaps not so comfortable.– perhaps not so comfortable.
ExamplesExamples
A hollow worldA hollow world Frames of Frames of
referencereference Rolling, sliding and Rolling, sliding and
spinning ballsspinning balls Snooker CollisionsSnooker Collisions TrebuchetTrebuchet
Circus see-sawCircus see-saw A bolaA bola Half pipeHalf pipe Rolling CoinRolling Coin Why does a Why does a
boomerang come boomerang come back?back?
Building BlocksBuilding Blocks
Speed, acceleration, force & inertiaSpeed, acceleration, force & inertia Having cycled on a bike or on a Having cycled on a bike or on a
skateboardskateboard Spinning a bucket of water.Spinning a bucket of water.
Hollow PlanetHollow Planet
Sun
EarthHollow planet
‘‘Forces’ on the surfaceForces’ on the surfaceThe world insideThe world inside
Letting a ball drop & Launching Letting a ball drop & Launching a ‘satellite’a ‘satellite’
Coriolis EffectCoriolis Effect
Ground frame of reference
Carousel frame of reference
x
y
x’
y’
wt
G
G
x’
y’
Ground frame of reference
Carousel frame of reference
x
y
x’
y’
carpet
ice
Frame of ReferenceFrame of Reference
Moving frame of reference
Inertial frame of reference
If stationary, only involves a
centrifugal force.
centrifugal force
Coriolis force
stars
stars
Initial Velocity v’
movement
Coriolis Force
Transit time 1 hr 20minsTransit time 1 hr 20mins Same time no matter what arc is chosen.Same time no matter what arc is chosen.
Bug moving at steady speed along radial spoke of a wheel rotating at constant
angular speed
'2 vw
v’
v’
w(r-v’Dt/2)
w(r+v’Dt/2)
wDt
Can show the radial acceleration is
And the transverse acceleration is
r2w
Rolling, sliding & Spinning Rolling, sliding & Spinning BallsBalls
v
O
Pure rolling
w
friction
Sliding
friction
Spinning
Slowing down while RollingSlowing down while Rolling
v
Rolling
Friction
v
Rolling
Friction
v
Rolling
Friction
Normal force
v
Skidding to
Rolling
v
Rolling
v
Sliding to
Rolling
v
Yoyo Rolling
What would the rolling curve be like What would the rolling curve be like if you were inside a light rubber tyre if you were inside a light rubber tyre that was rolling on the flat?that was rolling on the flat?
And if you were at the edge of a Yo-And if you were at the edge of a Yo-yo that was falling under gravity?yo that was falling under gravity?
What would it feel like in both cases?What would it feel like in both cases?
QuestionsQuestions
CollisionsCollisions
Snooker CollisionsSnooker Collisions
w0
U
v1
v2Sliding only
2
2
1
1
v2
y
x
Rolling only
With under spin
With top spin
Usually no spin Usually no spin given to the given to the target balltarget ball
Height of Rebounding EdgeHeight of Rebounding Edge
Circus See Saw – Circus See Saw – TeeterboardTeeterboardM
Vi
m
rM
rm
The angular The angular acceleration on both acceleration on both sides must the same:sides must the same:
The work done on The work done on mm must equal the work must equal the work expended by expended by M M ::
Combining these we Combining these we get get
So the moments on So the moments on inertia must the inertia must the same.same.
And not -And not -
M
M
m
m
rM
F
rm
F 11
MMmm rFrF
22Mm
m
M
M
m
M
m Mrmrr
r
r
r
M
m
F
F
Mm Mrmr FM
rM
rmFm
A BolaA Bola
O
vr
1
2
3
1
3
2
R
L
V
Fv
O
r
No gain or loss of energy, until the No gain or loss of energy, until the balls strikeballs strike
But angular momentum of the balls But angular momentum of the balls decreases as it’s imparted to the decreases as it’s imparted to the earth.earth.
Balls strike post radially.Balls strike post radially.
TrebuchetTrebuchet
m1
m2
R1
R2
Line of centre of mass
M
m
m
m
M
M
vm
vM
Trebuchet frame moves in this
direction
Mm Mvmv
If both arms have If both arms have the same length:the same length:
Without wheels, Without wheels, we getwe get
With wheels, we With wheels, we getget
Mm vv
Half Pipe – pumping for Half Pipe – pumping for speedspeed
R1R2
v1v2
Rolling CoinRolling Coin
Balance in the vertical Balance in the vertical givesgives
‘‘Balance’ in the Balance’ in the horizontalhorizontal
This gives This gives
If the basin is the If the basin is the bottom part of a shell: bottom part of a shell:
thenthen
mgN cos
r
mvN
2
sin
gr
v2
tan
sinRr
tansin gRv
00 ln
r
rzz If the basin has a log If the basin has a log
shape i.e.shape i.e.
Then Then
We then get We then get
And soAnd so
i.e. a constant i.e. a constant
r
z
dr
dz 0
0tantan zrdr
dz
0gzv N
mg
r
z
r
BoomerangBoomerang
V V
RV w RV w
RV w RV w
Bottom end
Ft
Fb
ConceptsConcepts Stimulates 3D visualisation, through Stimulates 3D visualisation, through
not essentialnot essential Doesn’t need to be mathematical – Doesn’t need to be mathematical –
simple algebra is enoughsimple algebra is enough Don’t need to specify moment of Don’t need to specify moment of
inertiainertia Experience with a bucket of waterExperience with a bucket of water
Don’t need to specify torqueDon’t need to specify torque Just the example of a spanner and nutJust the example of a spanner and nut
THANK YOU FOR YOUR THANK YOU FOR YOUR ATTENTIONATTENTION
Questions, comments, other Questions, comments, other experiences?experiences?
[email protected]@optusnet.com.au