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Circular Thinking
Ken McGregor
Why?
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It’s a challenge.
Can see & feel what’s going on. Instinctive
gut reaction that might be right or wrong,
but probably needs explanation.
Allows for creative & critical thinking –
perhaps not so comfortable.
Examples
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A hollow world
Frames of reference
Rolling, sliding and
spinning balls
Snooker Collisions
Trebuchet
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Circus see-saw
A bola
Half pipe
Rolling Coin
Why does a
boomerang come
back?
Building Blocks
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Speed, acceleration, force & inertia
Having cycled on a bike or on a
skateboard
Spinning a bucket of water.
Hollow Planet
Sun
Hollow
planet
Earth
‘Forces’ on the surface
The world inside
Letting a ball drop & Launching a
‘satellite’
Coriolis Effect
Ground frame of
reference
Carousel frame of
reference
y
y’
x’
x
carpet
ice
Ground frame of
reference
y’
Carousel frame of
reference
y
y’
x’
G
wt
x
G
x’
Frame of Reference
Moving frame of
reference
If stationary, only
involves a
centrifugal force.
stars
movement
Coriolis
force
Initial
Velocity v’
centrifugal
force
stars
Inertial frame of
reference
Coriolis Force
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Transit time 1 hr 20mins
Same time no matter what arc is chosen.
Bug moving at steady speed along radial spoke of
a wheel rotating at constant angular speed
w(r+v’Dt/2)
v’
wDt
w(r-v’Dt/2)
v’
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Can show the radial acceleration is
w r
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And the transverse acceleration is
2w v '
2
Rolling, sliding & Spinning Balls
Pure
rolling
v
w
O
Sliding
Spinning
friction
friction
Slowing down while Rolling
Rolling
Rolling
v
v
Friction
Friction
Rolling
v
Normal
force
Friction
Skidding
to
Rolling
Rolling
v
v
Sliding
to
Rolling
v
Yoyo
Rolling
v
Questions
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What would the rolling curve be like if you
were inside a light rubber tyre that was
rolling on the flat?
And if you were at the edge of a Yo-yo
that was falling under gravity?
What would it feel like in both cases?
Collisions
Snooker Collisions
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Usually no spin
given to the target
ball
Sliding only
1
v2
v2
2
U
w0
2
y
x
1 v1
With top spin
Rolling only
With under spin
Height of Rebounding Edge
Circus See Saw – Teeterboard
M
Vi
rM

rm
m
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The angular acceleration
Fm 1 FM 1

on both sides must the
m rm M rM
same:
The work done on m
Fm rm  FM rM
must equal the work
expended by M :
Combining these we get Fm  m rm  rM  mr 2  Mr 2
m
M
FM M rM rm
So the moments on
inertia must the same.
r
And not - mrm  MrM
F
M
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Fm
rm
M
2
1
3
A Bola
2
1
3
V
F
v
L
O
O
r
v
R
r
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No gain or loss of energy, until the balls
strike
But angular momentum of the balls
decreases as it’s imparted to the earth.
Balls strike post radially.
Trebuchet
R1
R2
m1
m2
Line of centre of
mass
M
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m
vm  vM
M
m
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m
vm
M
vM
Trebuchet frame
moves in this
direction
If both arms have
the same length:
Without wheels, we
get
With wheels, we get
mvm  MvM
Half Pipe – pumping for speed
R2
v2
R1
v1
Rolling Coin
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Balance in the vertical
gives
N cos   mg
2
mv
‘Balance’ in the horizontal N sin   
r
This gives
v2
tan   
gr
If the basin is the bottom
part of a shell:
then
r  R sin  
v  R sin  g tan  
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If the basin has a log
shape i.e.
Then
dz z0

dr r
We then get
And so
r
z  z0 ln  
 r0 
tan   
v  gz0
dz
 r tan    z0
dr
r
z
N
i.e. a constant

mg
r
Boomerang
V  wR
V  wR
V
V
V  wR
V  wR
Ft
Fb
Bottom
end
Concepts
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Stimulates 3D visualisation, through not
essential
Doesn’t need to be mathematical – simple
algebra is enough
Don’t need to specify moment of inertia
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Experience with a bucket of water
Don’t need to specify torque
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Just the example of a spanner and nut
THANK YOU FOR YOUR
ATTENTION
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Questions, comments, other experiences?
[email protected]