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?

kwmcgregor@optusnet.com.aukwmcgregor@optusnet.com.au