Pylon of the Month

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Pylon of the Month
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Rigidity
?
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Division Can be Dangerous
? cut
Division leads to more boundary
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Fire Storms Ignition of dust produces explosive
spread of fire
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Why use the Fosbury Flop?
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High Jump
Centre Of Gravity
Centre of gravity
Launch Kinetic Energy M ? g ? H
H height of centre of gravity is raised
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Why Do Tightrope Walkers Carry Long Poles ?
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Inertia

• High inertia
• Slow to move
• Low inertia
• Quick to move
• Mass M
• Average radius R
• mass spread C

high
low
Inertia C ? Mass ? (Radius)2
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Large inertia means you wobble more slowly
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Stable
Unstable
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MIR Space Station Disaster 1977
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Damaged Hit by the cargo Spaceship!
What are MIRs Principal Moments of Inertia ??
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Rigging Rowing Eights
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Rowing Has Its Moments
Forces on boat
N
F
then
F
-N
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The Wiggling Boat
s
Moment on boat?? -Ns N(sr) - N(s2r)
N(s3r) 2Nr Then, half a stroke laterN
reverses to N and.. Moment on boat -2Nr
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The Italian Rig
Moment Ns - N(sr) - N(s2r) N(s3r) 0
No wiggle!

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Three no-wiggle rigs for eights
Italian tandem Rig
German Rig
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The Art Gallery Problem
camera
How many cameras are needed to guard a gallery
and where should they be placed?
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Simple Polygonal Galleries
Regions with holes are not allowed and no self
intersections
convex polygon
one camera is enough
an arbitrary n-gon (n corners) ? cameras might
be needed
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Triangulate! Triangulate!
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Triangulation
To make things easier, we decompose a polygon
into pieces that only need one guard
Guard the gallery by placing a camera in every
triangle
Join pairs of corners by non-intersecting lines
that lie inside the polygon
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3-Colouring the Gallery
Assign each corner a colour pink, green, or
yellow.
Any two corners connected by an edge or a
diagonal must have different colours.
Thus the vertices of every triangle will be in
three different colors.
A 3-colouring is always possible. Put guards at
corners of same colour Pick the smallest of the
coloured corner groupings to locate the
cameras. You will need at most n/3 cameras.
Here it is 19/3 6.
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The Chvátal Art Gallery Theorem
For a simple polygon with n corners, n/3
cameras are sufficient and sometimes necessary
if every interior point is to be visible from at
least one of the cameras.
For n 100, n/3 33.33 and n/3 33 x is
the integer part of x
Note that n/3 cameras may not always be
necessary
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The Worst Case Scenario
n/3 V-shaped rooms
A camera can never be positioned so as to watch
over two Vs

The maximum of n/3 cameras are required
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Orthogonal galleries
All corners are right angles Only n/4 guards
are needed, and are always sufficient n 100
needs only 25 guards now
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Rectangular galleries
All adjacent rooms have connecting archways
8 rooms and 4 guards in the arches
In a rectangular gallery with r rooms, r/2
guards are needed to guard the gallery
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The Global Village
57 Asians 21 Europeans 14 Western hemisphere 8
Africans 70 are non-white 30 are white 6
possess 60 of all wealth all from the USA 30
live in substandard housing 70 are illiterate 50
are malnourished 1 owns a computer 1 has a
college education
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