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Pylon of the Month

Pylon of the Month. Rigidity. . Division Can be Dangerous.  cut. Division leads to more boundary. Fire Storms Ignition of dust produces explosive spread of fire. Why use the Fosbury Flop?. High Jump. Centre Of Gravity. Centre of gravity. Launch Kinetic Energy = M  g  H.

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Pylon of the Month

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  1. Pylon of the Month

  2. Rigidity

  3. Division Can be Dangerous  cut Division leads to more boundary

  4. Fire Storms Ignition of dust produces explosive spread of fire

  5. Why use the Fosbury Flop?

  6. High Jump Centre Of Gravity Centre of gravity Launch Kinetic Energy = M  g  H H = height of centre of gravity is raised

  7. Why Do Tightrope Walkers Carry Long Poles ?

  8. Inertia • High inertia • Slow to move • Low inertia • Quick to move • Mass M • Average radius R • mass spreadC high low Inertia = C  Mass  (Radius)2

  9. Large inertia means you wobble more slowly

  10. Stable Unstable

  11. MIR Space Station Disaster 1977

  12. Damaged Hit by the cargo Spaceship! What are MIR’s Principal Moments of Inertia ??

  13. Rigging Rowing Eights

  14. Rowing Has Its Moments Forces on boat +N F then F -N

  15. The Wiggling Boat s Moment on boat = -Ns + N(s+r) - N(s+2r) + N(s+3r) = +2Nr Then, half a stroke later…N reverses to –N and.. Moment on boat = -2Nr

  16. The Italian Rig Moment = +Ns - N(s+r) - N(s+2r) + N(s+3r) = 0 No wiggle!

  17. Three no-wiggle rigs for eights ‘Italian tandem Rig ’ ‘German Rig’

  18. The Art Gallery Problem camera How many cameras are needed to guard a gallery and where should they be placed?

  19. 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

  20. Triangulate! Triangulate!

  21. 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

  22. 3-Colouring the Gallery Assign each corner a colour: pink,green,oryellow. 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.

  23. 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

  24. 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

  25. Orthogonal galleries All corners are right angles Only [n/4] guards are needed, and are always sufficient n = 100 needs only 25 guards now

  26. 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

  27. 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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