TEASER - 04
THE WALL
The dimension of a wall is measured by its length (Y), width (X) and thickness or depth (Z). The width (X) of a wall can also be equated to its Height (H).
Now, a wall of exact 5 m height (H), not more & not less, is to be erected across an undulating earth’s surface, similar to that of China wall. Indicate whether the width (X) of the wall, shall remain constant throughout its length over an undulating earth surface?
(a) Yes
(b) No
GREAT WALL OF CHINA CRISS-CROSSING MOUNTAINS
THE EXPLANATION:
When a wall is standing on a flat ground its dimension is defined by three elements, they are - width (X), length (Y) and thickness (Z). However, when same wall is build over an inclined ground surface, a fourth element or fourth dimension - height (H) comes into play. This fourth element is same as width (X) when on flat ground.
GEOMETRICAL ELEMENTS OF A WALL
The width of a wall (X) is measured along a line normal to ground surface which may not be parallel to equi-potential gravity field. On the other hand the height of a wall (H) is measured along a vertical line (plumb line) which is perpendicular to the equi-potential surface of earth’s gravity field. Normally, sea surface can be considered to be parallel to equi-potential gravity field and plumb line is always perpendicular (normal) to sea surface. However, ground surface is not always parallel to equi-potential surface and is highly undulating. On an average, trees are always perpendicular to equi-potential surface, irrespective of ground slope on which it grows, therefore can be considered to be parallel to Plumb Line.
TREES GROWING VERTICAL (PARALLEL TO PLUMB LINE) (OBLIQUE TO STEEP SLOPE)
If height (H) of a wall (measured along plumb line) is kept constant throughout its length over an undulating ground, the width (X) will not remain same every where. Width (X) would be maximum on a flat ground and would decrease proportionately along sloppy ground.
HEIGHT OF A WALL IS INDICATED BY ARROWS
Alternatively if width (X) is kept constant, the height (H) of wall would not remain same everywhere. Wall would have maximum height along slopes and minimum on flat surface.
WIDTH OF A WALL IS INDICATED BY ARROWS
There can be varying degree of ground slopes. For example a stair case type slope can be likened to combination of vertical and flat surfaces. Over flat part of staircase both plumb-line and width of wall would be parallel, therefore H and X would be same. However, over vertical part, the width (X) would be zero while height (H) would be 100%. Geometrically it would amount to putting a new staircase over the ground surface.
VARIOUS SHAPES OF WALL RESULTING DUE TO COMBINATION OF WIDTH & HEIGHT OF THE WALL
Over a slopping ground, the width (X) of wall can vary from 0% to 100%. The relationship between H and X can be roughly established as follows:
ANGLE (Q) BETWEEN GROUND SLOPE AND PLUMB LINE
If Q is 90 degree (in case of flat ground) X would be equal to H. The X would progressively decrease with decreasing Q and eventually become zero if Q is zero (in case of vertical ground).
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