REASONING FOR DISTINCT ACCLERATION
DUE TO GRAVITY VALUES ON POLYGONS
INTRODUCTION
The
gravitational field of Polygons viz. Pentagon, Hexagon, Heptagon and Octagon
were determined in previous articles. One of the profound observations was, the
magnitude of gravitational field at corner and mid-point were mathematically
same for all the above mentioned four polygons. Although the equations for
gravitational field at corner and mid-point were mathematically same, it is
worthy to note that they do not yield same results for every polygon. The two
predominant metrics defining the gravitational field of polygon are the side
length of the polygon ‘a’ and its height ‘h’. This article aims at analyzing
the equations to prove that the magnitude of gravitational field for all
polygons will be distinct.
ASSUMPTIONS
1. All
polygons considered are regular.
2. The
thickness of the polygon is negligible compared to its length and width.
3. The
polygon is not under the influence of an external gravitational field.
4. The
polygon is homogeneous in nature.
5. All
the mass of the polygon is assumed to be concentrated at the center.
CALCULATION
Consider
a Polygonal plate of side length ‘a’ [m] and mass M [Kg]. The types of polygons
considered are Pentagon, Hexagon, Heptagon and Octagon respectively. From the
calculations of center of gravity of polygons, it is evident that every polygon
can be divided equally into isosceles triangles with the exception of Hexagon
which can be divided equally into equilateral triangles.
Figure .1
Consider
one such isosceles triangle of side length ‘a’, height ‘h’ and slant length ‘b’
as depicted in figure .1. Assume all polygons considered have same side length
‘a’.
In
∆ABC by simple trigonometry,
θ
– Angle between side ‘b’ and side ‘a’
Angle
‘θ’ is also half of the interior angle of the polygon. Hence θ = ϕ/2 where ‘ϕ’
is the interior angle of the polygon. The interior angle of the polygon
increases with the increase in the number of sides of the polygon. Hence a
pentagon has smaller interior angle compared to an Octagon. From equation (2)
height ‘h’ is directly proportional to side ‘a’ and tan (θ). As angle ‘θ’
increases, tan (θ) will also increase. This implies that height ‘h’ would
change for every polygon despite the same side ‘a’ since each polygon has
different interior angle.
The
magnitude of gravitational field at corner (Point A) and mid-point (Point B)
are as follows:
The
gravitational field magnitude only depends on side length ‘a’ and height ‘h’.
Since it is proved that height ‘h’ cannot be the same for every polygon, it is
clear that gravitational field magnitude for every polygon at both corner and
mid-point will not be the same despite the same equation.
INSIGHTS
The
two most important observations regarding the dependency of height ‘h’ on the
gravitational field magnitude are as follows:
- For a given polygon of side length ‘a’, the height
‘h’ will always be different for every type of polygon.
- The height ‘h’ is proportional to tangent of the
angle of the triangle. Hence ‘h’ will increase as the angle increases or in
other words ‘h’ will be greater for multi sided polygons assuming constant
length ‘a’ for every polygon.
