GRAVITATIONAL
FIELD LINES OF A RECTANGULAR HYPERBOLA
INTRODUCTION
The force of gravity as
described by Sir Isaac Newton is a force of attraction. The gravitational force
acting on a body of mass ‘m’ is equal to the product of its mass ‘m’ and the acceleration
due to gravity acting on that body. Now Force is a vector quantity since it is
a product of mass [scalar] and acceleration [vector]. By definition a scalar
quantity can be represented by magnitude alone whereas a vector quantity must
be represented by magnitude and direction. Thus while solving a vector
quantity, it is important to obtain both magnitude and direction to obtain
complete information of the vector. Acceleration due to gravity being a vector
quantity has both magnitude and direction. The direction of gravitational force
is termed as gravitational field lines or orthogonal trajectories which are
always orthogonal [perpendicular] to the surface as explained in detail in the
calculation section. This article intends to determine the gravitational field
lines of a rectangular hyperbolic arc or plate.
ASSUMPTIONS
1. The plate has
negligible thickness hence is assumed to be two dimensional
2. The plate is
homogeneous in nature meaning composed of only one material
3. The plate is not under
the influence of an external gravitational field
4. The whole mass of the
plate is assumed to be concentrated in the center
CALCULATION
Consider a rectangular
hyperbola of major axis ‘a’ [m], minor axis ‘b’ [m] and mass ‘m’ [Kg].
There is a 3 step
procedure to determine the field lines
Step1 Determine differential equation of
Rectangular hyperbola and its slope
For a rectangular
hyperbola, the major and minor axis are equal. The equation of a regular Rectangular
hyperbola with major and minor axes ‘c’ is,
The equation is
perfectly arranged with the constant being isolated.
Differentiating equation
(1) with respect to x,
Equation (2) represents
the slope of equation (1) [Rectangular hyperbola]. Thus an orthogonal
trajectory to the Rectangular hyperbola must have a slope that is negative
inverse of the slope in equation (2).
Step2 Determine the slope of the
orthogonal trajectory
Thus the new slope or
the slope of the orthogonal trajectory which is a negative inverse of equation (2)
is
Step3 Determine equation of the
orthogonal trajectory by integration
To obtain the equation
of the orthogonal trajectory, integrate equation (3) by separating the
variables. Rearranging equation (3)
{‘k’ is a constant of
integration}
Raising
exponentials to the above equation,
where c = ek
Equation (4) represents
the family of orthogonal trajectories of a rectangular hyperbola. The
orthogonal trajectories which represent hyperbolas are pictorially represented
in figure 1.
REPRESENTATION
The graph of a rectangular
hyperbola and its orthogonal trajectories are represented in Figure 1. The
hyperbolas which are orthogonal trajectories to the rectangular hyperbola
appear emanating outward and perpendicular to them. Thus a rectangular
hyperbola will have its gravitational field directed as regular hyperbolas.
 |
| Fig .1 Gravitational field lines of Rectangular hyperbola |
EXPLANATION
Since a rectangular
hyperbola is only symmetric about one axis x or y but not both, the
gravitational field lines are not equidistant from the center and hence the
magnitude of gravitational field is not the same at all points on the perimeter
of rectangular hyperbola. From figure 2, it is evident that the field lines are
non-uniform since they are not parallel to each other. The non-uniformity stems
from the curvature of the curve [rectangular hyperbola]. The field lines in
general represent the direction of the field.
CONCLUSION
The final equation (5)
implies that the orthogonal trajectory of a rectangular hyperbola is a family
of hyperbolas. It is important to note that the constant ‘c’ or ‘ek’
can have infinite values hence there are infinite orthogonal trajectories for a
given shape in this case the Rectangular hyperbola.
