GRAVITATIONAL
FIELD LINES OF AN ASTROID
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 two dimensional astroid or an astroidal plate.
ASTROID
DEFINITION
An astroid is a
hypocycloid with four cusps. A hypocycloid is a special plane curve generated
by the trace of a fixed point on a small circle that rolls within a larger
circle. This can be explained as follows: Consider a small and a large circle
where the small circle is residing inside the large circle. The large circle is
stationary while the small circle will roll inside by traversing the perimeter
of the large circle. Any point on the small circle will trace out a curve
inside the large circle. This curve generated depends on many factors, one such
factor being the radius of the small circle ‘a’. This radius is often referred
to as the radius of the rolling circle. When the point on the small circle
completes four rotations, the curve generated is a hypocycloid with four sharp
corners or in other words an astroid.
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 two
dimensional astroidal plate of rolling radius ‘a’ [m] and mass ‘m’ [Kg].
There is a 3 step
procedure to determine the field lines
Step1 Determine differential equation of
Astroid and its slope
The equation of a
regular Astroid with ‘a’ as the radius of rolling circle is,
There is no need to
rearrange the equation since it is already arranged in a form where the
constant is isolated.
Differentiating equation
(1) with respect to x,
Equation (2) represents
the slope of equation (1) [Astroid]. Thus an orthogonal trajectory to the Astroid
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}
Multiplying
throughout by (4/3) to the above equation,
Where c = -4k/3
Equation (4) represents
the family of orthogonal trajectories of an Astroid. The orthogonal
trajectories which represent an algebraic curve are pictorially represented in
figure 1.
REPRESENTATION
The plot of an Astroid
and its orthogonal trajectories is represented in Figure 1. The algebraic
function appear emanating outward and perpendicular from the Astroid. Thus an
astroid will have its gravitational field directed as given by the algebraic
curve in equation (4).
 |
| Fig . 1 Gravitational field lines of Astroid |
EXPLANATION
The gravitational field
lines represented by black dotted lines appear curved. But one such field line
represented by green dotted line appears like a straight line. In equation (4),
if ‘x’ and ‘y’ values are unique, the right hand side will be some non-zero
value ‘c’, meaning the curve will pass through this non-zero ‘c’ on either axis
based on the equation format. Suppose if x = y, the right hand side of (4) will
become zero. Hence the curve will satisfy this condition by passing through the
origin. Thus the green dotted line passes through the origin and it appears
like a straight line since x = y, while the other black dotted lines pass
through some non-zero value on either of the axes. Although all the field lines
appear parallel to each other, they are not and this is a clear representation
of a non-uniform field.
CONCLUSION
The final equation (4)
implies that the orthogonal trajectory of an Astroid is an algebraic curve. It
is important to note that the constant ‘c’ can have infinite values hence there
are infinite orthogonal trajectories for a given shape in this case the Astroid.
