Gravitational field lines of Astroid

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.