Gravitational field lines of a circle

GRAVITATIONAL FIELD LINES OF A CIRCLE

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 circular 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 two dimensional circular plate of radius ‘r’ [m] and mass ‘m’ [Kg].

There is a 3 step procedure to determine the field lines

Step1 Determine differential equation of circle and its slope

The equation of circle with radius ‘r’ is   

Differentiating equation (1) with respect to x, 

Equation (3) represents the slope of equation (1) [Circle]. Thus an orthogonal trajectory to the Circle must have a slope that is negative inverse of the slope in equation (3).

Step2 Determine the slope of the orthogonal trajectory

The new slope or the slope of the orthogonal trajectory which is a negative inverse of equation (3) is

Step3 Determine equation of the orthogonal trajectory by integration

To obtain the equation of the orthogonal trajectory, integrate equation (4) by separating the variables. Rearranging equation (4)

On solving the above integral, the solution is  

Raising exponential to above equation, 

It is of the form  

REPRESENTATION

The graph of a circle and its orthogonal trajectories are represented in Figure 1. The family of straight lines which are orthogonal trajectories to the circle appear emanating from the circle radially thereby satisfying the mathematical condition. Thus a circular plate will have its gravitational field directed as infinite straight lines.

EXPLANATION

A circle is symmetric about both x and y axes, thereby the gravitational field lines are equidistant from the center and hence the magnitude of gravitational field is same at all points on the circumference of circle. This statement is made assuming that field lines are uniform. Although this is an assumption, it is not necessarily true. The field lines of circle being straight lines are parallel to every other neighboring lines in close proximity of the circle. The farther the lines are away from the circle, the more spread out they are, which means the field lines are non-uniform at larger distances.

Fig .1 Gravitational field lines of circle

CONCLUSION

The final equation (6) implies that the orthogonal trajectory of a circle is a straight line of slope ‘M’ where M is equal to e^k which is a constant. It is important to note that the constant ‘k’ can have infinite values hence there are infinite orthogonal trajectories for a given shape in this case the circle.