Next Tech: Center of Gravity of Sector of an Ellipse

CENTER OF GRAVITY OF SECTOR OF AN ELLIPSE

INTRODUCTION

The center of gravity COG of an object is the point of action of the gravitational force. It is also known as the balancing point since all objects [simple or complex] have their COG within the object’s sphere of influence and this ensures stability. There are two values to be noted which is the geometrical center of gravity and the actual center of gravity. The geometrical COG is the exact center of the object. It can be calculated by various available methods like summation, moment of inertia, etc. The geometrical COG is valid as long as there is uniform distribution of mass. In case of uneven mass distribution or use of composite or heterogeneous materials, the actual COG will no longer coincide with the geometrical COG. This is because mass is distributed unevenly and the COG will shift where there is more mass. In this article we intend to determine the COG of sector of an ellipse and we assume uniform mass distribution for simplicity.

ASSUMPTIONS

1. Mass of the object is evenly distributed

2. Earth has uniform gravitational field

CALCULATION

Consider a sector of ellipse inscribed inside an actual ellipse. The ellipse has major axis ‘2a’ and minor axis ‘2b’. Consider a small element [a sliver of the sector]. Let the angle of this element be ‘dθ’. Let ‘θ’ be the angle between this element and one of the side of the ellipse.

Figure .1 Sector of an Ellipse

To determine the center of gravity, we need to follow three steps:

1. Determine the area of the element

2. Determine the area of the final object in question [sector]

3. Determine the distance between COG of element and the origin

Once we have the above three values, we can determine the COG coordinates.

Step1: Area of object [sector]

The sector of an ellipse is just a subset of the entire ellipse, hence the area will also be a part of the entire ellipse area. The area will depend on the angle of the sector. The equation is as follows:

Step2: Area of the element

From the figure, the element can be approximated as a triangle with height ‘h’ and base ‘hdθ’ respectively. Now the approximate perimeter of ellipse is given by,

The base of the triangle element which is also the arc length of the ellipse is given by,

But arc length is also equal to hdθ, so on comparison from both equations we obtain that height,

The area of the element dA is as follows,

Step 3: COG of the element

Since the element is a triangle, the COG of a triangle is distributed in the ratio 2/3 and 1/3. The COG is (2/3)*h away from the origin. We then resolve the distance into two components which is x and y. We get cosine term for the x component while the sine term for y component.

Let (x_s ,y_s) be the COG coordinates of the element.