Next Tech: Center of Gravity of a Parabolic Arc

CENTER OF GRAVITY OF A PARABOLIC ARC

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 an arc of a regular parabola 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 regular parabola. Let A≡ [h, k] be the vertex of the parabola and B ≡ [h + a, k] be its focus. Let ‘a’ be the distance between the focus and vertex. This is a parabola which is along the x axis. For simplicity let us assume that the vertex is the origin. Consider a small rectangular length element of width ‘dL’. This element when integrated traces out the arc length of the Parabolic arc. The equation of parabola or arc of a parabola is,

If vertex is origin, replace (h, k) by (0, 0) in the above equation,

The equation can be rearranged to represent in terms of y as,

Fig .1 Parabolic Arc

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

1 1. Determine the length of the element in terms of one of the known variables x or y

2 2. Determine the arc length or the total length of the circle by integrating the element dL

3 3. Determine the C.O.G coordinates (x, y) by integrating each variable and dividing by the total length

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

Step1: Length of the element dL

From fig.1, we can observe that dL can be represented in terms of ‘dx’ and ‘dy’ by using the Pythagorean Theorem.

On taking dx common out of the square root, we obtain

Step2: Total length of the element

We can solve the above equation by differentiating y with respect to x, which is differentiating the equation of circle.

On solving we obtain,

Integrate the above equation indefinitely to obtain the total length of the curve.

This is the arc length of a parabolic arc. When the lower and upper bounds of integration are set, we will obtain the perimeter of parabola or arc of parabola based on the choice of limits.

Step 3: COG of the object {parabolic arc}

We now can perform integral calculations to determine the x and y coordinates respectively.

First we integrate x and y with respect to x. Later divide each answer by ∫dL to determine the x and y coordinate of C.O.G

On performing numerical computation, we obtain the values of integration as follows:

The x integral is,