Gravitational field of a Heptagonal plate

GRAVITATIONAL FIELD OF A HEPTAGONAL PLATE

INTRODUCTION

Gravity is derived from the Latin word ‘gravitas’ meaning mass. The universal law of gravitation was coined by Sir Isaac Newton. According to the law, any two masses anywhere in the universe separated by a distance will attract each other. This force of attraction is proportional to the product of their masses and inversely proportional to square of the distance between them.

F = GM₁M₂/R² (Eq. 1)

The distance between two masses can be finite or infinite, which is why gravitational force is referred to as long range force but is also the weakest force among all the other fundamental forces. All objects that have mass will attract other masses. This means that each mass has its own gravitational field just like Earth. So this implies that all objects will attract each other since they will have their own field. This is not evident on Earth since Earth’s gravitational field outweighs all other mass’s field and hence all objects no matter how massive are attracted toward the Earth. In this article the gravitational field of a two dimensional Heptagonal plate will be determined and points on the plate would be identified where the plate’s own gravity is strong or weak.   

ASSUMPTIONS

1. The Heptagon is assumed to be a regular polygon.

2. The thickness of the plate is negligible compared to its length and width.

3. The plate is not under the influence of an external gravitational field.

4. The plate is a homogeneous material.

5. All the mass is assumed to be concentrated at the center.

CALCULATION

Consider a Heptagonal plate of side length ‘a’ [m] and mass M [Kg]. First the center of gravity of this plate will be determined, followed by the magnitude of the gravitational field at points of interest.

Center of gravity

Fig 1 Heptagonal plate

The center of gravity of Heptagon as determined from previous article is (x, y) ≡ (1.123a, h)

Points of interest and their distances from center

Consider two points namely Point A and Point B as shown in figure 1. Point A represents the corner point while point B represents the mid-point of a side. All the other points depicted in figure 1 are at the same distances as points A and B are from the center.

Point A ≡ (0.623a, 0)

The distance between Point A and C.G. can be calculated by the distance formula

Point B ≡ (1.123a, 0)

The distance between Point B and C.G. can be calculated by the distance formula

Gravitational field

From Eq. 1, the force of attraction between a mass and its own surface is given by,

g = GM/R²

g – Acceleration due to gravity of the mass (m/s²)

G – Universal constant of gravitation

G = 6.67*10^-11 Nm²/Kg²

M – Mass of the object (Kg)

R – Distance between the centers of two masses (m)

G field at point A

The gravitational field beyond the surface is obtained by adding the additional distance,

d – Distance between point A and any other point in space.

G field at point B

The gravitational field beyond the surface is obtained by adding the additional distance,

d – Distance between point B and any other point in space.

CONCLUSION

Thus the magnitude of gravitational field at two points on a Heptagonal plate was found successfully. On comparing the gravitational acceleration values at point A and B it is important to note that gravitational field is stronger at point B due to its closer distance to the center while it is relatively weaker at point A due to its greater distance from the center. This is true for every mid-point and corner point on the Heptagon.