Gravitational field of Pentagonal plate

GRAVITATIONAL FIELD OF A PENTAGONAL 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 Pentagonal 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 Pentagon 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 Pentagonal 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

The center of gravity of a Pentagon as determined from previous article is, (x, y) ≡ (0.809a, h)

 

Fig 1 Pentagon

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 ≡ (0.309a, 0)

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

Point B ≡ (0.809a, 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 Pentagonal 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 Pentagon.

Comparison of capillary rise on Cosmic bodies

COMPARISON OF CAPILLARY RISE ON COSMIC BODIES

INTRODUCTION

Capillarity or capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces. When a capillary tube is inserted into a beaker containing liquid, the level of liquid in the capillary tend to rise or fall depending on the angle of contact between liquid and the tube wall. We intend to study only the rise of liquid in a capillary tube. Since the liquid rises against gravity, the extent to which it can rise also directly depends on the local acceleration due to gravity. We thus intend to compare the capillary rise on each cosmic body such as the Moon, Earth, Jupiter, Sun and Neutron Star.

ASSUMPTIONS

1. The local acceleration due to gravity is constant and not varying continuously

2. The angle of contact is always acute

3. The liquid is free from impurities

4. Temperature of the liquid is constant

5. The capillary is indestructible

6. Space time distortion does not impede capillary motion

CALCULATION

Consider a liquid [pure water] of density ‘ρ’ in a glass beaker with an inverted glass capillary tube inserted in a beaker as shown in figure.1. Let ‘r’ be the radius of the capillary tube, ‘θ’ be the angle of contact between pure water and the wall of capillary tube and ‘T’ be the surface tension of water.

The equation for capillary rise ‘h’ in the tube is given by,

h – Capillary rise in the tube (m)

T – Surface tension of water [T = 7.28*10^-2 N/m]

θ – Angle of contact between pure water and glass tube [θ = 0°]

ρ – Density of water [ρ = 1000 Kg/m³]

r – Radius of capillary tube [r = 1.5 *10^-3 m]

g – Acceleration due to gravity on a cosmic body [m/s²]

As discussed in the previous posts, the capillary rise of water is different for each planet or star since the rise is inversely proportional to the acceleration due to gravity. The capillary rise of water on each cosmic body is given in table.1

	COSMIC BODY	CAPILLARY RISE [m]
1	Moon	5.99E-02
2	Earth	9.90E-03
3	Jupiter	3.96E-03
4	Sun	3.55E-04
5	Neutron Star	1.39E-14

            Table.1 Capillary rise of water on each cosmic body

GRAPH

The acceleration due to gravity is plotted against the capillary rise on each cosmic body:

This graph allows us to study the effect of acceleration due to gravity on the capillary rise in each cosmic body. The maximum capillary rise is on Moon where the acceleration due to gravity is less while the minimum capillary rise is on Neutron star. Practically, the observable difference in capillary rise between Sun and Neutron star is negligible.

CONCLUSION

Thus the effect of acceleration due to gravity on capillary rise was successfully studied