Newton's Law of Gravitation: Detailed Law, Gravitational Constant

INTRODUCTION

Since ancient times, the curiosity of scientists about planets and stars is immense. Nicolaus Copernicus was the first to say that the sun is fixed and the earth revolves around the sun. Later, the German scientist Johannes Kepler discovered three laws of motion of the planets around the Sun. By analyzing these formulas, the scientist Sir Isaac Newton determined the nature of the force of attraction acting between the Sun and the planets and also established the rules for measuring this force. The force of attraction is called gravitation, which acts not only between the sun and the planets, but also between any two objects or particles of matter in the universe. The law for measuring the force of attraction is known as Newton’s law of gravitation or the law of universal gravitation.

NEWTON'S LAW OF GRAVITATION

Definition: The force by which any two particles of matter in this universe attract each other is called gravitation.

Law: Any two particles in the universe attract each other along their connecting straight lines. The value of this attractive force is proportional to the product of the masses of the two particles and inversely proportional to the square of the distance.

Let, r be the distance between two particles of masses m1 and m2. If the mutual attraction force between two particles is F, then according to Newton’s law of gravitation,

F ∝ m1·m2; When R is unchanged,

F ∝ 1/r²; When m1 and m2 are unchanged,

So, F ∝ m1·m2/r² or, F = G·(m1·m2/r²) ……….(I)

Here G is called universal gravitational constant or in short gravitational constant. (I) Equation No. is the mathematical form of the law of gravitation. Incidentally, the mass (m1, m2) used in Newton’s law of gravity is called gravitational mass. On the other hand, the mass used in Newton’s second law of motion is called inertial mass. Although these two masses come from two different sources, it has been shown through many experiments that they are two identical physical quantities. Hence there is no need to mention these two masses separately.

GRAVITATIONAL CONSTANT

Definition: The force with which two particles of unit mass attract each other at a distance of unit distance is called the gravitational constant.

Unit: (I) From equation no, it can be written, Unit of G = Unit of F X (Unit of r)²/ (Unit of m)².

So, the units of G in various ways are:

CGS – dyn·cm²/g²

FPS – poundal·ft²/lb²

SI – N·m²/kg²

Dimension: (I) From equation no, it can be written,

Dimension of G = Dimension of F X (Dimension of r)²/ (Dimension of m)² = [MLT^-2X L²]/[M²] = [M^-1L³T^-2]

Value of G: The values of gravitational constant determined by various experiments are:

G = 6.67 X 10^-8 dyn·cm²/g² (CGS)

1.07 X 10^-9 poundal·ft²/lb² (FPS)

6.67 X 10^-11 N·m²/kg² (SI)

According to definition of G, if two objects of mass 1kg are placed 1m apart, they attract each other by 6.67 X 10^-11 N. Obviously this is a very low amount of force. That is why we cannot feel the force of gravity acting in our everyday objects. But we can easily feel the gravitational force of the earth. Because the mass of the earth is very high, this force of attraction is never negligible.

UNIVERSALITY OF THE LAW OF GRAVITATION

According to Newton’s law of gravitation, the force of attraction between two objects depends only on the mass of the two objects and their distance. This force of attraction—the state of the two objects (solid, liquid, or gas), chemical composition, temperature, medium, etc.—does not depend on anything. The law of gravitation applies equally to the small distances between earthly objects and the vast distances between galaxies in space. With the help of this formula, it is possible to explain the movement of the planets around the Sun beautifully. For these reasons the law of gravitation is considered universal and the gravitational constant is called the universal gravitational constant.

But according to Einstein’s theory of relativity,

(i) the mass of an object depends on its velocity.

(ii) the amount of distance between two objects depends on the observer; The distance between two objects measured by a stationary observer will not have the same value as measured by a moving observer.

(iii) A heavy object (e.g. Sun) located near the straight line joining two objects affects the distance between them.

(iv) Newton’s law of gravitation does not apply to very small distances like atomic distance (10^-9 m). For these reasons, Newton’s law of gravitation can no longer be called a universal law. However, even in Einstein’s theory, the universality of the gravitational constant G is not undermined in any way.

GRAVITATIONAL FIELD

Due to its own mass, the region around which the gravitational effect of the object is felt, i.e. the region on which the object first exerts gravitational force when brought to another object, is called the gravitational field of the object. In fact, the gravitational field of any object extends to an infinite distance, but the region beyond which the gravitational force becomes practically negligible may not be considered as the object’s gravitational field. It is very important not only to identify a region as the gravitational field of the object, but also to measure the gravitational effect at each point of that region. Therefore, the physical quantity that is indicated is the definition of the gravitational field intensity, sometimes referred to simply as the gravitational field.

Definition: The gravitational force acting on an object of unit mass at any point in a gravitational field is called the gravitational field strength at that point.

Let the mass of the object be M; Determine the gravitational field due to the object at a point at a distance r from the center of the object. If an object of unit mass is placed at that point, according to the law of gravity,

gravitational force = GM·1/r² = GM/r²

Hence, the gravitational field strength at that point, g = GM/r²

Gravitational acceleration at any point in Earth’s gravitational field, i.e. gravitational field, g = GM/r². From this analogy, the gravitational field strength and the gravitational acceleration at a point in the Earth’s gravitational field are identical physical quantities. Note that both of these equations are forces acting on a unit mass.

Unit: The units of gravitational field intensity in CGS system and SI are cm·s^-2 and m·s^-2 respectively.

GRAVITATIONAL POTENTIAL

Definition: The amount of work done by an external agent to bring an object of unit mass from an infinite distance to a point in the gravitational field is called the gravitational field at that point.

Unit: The units of gravitational force in CGS system and SI are erg·g^-1 and J·kg^-1 respectively.

At infinite distance, i.e. where there is no gravitational field, the gravitational field is assumed to be zero. An object of unit mass is acted upon by the force of gravitational attraction when it is brought from there to a point in the gravitational field. Since the work done by the force is negative, the gravitational potential is negative. That is to say, since the gravitational force is always an attractive force, the gravitational potential is negative.

Gravitational field strength i.e. gravitational force acting on a unit mass at a distance r from the center of an object of mass M is g = GM/r². Work done by the external agent if the displacement of the unit mass is →dr = – →g · →dr = -(- gdr) = gdr (since →g and →dr are inverse, so →g · →dr = -gdr). Hence, the work done to bring a unit mass from an infinite distance to a position r or the gravitational potential at a distance r is,

Hence, the relationship between gravitational field intensity and potential is,

REMINDER ON THIS TOPIC

The force by which any two particles of matter in this universe attract each other is called gravitation.
Any two particles in the universe attract each other along their connecting straight lines. The value of this attractive force is proportional to the product of the masses of the two particles and inversely proportional to the square of the distance.
The force with which two particles of unit mass attract each other at a distance of unit distance is called the gravitational constant.
The gravitational force acting on an object of unit mass at any point in a gravitational field is called the gravitational field strength at that point.
The amount of work done by an external agent to bring an object of unit mass from an infinite distance to a point in the gravitational field is called the gravitational field at that point.
The force with which the earth attracts any object on or near the surface is called gravity. The acceleration due to the force of gravity on a freely falling object is called gravitational acceleration.
Kepler’s Laws of Motion of Planets and Satellites:

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at one focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the orbital period of a planet around the Sun is proportional to the cube of the semi-radius of the orbit.

The minimum velocity at which an object can be thrown from the surface of the Earth or any other planet or satellite beyond its gravitational attraction is called the release velocity.
If the relative angular velocity of an artificial satellite with respect to the Earth’s angular momentum is zero and the satellite is always at the equator, it appears from the surface that the satellite is fixed in the same place in the sky. Such satellites are called geosynchronous satellites.

Newton’s Law of Gravitation: Gravitational Law, Constant, Field, Potential

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