Newton’s Law of Gravity

In the late 1600s, Sir Isaac Newton came up with the law of gravity which is also known as the universal law of gravitation. Sir Isaac Newton’s inspiration for deducing the revolutionary law of gravity was an apple falling from a tree. We are all pretty familiar with the story of Newton and how he discovered gravity. The falling of an apple made him discover Newton's gravity and the law of gravitation. Newton’s law of gravity plays an important role in mechanics. 

Newton had a simple question out of curiosity is why an apple was falling instead of either sideways or upward!!! Later Newton realized that the earth must be responsible for the apple to fall downwards perpendicular to the ground. This was the major turning point and then he developed the law of gravity.

Gravity is the force of attraction happening between any two bodies. Basically, all the objects in the universe attract each other with a certain amount of force, but in most cases, the force is either too weak or too small to be observed due to the very large distance of separation. 

So Newton’s law of gravitation was introduced, and it states that any particle of matter in the universe attracts any other particle with a force varying directly as the product of the masses and inversely as the square of the distance between them. Newton’s law of gravitation is the magnitude of the attractive force F is equal to G multiplied by the product of the masses (\[m_{1}~and~m_{2}\]) and divided by the square of the distance R :

\[F = \frac{G(m_{1} m_{2})}{R^{2}}\]. 

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What is Newton's Law of Gravitation?

The law of gravity is an important discovery in the field of physics. It gives an insight into the relationship between mass and force. The law of gravitation states that- every object in the universe attracts every other object such that the force exerted will be proportional to the product of the masses and inversely proportional to the square of the distance between them. 

The formula of Newton’s Law of Gravity

Newton’s Law of Gravitation is formulated as : 

\[F_{G} = \frac{G(m_{1} m_{2})}{r^{2}}\]

In the above equation, the values are defined as:

Fg is the force of gravity that is typically in newtons.

G is the gravitational constant that adds the proper level of proportionality to the equation. 

The value of the gravitational constant is \[6.67259 * 10^{-11} N * m^{2} / kg^{2}\], the value will change if other units are being used.

(\[m_{1}~and~m_{2}\]) are the masses of the two particles that are typically in kilograms.

r is the straight-line distance between the two particles that are typically in meters.

According to Newton’s Law of Gravitation

• The magnitude of the force acting between two point masses is directly proportional to the product of their masses.

• The magnitude of the force acting between two point masses decreases rapidly as distance increases.

Mathematically we write,

Consider two objects having masses \[m_{1}~and~m_{2}\] separated by a distance r, as shown in the figure. 

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According to the Statement of the Law of Gravitation,

The magnitude of the force acting on the body is directly proportional to the product of the masses of interacting bodies, then we get:

\[\Rightarrow F \alpha m_{1}m_{2}.....(1)\]

The value of the proportionality constant is found to be \[G = 6.673 \times 10^{-11} Nm^{2}/kg^{2}\]

Equation (4) is known as the mathematical form of Newton’s law of gravitation or the law of gravitational force. From equation (4) we find that the force acting on each other will be directly proportional to the product of point masses and inversely proportional square of the distance between them. It is also known as the inverse square law. In some articles, it is also referred to as the first law of gravity.

The gravitational force acting between two objects is only due to their masses. The gravitational force is one of the four basic forces of physics. Sometimes it is also referred to as Newton gravity or Newton's gravity. The gravitational force is valid throughout the universe. For significant gravitational force, one among the two objects must be larger than the other. 

Characteristics of Gravitational Force 

Following are the characteristics of Gravitational Force : 

• Gravitational force is a central force. 

• Gravitational force is a mutual force.

• Gravitational force is mass-dependent. 

• Gravitational force is an attractive force.

• Gravitational force is independent of the presence of other mass bodies.

• Gravitational force is a long-range force.

• Gravitational force is a universal force.

• Gravitational force is the weakest among the basic forces of nature.

• The gravitational force is always attractive and it is directed along with the line joining of two interacting bodies.

• The gravitational force is independent of the medium and the surrounding environment.

• The gravitational force is valid for long distances like the distance between two planets and for short distances like interatomic distances.

• The force of gravitation is conservative. Thus the work done gravitational force will be zero.

• If a particle is acted by n particles then the net force exerted on it will be equal to the vector sum of the forces due to surrounding particles. i.e.,

• \[F_{net}~=~\sum_{i}~=~1^{n}F_{i}\] where \[F_{i}\] is the force acting on the object by \[i^{th}\] particle.

Weight

• From the law of gravity or Newton’s law of gravitation, we understood that mass is a crucial entity. There is always confusion between mass and weight, we consider mass and weight to be the same, but in reality, they are interrelated but are different from each other. 

• Weight is the gravitational force exerted on any object of a certain mass. The weight of an object can be estimated by multiplying the mass m of the object by the acceleration due to gravity, g, at the surface of the Earth. The measured acceleration due to gravity at the Earth’s surface is found to be about \[9.8 m/s^{2}\] or \[980 cm/s^{2}\].

• The measure of how much matter is in an object is known as mass, while weight is the measure of the gravitational force exerted on the material in a given gravitational field; thus, mass and weight are proportional to each other.

• W ∝ m

• Where,

• m - The mass of the object

• W = mg

• Where,

• g - acceleration due to gravity.

• It is observed that the mass of the given object will be constant, but the weight depends on the position of the object.

Solved Examples 

1. Define the force of gravity acting on an object of mass 2000 kg at the Earth’s surface?

Given: Mass of Earth \[m_{1}\] = 5.98 × 1024 kg

Mass of object \[m_{2}\] = 2000 kg

The radius of the Earth r = 6.38 × 106 m

Acceleration due to gravity \[ g = 9.8 m/s^{2}\]

Universal constant \[ G = 6.67 \times 10^{-11} N m^{2} / kg^{2}\]

Now, 

\[ F = \frac{Gm_{1}m_{2}}{r^{2}}\]

F = \[\frac{(6.67 \times 10^{-11}) (5.98 \times 10^{24})(2\times 10^{3})}{(6.38 \times 10^{6})^{2}}\]

F = \[\frac{(7.978 \times 10^{17})}{(4.07044 \times 10^{13})}\]

F = \[1.959 \times 10^{4}\] 

F = 19.59 N

Therefore, the force of attraction between the earth and a man is 882.3N.

2. Why Doesn’t the Moon Crash Into the Earth? What is the Value of Gravity on the Moon in Newtons?

Ans: Moon is the natural satellite of the earth. The forces of speed and gravity keep the moon in a constant orbit around the earth. The Moon seems to revolve around the earth, unaffected by gravity. However, the reason the Moon stays in orbit is precise because of gravity. Now the value of gravity on the moon can be calculated by using Newton’s law of gravitation.

This is all about Newton’s Laws of Gravitational forces explained with solved examples. Focus on how the terms are used to determine the formula and the value of the gravitational constant.