Gravitational Force
What is Gravitational Force?
All objects having mass attract each other with force known as the gravitational force. It is quite noticeable in astronomical objects such as Sun, Earth, and Moon that have enormous masses. The reason is that the force is proportional to the products of the objects’ masses. It is responsible for keeping the planets in motion around the Sun and the Moon around the Earth. Even human beings exert a force on each other, but it is quite insignificant because of relatively low masses.
Gravitational force is noncontact since there is no contact between the objects. It is centripetal since it is directed towards the center of the orbit around which the object moves. It is responsible for keeping the body in orbit. The revolving body feels a tug that is directed away from the center. This tug is called the centrifugal force. The gravitational force is the weakest of all fundamental forces.
Many famous scientists have made significant contributions to the field of gravitation. Among them were Italian astronomer Galileo Galilei, who, in the early 17^{th} century, found that all objects accelerate equally towards the center of the Earth. British mathematician Isaac Newton was the first to discover the laws of gravitation in his 1687 seminal work.
Universal Laws of Gravitation
Also known as Newton’s Law of Universal Gravitation, the law states that allobjects with a mass in the universe attract each other with a force that is
 directly proportional to the product of the masses
 inversely proportional to the square of the distance between their centers
Theseare the two factorsthat affect the gravitational force. The value is high for massive objectsand when the bodies are closer to one another.
General Formula for Gravitational Force
Suppose M_{1} and M_{2} be the masses of the two bodies, and R be the distance of separation between their centers. The following equation gives the gravitational force between the two objects.
F = G M_{1} M_{2} / d^{2} (1)
Unit of Gravitational Force: N or Newton
Here, G is called the universal gravitational constant. It is an empirical physical constant, which has a value of 6.67 X 10^{11}N.m^{2}/kg^{2}. Its dimensional formula is M^{1}L^{3}T^{2}. By knowing the masses M_{1} and M_{2} and their distance of separation d, it is possible to calculate the magnitude of F.
Properties of the Gravitational Force
Here are some facts and characteristics of the gravitational force.
 Attractive
 Noncontact
 Longrange
 Does not require a medium
 Directly proportional to the product of the masses of the objects
 Inversely proportional to the square of the distance of separation between the objects
 A constant value on the surface of the Earth
 Weakest of all fundamental forces
 Acts along the line joining any two bodies
Examples of Gravitational Force
An example of gravitation force in our daily lives is that when an object is thrown in the air, it comes back to the surface due to Earth’s gravitation. Below are some more examples.
1. Gravitational Force of the Earth
Earth exerts a gravitational force on every object, a phenomenon known as gravity. Gravity holds every entity, including us, on the surface and not allow to float freely in the air. We exert the same force on Earth that the Earth exerts on us. However, the Earth is so massive that it is unperturbed. An object suspended in air, if released, will fall freely towards the center of the Earth.
The difference between gravitational force and gravity is that the former is applied to any two objects in the universe. Gravity is the force between Earth and any other object close to Earth, including one on its surface.
The force of gravity is given by,
F = G M_{E} m / R_{E}^{2} (2)
Where M_{E} is the mass of the Earth, m is the mass of an object, and R_{E} is the Earth’s radius. If the object is at an altitude h above the surface of the Earth, then the equation modifies to,
F = G M_{E} m / (R_{E }+ h)^{2} (3)
From the above equation, it is quite clear that the Earth’s gravitational force vanishes when h →∞, i.e., at large distances from the surface.
According to Newton’s second law of motion, force is given by mass m multiplied by acceleration a. Therefore,
F = ma (4)
Comparing equations (2) and (4),
a =g = G M_{E} / R_{E}^{2} (5)
The above term is called the acceleration due to gravity. It has a value of 9.81 m/s^{2} on the surface of the Earth. Multiplying g by the mass of a person gives the weight W of the person.
W = mg (6)
The acceleration due to gravity in space is zero, which is why astronauts feel weightlessness and float freely.
The following expression gives the work done by Earth’s gravitational force when an object of mass m falls from a height h above the Earth’s surface.
Work done = mgh (7)
This work is the change in the object’s potential energy as the object falls through the air.
Satellites orbit around the Earth in fixed orbits due to the gravitational pull. Had there been no gravity, the satellites would have flung away into space.
The gravity in the polar region is higher than that at the equator. The reason is that the poles are closer to the center of the Earth than the equator.
Gravitational Force Between Earth and Moon
The Moon revolves around the Earth because gravitational forces hold them together. We put their masses and the distance between their two centers in equation (1) to calculate this force.
Mass of the Earth: M_{E} = 6.0 x 10^{24} kg
Mass of the Moon: M_{M} = 7.35 x 10^{22} kg
The average distance between the Earth and Moon, R_{EM} = 3.844 x 10^{8} m
Universal gravitational constant, G = 6.67 x 10^{11} Nm^{2}/kg^{2}
From equation (1),
F = G M_{E} M_{M} / R_{EM}^{2}
= 6.67 x 10^{11} N.m^{2}/kg^{2} x 6.0 x 10^{24} kg x 7.35 x 10^{22} kg / (3.844 x 10^{8})^{2} m^{2}
= 2 x 10^{20} N
Thus, the gravitational force between the Earth and Moon is 2 x 10^{20} N.
2. Gravitational Force of the Sun
The Sun has a gravitational force due to its mass, which is so large that its influence extends vastly. All planets revolve around the Sun in elliptical orbits due to this attractive force. The gravitational force between the Sun and Earth can be calculated using equation (1).
Mass of the Sun: M_{s} = 2.0 x 10^{30} kg
Mass of the Earth: M_{E} = 6.0 x 10^{24} kg
The average distance between the Sun and Earth, R_{SE} = 1.5 x 10^{11} m
Universal gravitational constant, G = 6.67 x 10^{11} Nm^{2}/kg^{2}
From equation (1),
F = G M_{S} M_{E} / R_{SE}^{2}
= 6.67 x 10^{11} N.m^{2}/kg^{2} x 2.0 x 10^{30} kg x 6.0 x 10^{24} kg / (1.5 x 10^{11})^{2} m^{2}
= 3.5 x 10^{22} N
Thus, the gravitational force between the Sun and Earth is 3.5 x 10^{22} N.
Likewise, the gravitational force between the Sun and other planets will depend upon their masses and the distances.
3. Gravitational Force of the Planets
All planets of the solar system have their gravitational force. The values of the acceleration due to gravity are given below.
 Mercury: 3.7 m/s^{2}
 Venus and Uranus: 8.87 m/s^{2}
 Mars: 3.71 m/s^{2}
 Jupiter: 24.79 m/s^{2}
 Saturn: 10.44 m/s^{2}
 Neptune: 11.15 m/s^{2}
Besides, the gravity of the Moon is 1.62 m/s^{2}. When compared to Earth, it is about 1/6^{th}.
Advantage and Disadvantage of the Gravitational Force
The gravitational force has significant importance that has impacted our daily lives. Here are some of its benefits.
 Constant on the surface of the Earth
 Keeps the muscles and bones working
 Allows Earth to retain its atmosphere
 Allows water dams to store energy
Here are some of its disadvantages.
 Makes it difficult to travel to outer space as rockets have to overcome the force of gravity
 Makes us fall and get hurt
 Limits the height of tall buildings during construction
 Hard on the bones and joints as a person ages
FAQs
Ans. Yes. The gravitational force is conservative since the work done by it around a close path is zero.
Ans. Yes. There can be a gravitational force in a vacuum.
Ans. The range of the gravitational force is infinite, although it becomes weaker as the distance between objects increases.
Ans. Weight. A person’s weight on Earth is not the same on the Moon due to the difference in the gravitational forces.
Ans. The electrical force is stronger than the gravitational force.
Gravitational Force Problems and Solutions
Here is an example problem with the solution.
P.1. What is the gravitational force between two solid objects whose masses are 110 and 130 kgs, and their distance of separation is 80 cms.? Given, G = 6.67 x 10^{11} Nm^{2}/kg^{2}.
Soln. Given
M_{1} = 110 kg
M_{2} = 130 kg
d = 80 cm or 0.8 m
F = G M_{1} M_{2} / d^{2}
= 6.67 x 10^{11} N.m^{2}/kg^{2} x 110 kg x 130 kg / (0.8)^{2}
= 1.5 X 10^{6} N

References
Article was last reviewed on Friday, November 6, 2020