Gravitational Force: Definition, Formula, and Examples (2023)

Also known as Newton’s Law of Universal Gravitation, the law states that all objects 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

These are the two factors that affect the gravitational force. The value is high for massive objects and when the bodies are closer to one another.

General Formula for Gravitational Force

Suppose M₁ and M₂ 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.

\[ F = \frac{GM_1M_2}{d^2} \hspace{2 cm} (1) \]

Unit of Gravitational Force: N or Newton

Here, G is called the universal gravitational constant. It is an empirical physical constant with a value of 6.67 X 10^-11N.m²/kg². Its dimensional formula is M^-1L^-3T^-2. By knowing the masses M₁ and M₂ and their distance of separation d, it is possible to calculate the magnitude of F.

Here are some facts and characteristics of gravitational force.

• Attractive
• Non-contact
• Long-range
• It 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

An example of gravitation force in our daily lives is that when an object is thrown in the air, it returns 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 us on the surface and does not let us 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 will fall freely towards the center of the Earth when released.

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 on its surface.

The force of gravity is given by,

\[ F = \frac{G\hspace{0.1 cm}M_E\hspace{0.1 cm} m}{R_{E}^2} \hspace{2 cm} (2)\]

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

\[ F = \frac{G\hspace{0.1 cm}M_E\hspace{0.1 cm} m}{(R_{E}+h)^2} \hspace{2 cm} (3)\]

From the above equation, it is quite clear that the Earth’s gravitational force vanishes when h →∞, or 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,

\[ F = ma \hspace{2 cm} (4) \]

Comparing equations (2) and (4),

\[ a = g = \frac{G\hspace{0.1 cm}M_E\hspace{0.1 cm}}{R_{E}^2} \hspace{2 cm} (5) \]

The above term is called acceleration due to gravity. It has a value of 9.81 m/s² on the surface of the Earth. Multiplying g by the mass of a person gives the weight W of the person.

\[ W = mg \hspace{2 cm} (6) \]

The acceleration due to gravity in space is zero, which is why astronauts feel weightless 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.

\[ \text{Work done} = mgh \hspace{2 cm} (7) \]

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. Without gravity, the satellites would have been 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. To calculate this force, we put their masses and the distance between their two centers in equation (1).

Mass of the Earth: M_E = 6.0 x 10²⁴ kg

Mass of the Moon: M_M = 7.35 x 10²² kg

The average distance between the Earth and Moon, R_EM = 3.844 x 10⁸ m

Universal gravitational constant, G = 6.67 x 10^-11 Nm²/kg²

From equation (1),

\[ F = \frac{G M_E M_M}{R_{EM}^2} \\\Rightarrow F = \frac{6.67 \times 10^{-11} \hspace{0.1 cm} N \cdot m^{2} \cdot kg^{-2} \times 6.0 \times 10^{24} \hspace{0.1 cm} kg \times 7.35 \times 10^{22} \hspace{0.1 cm} kg}{ (3.844 \times 10^{8} \hspace{0.1 cm} m)^2} \\\Rightarrow F = 2 \times 10^{20} \hspace{0.1 cm} N \]

Thus, the gravitational force between the Earth and Moon is 2 x 10²⁰ 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 10³⁰ kg

Mass of the Earth: M_E = 6.0 x 10²⁴ kg

The average distance between the Sun and Earth, R_SE = 1.5 x 10¹¹ m

Universal gravitational constant, G = 6.67 x 10^-11 Nm²/kg²

From equation (1),

\[ F = \frac{G M_S M_E}{R_{SE}^2} \\\Rightarrow F = \frac{6.67 \times 10^{-11} \hspace{0.1 cm} N \cdot m^{2} \cdot kg^{-2} \times 2.0 \times 10^{30} \hspace{0.1 cm} kg \times 6.0 \times 10^{24} \hspace{0.1 cm} kg}{ (1.5 \times 10^{11} \hspace{0.1 cm} m)^2} \\\Rightarrow F = 3.5 \times 10^{22} \hspace{0.1 cm} N \]

Thus, the gravitational force between the Sun and Earth is 3.5 x 10²² N.

Likewise, the gravitational force between the Sun and other planets will depend upon their masses and 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/s²
• Venus and Uranus: 8.87 m/s²
• Mars: 3.71 m/s²
• Jupiter: 24.79 m/s²
• Saturn: 10.44 m/s²
• Neptune: 11.15 m/s²

Besides, the gravity of the Moon is 1.62 m/s². When compared to Earth, it is about 1/6^th.

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
• It keeps the muscles and bones working
• Allows Earth to retain its atmosphere
• Allows water dams to store energy

Here are some of its disadvantages.

• It makes it difficult to travel to outer space as rockets have to overcome the force of gravity
• It makes us fall and get hurt
• Limits the height of tall buildings during construction
• Hard on the bones and joints as a person ages

Problem 1. What is the gravitational force between two solid objects whose masses are 110 and 130 kgs and whose distance of separation is 80 cm.? Given, G = 6.67 x 10^-11 Nm²/kg².

Solution:

Given

M₁ = 110 kg

M₂ = 130 kg

d = 80 cm or 0.8 m

F = G M₁ M₂ / d²

= 6.67 x 10^-11 N.m²/kg² x 110 kg x 130 kg / (0.8)²

= 1.5 X 10^-6 N

Article was last reviewed on Friday, February 3, 2023