Gravitational Force

Have you ever wondered why the Earth revolves around the Sun and not the other way around? Or why does the Moon remain in orbit instead of crashing into Earth? If the Earth pulls the Moon and the Moon pulls the Earth, shouldn’t they just come together? What keeps them apart?

All these questions can be understood through the concept of gravitation. The Gravitational force is a universal force of attraction that acts between objects with mass and is one of the fundamental forces shaping the universe.

What is Gravitational Force?

According to Newton's Universal Law of Gravitation

"The attractive force between any two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them."

Let's Read more about -Acceleration due to Gravity.

Newton's Law of Gravitation 

Newton's Law of Gravitation or Newton’s Law of Universal Gravitation (or Universal Laws of Gravitation) is the Law that leads to the further study of Gravitation and states that all the objects in the universe having any masses always attract each other with a force of attraction. This force of attraction is called the Gravitational Force (F) which is,

• Directly proportional to the product of the masses (m₁ and m₂) of the two objects in contact with other, and 
• Inversely proportional to the square of the distance (r) between their centres.

The expression or the relation for the above-stated law is given by the gravitational force formula, discussed below:

Gravitational Force Formula

The Law of Gravitation gives the Gravitational Force (F) between two bodies of masses (m₁ and m₂) at a distance r, apart from their centers, is given as:

F ∝ m₁m₂

F ∝ 1/r²

Now, combining the above two relations as,

F ∝ m₁m₂ / r²

F = Gm₁m₂ / r²

where G is the proportionality constant known as Gravitational Constant (= 6.67 ×10⁻¹¹N⋅ m²/kg²). 

Gravitational Force Unit and Dimensional Formula

• SI unit of Gravitational Force is Newton (N). 
• Dimensional Formula of Gravitational Force is [M¹L¹T^-2].

Properties of Gravitational Force

Here are some important characteristic features of Gravitational Force,

• Gravitational forces are always attractive and the weakest of all the fundamental forces.
• It is a type of Non-Contact Force, as it does not require any physical contact or touch to be experienced by a system of objects.
• Gravitational Force is a Long-range force and does not require any medium.
• The Gravitational Force value at the surface of the Earth is constant.

Newton’s Thought Experiment on Gravitation

Sir Isaac Newton imagined a thought experiment to illustrate the function of Gravity beyond just objects falling to the ground. He visualized a cannon situated on top of a very tall mountain, high above the Earth's atmosphere, and thought about what would happen if it fired cannonballs with various speeds.

1. Firing at Low Speed:

• When the cannonball is fired at a low speed, it will travel in a curved path and return back to Earth, similar to a regular projectile.

2. Firing at a Higher Speed:

• If fired with increased speed, the cannonball will cover more distance before hitting the ground, since Earth’s curvature makes the surface "fall away" beneath it.

3. Firing at Just the Right Speed (Orbital Motion):

• If the cannonball is fired quickly enough, it will continuously "fall" toward Earth without ever touching the surface. This is due to the Earth's surface curving away at the same rate. The cannonball will continue to travel in orbit around Earth, similar to how the Moon operates.

4. Firing Even Faster (Escape Velocity):

• If the cannonball is fired at a higher speed, it will entirely escape Earth’s gravitational pull and move into outer space. This speed is called Escape velocity.

What This Thought Experiment Proved:

• Gravity draws objects toward the Earth, but if an object possesses sufficient sideways velocity, it will keep falling without ever touching down, creating an orbit.

• The Moon stays in orbit around Earth because it is constantly "falling" toward it while moving forward fast enough to avoid hitting the surface.

• Planets revolve around the Sun for the same reason—gravity pulls them toward it, but their motion keeps them from falling in.

Development of Gravitational Theory

For hundreds of years, Humans have wondered why things drop to the Earth and why planets move in the sky. Throughout history, Scientists have tried to answer these questions, leading to our modern understanding of gravity. Let’s take a simple journey through the key discoveries:

Aristotle (4th Century BCE)

• Aristotle believed that heavier objects descend or fall more quickly than lighter ones.
• He believed that everything naturally moves toward its “proper place,” like stones falling to the ground and smoke rising in the air.
• However, his ideas or concepts were later proven wrong.

Galileo (1600s)

• Galileo challenged Aristotle’s theory by dropping objects of different weights from the Leaning Tower of Pisa.
• He demonstrated that all objects fall at the same speed when air resistance is ignored (e.g., a feather falls slower only due to the air resistance).
• He also discovered that objects moving forward continue moving unless stopped, laying the foundation for understanding orbits.

Newton (Late 1600s)

The famous story says Newton saw an apple fall from a tree and wondered why the Moon does not fall like the apple.

He realized that the same force that pulling the apple down also keeps the Moon in orbit around Earth.

Newton’s Law of Universal Gravitation stated:

• Every object in the universe attracts every other object toward it.
• The force of attraction depends on their masses and distance between them.

His theory clarified the reason why planets orbit the Sun and led to advancements in space science.

Albert Einstein (1915)

• Newton’s theory worked well, but it could not explain everything, like bending of light around massive bodies.
• Einstein suggested that gravity is not a force, but a bending of space and time.
• Imagine placing a heavy ball on a stretched rubber surface—smaller balls roll towards it. This is how massive objects like the Sun bend space-time, causing planets to orbit.
• Einstein’s theory predicted things like black holes and gravitational waves, both of which have now been observed.

Modern Science

• Scientists continue to examine gravity, particularly in quantum physics, where gravity behaves differently at tiny scales.
• Gravitational waves, detected in 2015, validated Einstein’s predictions and opened new ways of studying space.
• Space agencies use gravitational physics to send satellites, land on the Moon, and explore other planets.

Also Read,

• Gravity
• Contact and Non-Contact Forces

Gravitational Force Examples

Some everyday life examples of gravitational force can be discussed as,

Gravitational Force of Earth

Every object is subject to the gravitational pull of Earth, a phenomenon known as gravity. We cannot freely float in the air because of gravity, which keeps us on the ground. The force that the Earth and we both apply to the planet is equal. The Earth, however, remains unaffected because of its immense size. If a hung object is let go, it will fall naturally in the direction of the Earth's centre.

Gravitational Force Between Earth and Moon

Due to the gravitational pull of the Earth and the Moon, the Moon revolves around the Earth. To compute this force, we put their masses and the separation between their two centres into the gravitational force formula. Then, the gravitational force between the earth and the moon was found to be 2 × 10²⁰ N.

Gravitational Force of the Sun

Because of its massive mass, the Sun exerts a gravitational force whose range is extremely wide. This attracting force causes all planets to orbit the Sun in an elliptical shape. The gravitational force formula can be used to determine the gravitational force acting on Earth from the Sun and was found to be 3.5 × 10²² N.

Also Read,

• Einstein Theory of Gravitation
• Escape Velocity
• Gravitational Potential Energy

Difference between Gravity and Gravitational Force

Let's discuss the key differences between gravity and gravitational force in detail as mentioned in the table below:

Read More,

• Difference between Gravitational Force and Gravity
• Difference between Mass and Weight
• Kepler’s Law of Planetary Motion
• Factors affecting Acceleration due to Gravity

Practice Questions on Gravitational Force

Example 1: Find the gravitational force of attraction between two elephants, one of mass 1000 kg and the other of mass 800 kg, if the distance between them is 5 m.

Solution:

Given: m₁ = 1000 kg, m₂ = 800 kg, r = 5 m

The formula for gravitational force is given as: F_g = \frac{Gm_1m_2}{r^2}

Here, G = 6.67 ×10⁻¹¹N⋅ m²/kg²

Substituting the values in the formula, we have:

F_g = \frac{6.67 ×10^{−11}N⋅ m^2/kg^2)(1000 kg)(800 kg)}{5^2}

F_g= 2.1 × 10^-6 N

Example 2: Find the gravitational force of attraction between a man of mass of 50 kg and a bus of mass 1500 kg, if the distance between them is 10 m.

Solution:

Given: m₁ = 50 kg, m₂ = 1500 kg, r = 10 m

The formula for gravitational force is given as: F_g = \frac{Gm_1m_2}{r^2}

Here, G = 6.67 ×10⁻¹¹N⋅ m²/kg²

Substituting the values in the formula, we have:

F_g = \frac{6.67 ×10^{−11}N⋅ m^2/kg^2)(50 kg)(1500 kg)}{10^2}

F_g = 5.0025 × 10^-8 N

Example 3: Suppose the gravitational force between two bodies at a certain distance is 4 N. Find the force of attraction if the distance between them is doubled.

Solution:

Newton’s law of gravitation states that the gravitational force between two point like objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F_g = \frac{Gm_1m_2}{r^2}

Equation shows that, for given masses, if r is replaced by 2r, the force becomes 1/4th original force. Therefore, force of attraction will become 4/4 =1 N.

Example 4: The mass of the Earth is 6 × 10²⁴ kg. The distance between the Earth and the Sun is 1.5 × 10¹¹m. If the gravitational force between the two is 3.5 × 10²²N, what is the mass of the Sun?

Solution:

Given: m_e = 6 × 10²⁴ kg, r = 1.5 × 10¹¹ m and F = 3.5 × 10²² N

Formula for gravitational force is given as: F_g = \frac{Gm_1m_2}{r^2}   .

⇒ 3.5 × 10²² N = \frac{6.67×10^{-11}×6×10^{24}×m_{sun}}{(1.5×10^{11})^2}

⇒ Mass of sun = \frac{3.5×10^{22}×2.25×10^{22}}{40.02×10^{13}}

= 1.967 × 10³⁰ kg

Interesting Facts About Gravitation

• Without Gravity, we would not have Planets, stars, or Galaxies.
• The Gravity of Sun is 28 times stronger than Earth's.
• The Gravity of Sun keeps Earth and other Planets in orbit
• The Gravity of Moon is six times weaker than Earth’s, so you’d feel much lighter!
• Astronauts grow taller in space because Gravity is not pulling on their Spine.
• Balck Holes: Places where Gravity is so strong, even light can't escape.