Gravitation - Gravitational Force and Newton’s Law of Gravitation

Gravitation, commonly known as gravity, is the force of attraction between any two objects. All objects in the universe attract one another with a certain amount of force, however, this force is usually too weak to be noticed due to the large distances between them. Additionally, gravity has an infinite range, but its effect lessens as objects move further away.

Sir Isaac Newton first observed the force of attraction known as gravitation in 1680, and presented it as Newton’s law of gravitation. This law of gravitation can generally exist in two main instances.

1. The Earth’s gravitation is the attraction it exerts on objects.

Example:

If a body (ball) is thrown upwards, it will reach a certain height and then fall back down due to the gravitational pull of the Earth.

2. Gravitation is the attractive force between objects in outer space.

The force of attraction between the other planets and the Sun.

Table of Contents

• What is Gravitational Force?
• Gravitational Theory Through the Ages
• Newton’s Law of Gravitation
• Vector Form of Newton’s Law of Gravitation
• Gravitational Force Formula
• Derivation of Newton’s Law of Gravitation from Kepler’s Law
• FAQs on Gravitation

What is Gravitational Force?

Gravitational Force is a force of attraction that exists between any two masses, any two bodies, or any two particles with mass. It is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The gravitational force is a phenomenon in which two masses interact with each other, resulting in a pull between them. The heavier mass is known as the source mass and the lighter mass is known as the test mass. Thus, the study of gravitation involves understanding the interaction between these two masses.

Gravitational force is a central force which depends only on the position of the test mass from the source mass and always acts along the line joining the centres of the two masses.

$\overrightarrow{F}(r)=f(r)\hat{r}$

The core issue of gravitation has always been comprehending the interaction between the two masses and their relativistic effects.

⇒ Also, Check Out: Gravitational field intensity

Gravitational Theory Through the Ages

Ptolemy proposed the geocentric model which failed to accurately explain planetary motions, leading to the development of the heliocentric model by Nicholas Copernicus. This model was based on the rotation of a test mass around the source mass in circular orbits and correctly predicted the position of planets and their motions. However, it failed to explain many aspects, such as the occurrence of seasons, prompting the construction of a model based on Kepler’s laws of planetary motion.

Newton’s Law of Gravitation

According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is proportional to the product of their masses and inversely proportional to the square of the distance between them.”

The force (F) is directly proportional to the product of the masses (M₁ and M₂), i.e., F ∝ (M₁M₂) . . . . (1)

Inversely proportional to the square of the distance between their centres, i.e., $(F\propto 1/r^2)$ . . . . (2)

By combining equations (1) and (2), we get.

$F\propto \frac{M_1{M}_2}{r^2}$

F = G × [M₁M₂]/r² . . . (7)

f(r) = GM₁M₂/r² [f(r) is a variable, Non-contact, and conservative force]

The inverse square law force can be expressed as f(r) varying inversely as the square of ‘r’. The proportionality constant (G) in this equation is known as the gravitational constant.

The dimension formula of G is [M^-1L³T^-2]. Also, the value of the gravitational constant is 6.67 x 10^-11 Nm²/kg².

In SI units: 6.67 x 10^-11 Nm²/kg²

In CGS units: 6.67 x 10^-8 dyne cm² g^-2

Vector Form of Newton’s Law of Gravitation

The vector form of Newton’s law of gravitation signifies that the gravitational forces between two particles form an action-reaction pair.

From the above figure, it can be seen that two particles of different masses are placed at a distance.

\begin{array}{l}\vec{r_{21}}=\vec{r_{2}}-\vec{r_{1}}\end{array}

The vector goes from M₁ to M₂.

Therefore, the force applied on M₂ by M₁ is

$\begin{array}{l}\overrightarrow{F_{21}}=-\frac{G{M}_1{M}_2}{r_{21}^2}\widehat{r_{21}}\end{array}$

The negative sign indicates that the force is attractive.

Similarly, the force on M₁ and M₂. \begin{array}{l}\vec{F_{12}} = - \frac{GM_1 M_2}{r_{12}^2}\hat{r_{12}}\end{array}

Since, $$\hat{r_{12}} = -\hat{r_{21}}$$

$$\vec{F_{12}} = \frac{GM_{1}M_{2}}{(-r_{21})^{2}}[\hat{r_{21}}]$$

$$\vec{F_{12}} = \frac{GM_1M_2}{(r_{21})^2}[\hat{r_{21}}]$$

$$\vec{F_{21}}$$

The applied forces are equal and opposite, thus adhering to Newton’s third law of motion: every action has an equal and opposite reaction.

Gravitational Force Formula

Newton’s law of gravitation explains the concept of gravitational force. This force determines our weight and how far an object will travel when thrown before it hits the ground.

According to Newton’s law of gravitation, every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Mathematically it can be represented as: $F=G{m}_1{m}_2/r^2$

F = Force
G = Gravitational Constant
m1 = Mass of Object 1
m2 = Mass of Object 2
r = Distance between Objects 1 and 2

The Gravitational force between two objects is measured in Newton (N) and is represented by the letter ‘F’.

The Universal Gravitational Constant, G, has a value of 6.674 × 10^-11 Nm²kg^-2.

The mass of one massive body, m₁, is measured in kg.

The mass of another massive body is measured in kg and is denoted by m₂.

The separation between them is measured in kilometres (Km).

Principle of Superposition of Gravitational Forces

Newton’s law of gravitation answers the interaction between two particles; if the system contains n particles, there are n(n - 1)/2 such interactions.

According to the principle of superposition, if each of these interactions acts independently and without being affected by the other bodies, the outcome can be expressed as the vector sum of these interactions.

$F=F_{12}+F_{13}+F_{14}+...+F_{1n}$

It states that:

The resultant gravitational force F acting on a particle is equal to the vector sum of the forces exerted by each of the individual point masses on the particle.

Derivation of Newton’s Law of Gravitation from Kepler’s Law

The centripetal force acting on a test mass in a nearly circular orbit of radius ‘r’, revolving around a source mass with a constant angular speed (ω), is given by:

$F=mr{\omega }^2=mr\times (2\pi /T{)}^2$

According to Kepler’s 3rd law, T² ∝ r³

Using this in the force equation, we get:

$F=\frac{4{\pi }^{2}mr}{K{r}^3}Where,K=\frac{4{\pi }^2}{GM}$

$\Rightarrow F=\frac{G\cdot M\cdot m}{r^2}$, which is the equation of Newton’s law of gravitation.

Solved Examples

What is the force of gravity acting on an object of mass 2000 kg at the Earth’s surface?

Response:

Mass of Earth (m₁) = 5.98 x 10²⁴ kg

Mass of object (m₂) = 2000 kg

The radius of the Earth (r) is 6.38×10⁶ m

Acceleration due to Gravity (g) = 9.8 m/s²

Universal Constant (G) = 6.67 x 10^-11 N m² / kg²

Solution:

$F=G\ast m_1\ast m_2/r^2$

$F=(6.67\ast {10}^{-11})(5.98\ast {10}^{24})(2\ast {10}^3)/(6.38\ast {10}^6{)}^2$

F = (7.978 x 10¹⁷) / (4.07044 x 10¹³)

F = 19,590 N

What is the magnitude of the gravitational force acting on an object of mass 1000 kg at 20,000 meters above the Earth’s surface?

Response:

Mass of Earth (m₁) = 5.98 x 10²⁴ kg

Mass of object (m₂) = 1000 kg

The radius of the Earth (r) = 6.38 x 10⁶ m

Acceleration due to Gravity (g) = 9.8 m/s²

Universal Constant (G) = 6.67 x 10^-11 N m² / kg²

h = 2 x 10⁴ m

Solution:

$F=G\ast m_1\ast m_2/(r+h{)}^2$

F = $$\frac{(6.67 \times 10^{-11})(5.98 \times 1024)(1 \times 10^3)}{(6.38 \times 10^6 + 2 \times 10^4)^2}$$

$$F = \frac{3.988 \times 10^{17}}{4.058 \times 10^{13}}$$

F = 9,827.50

F = 9827 x 10⁴

Frequently Asked Questions On Gravitation

Will your weight be constant when you are travelling to Greenland from Brazil?

No, your weight will not be constant when travelling from Brazil to Greenland.

Can you screen the effect of gravitation by any material medium?

No, it is not possible to screen the effect of gravitation by any material medium.

No. Gravitational effect cannot be screened, unlike electrostatic force, because the gravitational force does not depend on any medium. Why are space rockets launched eastward?

Why are space rockets launched eastward?

Earth spinning from west to east in 24 hours helps the relative velocity of a space rocket increase, enabling it to rise with less fuel. But why does a bouncing ball bounce higher on hills than on planes?

Why does a bouncing ball bounce higher on hills than on planes?

As the altitude increases, the acceleration due to gravity decreases. Therefore, the ball bounces higher in hills than on planes due to the increased gravitational potential energy. The gravitational potential energy is negative because it is a measure of the work done by the gravitational field in order to bring a mass from an infinite distance away to a particular point.

Why is Newton’s law of gravitation, which states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them, called a universal law?

Newton’s law of gravitation is applicable regardless of the type of interacting bodies, in all locations and at all times.

What is the Weight of the Body at the Centre of the Earth?

The weight of the body at the centre of the earth is zero.

W = mg = 0

The gravity at the centre of the earth is zero.

Does Gravitation Cause Friction?

Friction does not arise due to gravitation. Its origin is electrical in nature.