Newton’s Universal Law of Gravitation | Definition, Formula, and Example

Just look at this beautiful image of sun, planets, stars, and many other celestial objects.

You might have seen such scenes in movies or on television.

Are you curious to know what happens in this universe?

What happens between the sun, earth and moon? (Scroll down to see images)

I’ll explain to you about Newton’s Universal Law of Gravitation as well as other concepts, it’s examples and numerical problems in this article.

This is going to be amazing and very interesting. (You will love this)

Let me tell you a story behind Newton’s Universal law of gravitation.

You might be knowing this person.

Guess, who is he? (Sir Issac Newton)

I guess you might have heard this story somewhere. Or at least you have seen this image somewhere.

This is a story of the 17th century.

It’s a story, when Issac Newton was just 23 years old.

Once, he was sitting under an apple tree. Suddenly an apple from the tree falls on his head.

Well, this is just an ordinary phenomenon for all of us.

Everything falls downwards, Isn’t it !!

This is not a thing which Newton had seen for the first time.

Many thousands and lakhs of people had seen such phenomenon of falling things on the ground.

But, Issac Newton had a very curious mind.

He started thinking,

Why does this apple fall down only, Why doesn’t it go up?

Why do all the things fall downwards?

Why does this stone fall down only?

Issac Newton started thinking deeply on this.

After many years of study and research work,

He formulated one law that goes by his name which is known as Newton’s Universal Law of Gravitation.

With this law, he explained everything about why objects always fall down only?

Now we will see this law. (Keep reading…)

If you want to skip to your interested topics, feel free to check out from the below table.

Newton’s universal law of gravitation

With the help of this Universal Law of Gravitation,

Sir Isaac Newton explained why all the objects fall downwards only?

Newton’s universal law of gravitation states that:

“Every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers”

In mathematical terms,

Fg ∝ m1 m2

Fg ∝ 1/d2

Therefore,

Fg ∝ (m1 m2) / d2

Fg = G (m1 m2) / d2

Where,

Fg = Gravitational force

G = Universal gravitational constant

m1 , m2 = Masses of the two bodies

d = Distance between their centers

To understand the statement of Newton’s universal law of gravitation,

Let’s take one example.

Two Balls

Suppose that we have two objects over here.

These can be two identical objects, or different objects or it may have different size and shape.

Just simply consider two objects as shown in the above image.

Let,

m1 = mass of the 1st object

m2 = mass of the 2nd object

(As all the objects have some amount of mass)

Now,

According to Newton’s universal law of gravitation,

Both these objects will apply force on each other. (i.e. force of attraction)

See this,

The left side object will attract the right side object and in the same way, the right side object will attract the left side object.

In short, both the objects will attract each other.

This is Newton’s universal law of Gravitation.

4 points to remember in Newton’s law of gravitation

Now,

I want to give you some important points related to Newton’s law of gravity or Newton’s law of gravitation.

1. Two objects having mass attracts each other

2. Force on both the objects have the same value (action reaction pair)

3. Force acts along the line joining them

4. The value of force is Fg = G (m1 m2) / d2

I’ll explain all these points in short. (Keep reading…)

1. Two objects having mass attracts each other

Newton’s law of universal gravitation says that any two objects which possess mass, will attract each other.

It’s simple, right?

These bodies can be anything, including earth, sun, moon, stars, table, chair, television, you, me, etc…

Yes, table, chair, you and me also…

I’ll explain this concept later.

Don’t worry, you will come to know why the table and chair don’t get attracted towards each other in your room.

2. Forces on both the objects have the same value (Action – Reaction pair)

Just see this image, 10 N of force is experienced by the mass m1

And in the same way, the force experienced by mass m2 also remains the same. (i.e. 10 N)

Thus, the force experienced by both the bodies remains the same.

This pair of forces are also known as the action and reaction force pairs

You can refer more about action-reaction pairs from here: Detailed guide on Newton’s third law

3. Force acts along the line joining them

Just see this dotted line in the above image which is joining both these objects.

So I simply want to say that, the attraction forces will act along this straight line.

In other words, these forces will act along the line joining these two bodies.

4. The value of force is Fg = G (m1 m2) / d2

Now, this is most important.

What will be the value of this force of attraction?

This value is given by the formula:

Fg = G (m1 m2) / d2

Where,

Fg = Gravitational force

G = Universal gravitational constant

m1 , m2 = Masses of the two bodies

d = Distance between their centers

The value for Universal law of gravitation is:

G = 6.673 × 10-11 Nm² / kg²

This value is used for solving numericals based on Newton’s law of universal gravitation.

The value of force Fg is the same for both the masses m1 as well as m2.

This force is also known as the gravitational force Fg.

Why do all objects attract downwards?

Till now we have studied that two bodies attract each other.

And this is absolutely true, I agree with this.

The question is,

Why do all objects attract downwards?

Now, let’s take some practical examples to understand this.

Example 1: Pool

This is a billiards table having two balls on it.

Now both these two balls have masses, Say m1 and m2.

And they are also separated by some distance, Say d.

Now,

According to the statement/definition of Universal law of gravitation, there is a force of attraction Fg between these two balls.

See this,

But the question is,

Why do these two balls not come in contact with each other?

Yes, there is an attractive force between these two balls.

But, we know practically that they will not come towards each other.

I’ll tell you the reason behind this question, later.

Let’s take one more example of a chair.

Example 2: The chair

Consider two chairs in the room as shown in the above image.

Both these two chairs have some mass m1 and m2 and they are also at some distance d.

Now,

According to Newton’s law of gravity,

There should be attractive force between these two chairs and they should get attracted towards each other.

See this,

Have you seen two chairs coming towards each other due to such attractive force?

But, why do these two chairs not come towards each other?

Good question, right?

Don’t worry…

You know what,

The mass of all these objects like tables, chairs, balls, etc. is very small as compared to the mass of earth.

In other words,

Earth is a very very huge mass as compared to a small object like a ball.

Thus, the earth has more force of attraction on both of these balls as shown in the above picture.

So both the balls are attracted downwards towards the earth and not towards each other.

Similarly,

Here also the mass of earth is very large as compared to the mass of chairs.

Thus, earth pulls the chairs with a more force of attraction.

This is the reason why, both the chairs are attracted downwards and not towards each other.

Mathematical proof – Why do all objects attract downwards?

We can also get this proof from the formula of Newton’s law of gravitation.

Fg = G (m1 m2) / d2

Just look at this formula.

I will give you a mathematical proof by using this formula of Newton’s universal law of gravitation.

What is the force between the two chairs?

Consider the above example of two chairs.

Let,

Mass of 1st chair, m1 = 10 kg

Mass of 2nd chair, m2 = 10 kg

Distance between their centers, d = 1 m

Now,

Calculate the force between these two objects using the formula,

Fg = G (m1 m2) / d2

Fg = (6.673 × 10-11 × 102) / (1)2

Fg = 6.673 × 10-9 N …………… Equation (1)

What is the force between the chair and earth?

Now,

Consider the chair and earth as two objects.

Let,

Mass of the chair, m1 = 10 kg

Mass of the earth, m2 = 5.97 × 10²⁴ kg

Distance between their centers, d = 6371000 m

(Note: This is the average radius of earth)

Now,

Calculate the force between these two objects using the formula,

Fg = G (m1 m2) / d2

Fg = (6.673 × 10-11 × 10 × 5.97 × 1024) / (6371000)2

Fg = 98.14 N …………… Equation (2)

Conclusion

After comparing the Equation (1) and (2), which one is a larger value?

Obviously,

Equation (2) has a larger value, right?

It means that,

There is more force of attraction between earth and chair and less force of attraction between the two chairs.

You can see a huge difference between these values of gravitational force Fg.

The gravitational force between the two chairs is 6.673 × 10-9 N which is very very negligible.

(As compared to gravitational force between earth and the chair which is 98.14 N)

Therefore,

All the objects are attracted downwards towards the center of earth.

Newton’s law of gravitation problems

Numerical 1:
The mass of the earth is 6 × 10²⁴ kg and that of the moon is 7.4 × 10²² kg. If the distance between the centers of the earth and the moon is 3.84 × 105 km, Calculate the force exerted by the earth on the moon. (Take G = 6.7 × 10-11 Nm² / kg²)

Solution:

Given data:

Mass of the earth, m1 = 6 × 10²⁴ kg

Mass of the moon, m2 = 7.4 × 10²² kg

Distance between their centers, d = 3.84 × 105 km = 3.84 × 108 m

Force exerted by the earth on the moon, Fg =?

G = 6.7 × 10-11 Nm² / kg²

According to Newton’s law of gravitation formula,

Fg = G (m1 m2) / d2

Fg = (6.7 × 10-11 × 6 × 10²⁴ × 7.4 × 10²²) / (3.84 × 108)2

Fg = (6.7 × 6 × 7.4 × 10-11 × 10²⁴ × 10²²) / (3.84 × 108)2

Fg = (297.48 × 1035) / (14.74 × 1016)

Fg = 20.18 × 1019 N

Therefore, a force of 20.18 × 1019 N is exerted by the earth on the moon.

Numerical 2:
A 30 kg boy and a 28 kg girl are standing at a distance of 3 m. Calculate the magnitude of the gravitational force they exert on each other. (Take G = 6.7 × 10-11 Nm² / kg²)

Solution:

Given data:

Mass of the boy, m1 = 30 kg

Mass of the girl, m2 = 28 kg

Distance between their centers, d = 3m

Magnitude of the gravitational force, Fg =?

G = 6.7 × 10-11 Nm² / kg²

According to Newton’s law of gravitation formula,

Fg = G (m1 m2) / d2

Fg = (6.7 × 10-11 × 30 × 28) / (3)2

Fg = (297.48 × 1035) / (14.74 × 1016)

Fg = (6.7 × 30 × 28 × 10-11) / (3)2

Fg = (5628 × 10-11) / 9

Fg = 625.33 × 10-11 N

Fg = 6.25 × 10-9 N

Therefore, the magnitude of the gravitational force is 6.25 × 10-9 N.

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Don’t you think it is easy to remember the statement of Newton’s universal law of gravitation?

(Let me know by leaving a comment)

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