How to calculate the time the earth takes to go around the sun, using Newton’s Universal Law of Gravitation?

Here we will see how to calculate the time the Earth takes to go around the Sun once.

We will use Newton’s Universal Law of Gravitation to calculate this.

Calculating the time taken by the earth to go around the Sun Using Newton’s Universal Law of Gravitation

Applying Newton’s universal law of Gravitation, we will find the value of the gravitational force experienced by the Earth due to the presence of the Sun.

This formula is given by: F = Gm1m2/r2

Where
m1 = mass of the Sun = 1.99 x 1030 kg
m2 = mass of the Earth = 5.97 x 1024 kg
r = distance between the Sun and Earth = 1.49 x 1011 m

After calculation we get the value of this force as follows:

So F = 3.56 x 1022 N………………… (1)

We know that the Earth travels in an elliptical orbit around the Sun, but we can take this to be a circular orbit for the purposes of this calculation.

From our knowledge of circular motion, we know that the force acting on the Earth towards the center of the circle is the centripetal force given by the equation F = mv2 / r ……………….. (2)

And, here this centripetal force is supplied by the gravitational force. Hence the force in equations 1 and 2 are equal.

Using equations (1) and (2):
So the velocity of the earth, v = (F r/m)1/2 = [( 3.56 x 1022 N)( 1.49 x 1011 m)/( 5.97 x 1024 kg)]1/2 = 29846 m/s

The circumference of the orbit = 2πr = (2)(22/7)( 1.49 x 1011 )= 9.38 x 1011 m

Time taken for 1 orbit = circumference/velocity = 9.38 x 1011/29 846= 3.14 x 107 s

This is equal to 1 year.

Summary:

Thus, we can calculate the time taken by the earth to go around the Sun Using Newton’s Universal Law of Gravitation.

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How to calculate the time the earth takes to go around the sun, using Newton’s Universal Law of Gravitation?
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