Question

write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity ?

Answer 1

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Here is your answer,
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The three laws were the :
1. Law of orbits :
Each planet moves around the sun in an elliptical orbit with the sun as center.

2. Law of Areas :
The line joining the sun and a planet sweeps out equal are in equal intervals of time.

3. Law of periods :
Teb square of the time taken by a planet to complete a revolution around the sun is directly proportional to the cube of the semi major axis of the elliptical orbits.

Newton assumed that radius of circular path is r, velocity is v then acceleration =
v*v /r.

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Answer 2

Answer:

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Three laws given by Kepler is as follows:

First Law: The orbits of the planets are in the shape of ellipse, having the Sun at one focus.

Second Law: The area swept over per hour by the radius joining the Sun and the planet is the same in all parts of the planet’s orbit.

Third Law: The squares of the periodic times of the planets are proportional to the cubes of their mean distances from the Sun.

Newton used Kepler’s third law of planetary motion to arrive at the inverse-square rule. He assumed that the orbits of the planets around the Sun are circular, and not elliptical, and so derived the inverse-square rule for gravitational force using the formula for centripetal force. This is given as:-

F = mv²/ r ...(i) where, m is the mass of the particle, r is the radius of the circular path of the particle and v is the velocity of the particle. Newton used this formula to determine the force acting on a planet revolving around the Sun. Since the mass m of a planet is constant, equation (i) can be written as:

F ∝ v²/ r ...(ii)

Now, if the planet takes time T to complete one revolution around the Sun, then its velocity v is given as:

v = 2πr/ T ...(iii) where, r is the radius of the circular orbit of the planet

or, v ∝ r/ T ...(iv) [as the factor 2π is a constant]

On squaring both sides of this equation, we get:

v² ∝ r³/ T²...(v)

On multiplying and dividing the right-hand side of this relation by r, we get:

v²∝r³T²×1r ...(vi)

According to Kepler’s third law of planetary motion, the factor r³/ T² is a constant. Hence, equation (vi) becomes:

v²∝ 1/ r...(vii)

On using equation (vii) in equation (ii), we get:

F∝1r²

Hence, the gravitational force between the sun and a planet is inversely proportional to the square of the distance between them.