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Write the three laws given by Kepler ? how did they help Newton to arrive at the Inverse Square Law of Gravity?
Report by Hrudayraj 09.06.2018
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Shivansh1mishra
Shivansh1mishra · Virtuoso
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Kepler’s three laws of planetary motion can be stated as follows: (1) All planets move about the Sun in elliptical orbits, having the Sun as one of the foci. (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. (3) The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the Sun.
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Kepler's First law :
The orbit of a planet is an eclipse with the sun at one of the foci.
Figure 1.3 shows elliptical orbit of a planet revolving around the Sun. S denotes the position of the Sun.
Kepler's Second Law :
The line joining the planet sweeps equal areas in equal intervals of time
AB, CD and EF are distances covered by the planet in equal intervals of time.
The straight lines AS, CS, and ES sweep equal areas in equal intervals of time
Area ASB = Area CSD = Area ESF.
Kepler's Third law :
The square of the period of revolution of a planet around the sun is directly proportional to the cube of the mean of distance of the planet from the sun.
T² proportional r³, i.e. T²÷r³ = Constant = K
For simplicity we shall assume the orbit to be a circle. In fig.1.4, S denotes the position of the Sun, P denotes the position of the planet at a given instant and r denotes the radius of the orbit ( – the distance of a planet from the sun ). Here, The speed of the planet is uniform. It is
V = Circumference of the circle÷Period of revolution of the the planet
v = 2πr÷T
If m is the mass of the planet, the centripetal force exerted on the planet by the sun ( – gravitational force). F = mv²÷r
Therefore,
F = m(2πr/T)²÷r = 4π²mr²÷T²r = 4π²mr÷T²
According to Kepler's third law,
T² = Kr³
Therefore, 4π²mr÷Kr³ = 4π²m÷K (1÷r²)
Thus, F is proportional to 1÷r²
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