I'm unable to understand the 2nd law of kepler please give me some information about this and send me video which show the 2nd kepler law please help I'm new

Asked by kashmirsunil | 24th May, 2020, 09:06: PM

Expert Answer:

Kepler's second law:
Consider that a planet of mass 'm' revolving around the Sun of mass 'M' in a circular orbit of radius 'r'. Let 'v' be its orbital velocity. Suppose that at any isntant the planet is at point A in its orbit and after an in small time dt, it reaches point B. As such, the circular path of the planet between points A and B may be considered as straight.
 
 
If dA is small area swept by the line joining the planet to the Sun in time dt, then
dA=area of ΔABS=begin mathsize 14px style 1 half cross times base cross times perpendicular end style
or   dA=begin mathsize 14px style 1 half cross times AB cross times AS end style           .... (1)
If dθ is the angular displacement of the planet in time dt i.e., when it moves from point A to B, then AB=r dθ.
Substituting AB with rdθ and AS with 'r' in equation(1), we get,
begin mathsize 14px style dA equals 1 half cross times straight r space dθ cross times straight r dA equals 1 half straight r squared space dθ end style
Dividing the above equation by dt, we get,
begin mathsize 14px style dA over dt equals 1 half straight r squared dθ over dt or dA over dt equals 1 half straight r squared straight omega space space space space space space space space space space... space left parenthesis 2 right parenthesis end style
where ω=begin mathsize 14px style dθ over dt end styleis the angular speed of the planet in its orbit and begin mathsize 14px style dA over dt end styleis the areal velocity of the planet.
Multiplying and divind the R.H.S of the equation(2) by 'm' i.e., the mass of the planet, we get,

begin mathsize 14px style dA over dt equals fraction numerator 1 over denominator 2 space straight m end fraction cross times straight m space straight r squared space straight omega end style
Since, m r2 ω = L, the angular momentum of the planet about the axis through the Sun, we have,
begin mathsize 14px style dA over dt equals fraction numerator straight L over denominator 2 space straight m end fraction end style    ... (3)
As no external torque acts on the planet during its orbital motion, its angular momentum (L) must remain constant. Since both L amd m are constant, the equation(3) becomes, begin mathsize 14px style dA over dt equals constant end style    ... (4)
Hence, when a planet moves around the Sun, its areal velocity remains constant. It proves Kepler's second law of planetary motion.
 

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Answered by Shiwani Sawant | 24th May, 2020, 11:02: PM