CBSE Class 11-science Answered

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=

or   dA=           .... (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,

Dividing the above equation by dt, we get,

where ω=is the angular speed of the planet in its orbit and is 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,

Since, m r² ω = L, the angular momentum of the planet about the axis through the Sun, we have,

    ... (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,     ... (4)

Hence, when a planet moves around the Sun, its areal velocity remains constant. It proves Kepler's second law of planetary motion.