Question

Kepler's second law is another from the statement is. a. work energy
b. conservation of linear momentum
c. conservation of angular momentum
d. conservation of energy​

Answer 1

Kepler's Second Law is another form of law of conservation of angular momentum.

The second law, known by the name 'Law of Period', states that the radius vector drawn from the sun to a planet sweeps out equal areas in equal intervals of time, i.e., areal velocity of a time at a unit time always remains constant.

This law is possible if and only if the angular acceleration is conserved throughout the revolution of the planet. Let's prove the law.

Consider a planet of mass m revolving around the sun with a velocity v along an orbit of radius r. Then, for a small time ∆t, the linear displacement of the planet will be (v · ∆t), and the radius vector r may sweep some area which can be considered as a right triangle, since the displacement is perpendicular to the radius vector.

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\put (0,0){\vector (1,0){48}}\put (48,0){\vector (0,1){12}}\put (0,0){\vector (4,1){48}}\put (24,-4){ \mathbf {r} }\put (50,5){ \mathbf {v}\Delta t }\put (20,10){ \mathbf {r+v}\Delta t }\end{picture}$

Then the area of the triangle will be,

$\Delta\mathbf {A}=\dfrac {1}{2}(\mathbf {r\times v})\Delta t\\\\\\\dfrac {\Delta\mathbf {A}}{\Delta t}=\dfrac {\mathbf {r\times v}}{2}\\\\\\\dfrac {\Delta\mathbf {A}}{\Delta t}=\dfrac {m(\mathbf {r\times v})}{2m}\\\\\\\dfrac {\Delta\mathbf {A}}{\Delta t}=\dfrac {\mathbf {r\times}(m\mathbf {v})}{2m}$

Since linear momentum, $\mathbf {p}=m\mathbf {v},$

$\dfrac {\Delta\mathbf {A}}{\Delta t}=\dfrac {\mathbf {r\times p}}{2m}$

But, angular momentum, $\mathbf {L}=\mathbf {r\times p}.$ Then,

$\dfrac {\Delta\mathbf {A}}{\Delta t}=\dfrac {\mathbf {L}}{2m}$

where L is a constant since no external torque is acting on the planet. Since mass of the planet is also constant,

$\boxed {\dfrac {\Delta\mathbf {A}}{\Delta t}=\text {a constant}}$

Hence the Proof! The proof is based on the law of conservation of angular momentum.

Hence (c) is the answer.