Subjective Type

Using the conservation laws, demonstrate that the total mechanical energy of a planet of mass $$m$$ moving around the Sun along an ellipse depends only on its semi-major axis $$a.$$ Find this energy as a function of $$a.$$

Solution

From the previous problem, if $$r_1, r_2$$ are the maximum and minimum distances from the sun to the planet and $$v_1, v_2$$ are the corresponding velocities, then, say,
$$E=\dfrac 12 mv_2^2 - \dfrac {\gamma mm_2}{r_2}$$
$$=\dfrac {\gamma mm_s}{r_1 r_2}. \dfrac {r_1}{r_2}-\dfrac {\gamma mm_s}{r_2}=-\dfrac {\gamma mm_s}{r_1+r_2}=-\dfrac {\gamma mm_s}{2a}$$ [ Using Eq. (2) of 1.207]
where $$2d=$$ major axis $$=r_1+r_2$$. The same result can also be obtained directly by writing
$$E=\dfrac {1}{2}mr^2 +\dfrac {M^2}{2mr^2}-\dfrac {\gamma mm_s}{r}$$
( Here $$M$$ is angular momentum of the planet and $$m$$ is its mass ). For extreme position $$\dot r=0$$ and we get the quadratic
$$Er^2 +\gamma mm_s r -\dfrac {M^2}{2m}=0$$
The sum of the two roots of this equation are
$$r_1 +r_2 =-\dfrac {\gamma mm_s}{E}=2a$$
Thus $$E=-\dfrac {\gamma mm_s}{2a}=$$ constant

SIMILAR QUESTIONS

Laws of Motion

Define linear momentum and state its S.I. unit.

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