Physics M.C.Q (1 Marks)

If $T$ is measured in years, $r$ in $A.U$. and masses in Solar masses then $G = 4{\pi ^2}$.

$\therefore $ ${M_1} + {M_2} = \frac{{{r^3}}}{{{T^2}}} = \frac{{{{(30)}^3}}}{{{{(30)}^2}}} = 30$ .....$(i)$

Now ${r_1} + {r_2} = 30 \Rightarrow {r_1} + 5{r_1} = 60$

$ \Rightarrow {r_1} = 5$ and ${r_2} = 25$

Again ${M_1}{r_1} = {M_2}{r_2} \Rightarrow \frac{{{M_1}}}{{{M_2}}} = 5$ .....$(ii)$

After solving $(i)$ and $(ii)$ we get ${M_1} = 25$ and ${M_2} = 5$

$\frac{1}{2}mv_1^2 - \frac{{G{M_S}m}}{{{r_1}}} = \frac{1}{2}mv_2^2 - \frac{{G{M_S}m}}{{{r_2}}}$.

Angular momentum is conserved, that is $m{v_1}{r_1} = m{v_2}{r_2}$

or ${v_2} = {v_1}\frac{{{r_1}}}{{{r_2}}}$==> $\frac{1}{2}mv_1^2 - \frac{{G{M_S}m}}{{{r_1}}} = \frac{1}{2}m\,{\left( {\frac{{{v_1}{r_1}}}{{{r_2}}}} \right)^2} - \frac{{G{M_S}m}}{{{r_2}}}$

or ${v_1} = \sqrt {\frac{{2G{M_S}{r_2}}}{{{r_1}({r_1} + {r_2})}}} \Rightarrow \,L = m{v_1}{r_1} = \sqrt {\frac{{2G{M_S}{m^2}{r_1}{r_2}}}{{{r_1} + {r_2}}}} $