Practicing Success
A planet of mass m moves along an ellipse around the sun so that its maximum and minimum distance from the sun is equal to r1 and r2 respectively. Then the angular momentum of this plane relative to the centre of the sun. |
$m \sqrt{\frac{2 G M r_1 r_2}{r_1+r_2}}$ $m \sqrt{\frac{3 GM r_1 r_2}{r_1+r_2}}$ $m \sqrt{\frac{GM r_1 r_2}{r_1+r_2}}$ $m \sqrt{\frac{GM r_1 r_2}{2\left(r_1+r_2\right)}}$ |
$m \sqrt{\frac{2 G M r_1 r_2}{r_1+r_2}}$ |
The angular momentum $= L = m v_1 r_1$ ......(1) Conservation of angular momenta at 1 and 2; $m v_1 r_1=m v_2 r_2$ $\Rightarrow v_1 r_1=v_2 r_2$ ......(2) Conservation of energy at 1 and 2; $\frac{1}{2} m v_1^2-\frac{G M m}{r_1}$ $=\frac{1}{2} m v_2^2-\frac{G M m}{r_2}$ ......(3) Using (2) and (3) we obtain $v_1=\sqrt{2 G M\left(\frac{1}{r_1}-\frac{1}{r_2}\right)+v_2^2}$ $v_1=\sqrt{2 G M\left(\frac{1}{r_1}-\frac{1}{r_2}\right)+\left(\frac{v_1 r_1}{r_2}\right)^2}$ $\Rightarrow v_1^2\left[1-\left(\frac{r_1}{r_2}\right)^2\right]=2 G M\left(\frac{r_2-r_1}{r_1 r_2}\right)$ $\Rightarrow v_1=\sqrt{\frac{2 G M r_2}{r_1\left(r_1+r_2\right)}}$ ∴ $L=m v_1 r_1=m \sqrt{\frac{2 G M r_1 r_2}{r_1+r_2}}$ |